diff --git a/README.md b/README.md
--- a/README.md
+++ b/README.md
@@ -21,19 +21,45 @@
 that a polyline is a proper polyline, and thus has at least two vertices.
 
 Please note that aspect (2), implementing good algorithms, is much work in
-progress. HGeometry currently has only very basic types, and implements only
-a few algorithms:
+progress. Only a few algorithms have been implemented, some of which could use
+some improvements. Currently, HGeometry provides the following algorithms:
 
-* two (optimal) $O(n \log n)$ time algorithms for convex hull in
+* two \(O(n \log n)\) time algorithms for convex hull in
   $\mathbb{R}^2$: the typical Graham scan, and a divide and conqueror algorithm,
-* an $O(n)$ expected time algorithm for smallest enclosing disk in $\mathbb{R}^$2,
+* an \(O(n)\) expected time algorithm for smallest enclosing disk in $\mathbb{R}^$2,
 * the well-known Douglas Peucker polyline line simplification algorithm,
-* an $O(n \log n)$ time algorithm for computing the Delaunay triangulation
+* an \(O(n \log n)\) time algorithm for computing the Delaunay triangulation
 (using divide and conqueror).
-* an $O(n \log n)$ time algorithm for computing the Euclidean Minimum Spanning
+* an \(O(n \log n)\) time algorithm for computing the Euclidean Minimum Spanning
 Tree (EMST), based on computing the Delaunay Triangulation.
+* an \(O(\log^2 n)\) time algorithm to find extremal points and tangents on/to a
+  convex polygon.
+* An optimal \(O(n+m)\) time algorithm to compute the Minkowski sum of two convex
+polygons.
+* An \(O(1/\varepsilon^dn\log n)\) time algorithm for constructing a Well-Separated pair
+  decomposition.
 
+It also has some geometric data structures. In particular, HGeometry contans an
+implementation of
 
+* A one dimensional Segment Tree. The base tree is static.
+* A one dimensional Interval Tree. The base tree is static.
+* A KD-Tree. The base tree is static.
+
+HGeometry also includes a datastructure/data type for planar graphs. In
+particular, it has a `EdgeOracle' data structure, that can be built in $O(n)$
+time that can test if the graph contains an edge in constant time.
+
+Numeric Types
+-------------
+
+All geometry types are parameterized by a numerical type `r`. It is well known
+that Floating-point arithmetic and Geometric algorithms don't go well together;
+i.e. because of floating point errors one may get completely wrong
+results. Hence, I *strongly* advise against using `Double` or `Float` for these
+types. In several algorithms it is sufficient if the type `r` is
+`Fractional`. Hence, you can use an exact number type such as `Rational`.
+
 A Note on the Ext (:+) data type
 ---------------------------------
 
@@ -54,9 +80,9 @@
 
 ```haskell
 data Polygon (t :: PolygonType) p r where
-  SimplePolygon :: C.CList (Point 2 r :+ p)                         -> Polygon Simple p r
-  MultiPolygon  :: C.CList (Point 2 r :+ p) -> [Polygon Simple p r] -> Polygon Multi  p r
-  ```
+  SimplePolygon :: C.CSeq (Point 2 r :+ p)                         -> Polygon Simple p r
+  MultiPolygon  :: C.CSeq (Point 2 r :+ p) -> [Polygon Simple p r] -> Polygon Multi  p r
+```
 
 In all places this extra data is accessable by the (:+) type in Data.Ext, which
 is essentially just a pair.
@@ -66,4 +92,24 @@
 
 Apart from geometric types, HGeometry provides some interface for reading and
 writing Ipe (http://ipe.otfried.org). However, this is all very work in
-progress. Hence, the API is experimental and may change at any time!
+progress. Hence, the API is experimental and may change at any time! Here is an
+example showing reading a set of points from an Ipe file, computing the
+DelaunayTriangulation, and writing the result again to an output file
+
+```haskell
+mainWith                          :: Options -> IO ()
+mainWith (Options inFile outFile) = do
+    ePage <- readSinglePageFile inFile
+    case ePage of
+      Left err                         -> print err
+      Right (page :: IpePage Rational) -> case page^..content.traverse._IpeUse of
+        []         -> putStrLn "No points found"
+        syms@(_:_) -> do
+           let pts  = syms&traverse.core %~ (^.symbolPoint)
+               pts' = NonEmpty.fromList pts
+               dt   = delaunayTriangulation $ pts'
+               out  = [asIpe drawTriangulation dt]
+           writeIpeFile outFile . singlePageFromContent $ out
+```
+
+See the examples directory for more examples.
diff --git a/benchmark/Benchmarks.hs b/benchmark/Benchmarks.hs
new file mode 100644
--- /dev/null
+++ b/benchmark/Benchmarks.hs
@@ -0,0 +1,69 @@
+module Main where
+
+import           Control.DeepSeq
+import           Control.Lens
+import           Criterion.Main
+import           Data.Ext
+import           Data.Geometry.Interval
+import qualified Data.Geometry.IntervalTree as IT
+import           Data.Geometry.SegmentTree (I(..))
+import qualified Data.Geometry.SegmentTree as SegTree
+import qualified Data.List.NonEmpty as NonEmpty
+import           QuickCheck.Instances
+import           Test.QuickCheck
+
+
+-- | generates n random intervals
+genIntervals                  :: (Ord r, Arbitrary r)
+                              => proxy r -> Int -> IO [Interval () r]
+genIntervals _ n | n <= 0     = error "genIntervals: need n > 0"
+                 | otherwise  = generate (vectorOf n arbitrary)
+
+genQueries                      :: (Ord r, Arbitrary r)
+                                => proxy r -> Int -> IO [r]
+genQueries _ n | n <= 0     = error "genQueries: need n > 0"
+               | otherwise  = generate (vectorOf n arbitrary)
+
+
+-- genQuerySetup     :: (Ord r, Arbitrary r)
+--                   => proxy r -> Int -> IO (Int,IT.IntervalTree (I (Interval () r)) r, [r])
+-- genQuerySetup p n = (\is qs -> (n, IT.fromIntervals . fmap I $ is, qs))
+--                  <$> genIntervals p n
+--                  <*> genQueries   p n
+
+
+main = defaultMain [
+  bgroup "IntervalTree" [ env (genIntervals (I (5 :: Int)) (100000 :: Int)) benchBuild
+                         -- env (genIntervals (I (5 :: Int)) (100000 :: Int)) benchQueryIT
+                        ]
+  ]
+
+benchBuild    :: (Ord r, NFData r) => [Interval () r] -> Benchmark
+benchBuild is = bgroup "build" [ bench (show n) $ nf IT.fromIntervals (take n is')
+                               | n <- sizes is
+                               ]
+  where
+    is' = I <$> is
+
+benchQueryIT    :: (Ord r, Arbitrary r, NFData r) => [Interval () r] -> Benchmark
+benchQueryIT is = bgroup "queries"
+    [ env (setup n) (\(t,qs) ->
+                        bench ("queries on size" ++ show n) $ whnf (queryAll t) qs)
+    | n <- sizes is
+    ]
+  where
+    is'        = I <$> is
+    r          = is^.to head.start.core
+    setup n    = (IT.fromIntervals (take n is'),) <$> genQueries (I r) 100000
+    queryAll t = map (flip IT.search t)
+
+
+-- benchQueryIT          :: Ord r
+--                       => (Int, IT.IntervalTree (I (Interval () r)) r, [r]) -> Benchmark
+-- benchQueryIT (n,t,qs) = bgroup "queries" [ bench "query" $ whnf (flip IT.search t) q
+--                                          | q <- qs
+--                                          ]
+
+
+sizes    :: [a] -> [Int]
+sizes xs = let n = length xs in (\i -> n*i `div` 100) <$> [5,10..100]
diff --git a/doctests.hs b/doctests.hs
--- a/doctests.hs
+++ b/doctests.hs
@@ -19,13 +19,16 @@
           , "TupleSections"
           , "MultiParamTypeClasses"
           , "LambdaCase"
+          , "TupleSections"
 
+
           , "StandaloneDeriving"
           , "GeneralizedNewtypeDeriving"
           , "DeriveFunctor"
           , "DeriveFoldable"
           , "DeriveTraversable"
           , "AutoDeriveTypeable"
+          , "DeriveGeneric"
           , "FlexibleInstances"
           , "FlexibleContexts"
           ]
diff --git a/examples/BAPC2012/G.in b/examples/BAPC2012/G.in
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2012/G.in
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diff --git a/examples/BAPC2012/G.out b/examples/BAPC2012/G.out
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2012/G.out
@@ -0,0 +1,49 @@
+8001.941
+1.000
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+1.414
+1.414
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diff --git a/examples/BAPC2012/Gunslinger.hs b/examples/BAPC2012/Gunslinger.hs
--- a/examples/BAPC2012/Gunslinger.hs
+++ b/examples/BAPC2012/Gunslinger.hs
@@ -3,15 +3,16 @@
 import           Algorithms.Geometry.ConvexHull.GrahamScan
 import           Control.Lens
 import qualified Data.CircularSeq as C
-import qualified Data.Foldable as F
-import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Ext
 import           Data.Fixed
+import qualified Data.Foldable as F
 import           Data.Geometry.Point
 import           Data.Geometry.Polygon
-import qualified Data.List     as L
+import           Data.Geometry.Polygon.Convex
+import qualified Data.List as L
+import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Maybe
-import           Linear.Affine(distanceA)
+import           Linear.Affine (distanceA)
 
 {-
 
@@ -98,8 +99,8 @@
 escape (Input l h ds) = case convexHull . NonEmpty.fromList $
                                (l :+ Luke) : (h :+ Hatch) : map (:+ Dalton) ds of
     -- all positions are distinct, so the hull has at least two elements
-    ConvexHull poly -> case C.findRotateTo (\p -> p^.extra == Luke) $
-                              poly^.outerBoundary of
+    ConvexPolygon poly -> case C.findRotateTo (\p -> p^.extra == Luke) $
+                                 poly^.outerBoundary of
       Nothing -> Impossible
       Just h  -> (distanceToHatch $ C.leftElements h)
                  `min`
diff --git a/examples/BAPC2012/sampleG.in b/examples/BAPC2012/sampleG.in
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2012/sampleG.in
@@ -0,0 +1,12 @@
+2
+0 0
+2 0
+2
+1 1
+1 -2
+0 0
+2 0
+3
+1 1
+1 -2
+-1 0
diff --git a/examples/BAPC2012/sampleG.out b/examples/BAPC2012/sampleG.out
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2012/sampleG.out
@@ -0,0 +1,2 @@
+2.828
+IMPOSSIBLE
diff --git a/examples/BAPC2014/Armybase.lhs b/examples/BAPC2014/Armybase.lhs
--- a/examples/BAPC2014/Armybase.lhs
+++ b/examples/BAPC2014/Armybase.lhs
@@ -14,15 +14,14 @@
 > {-# LANGUAGE GeneralizedNewtypeDeriving #-}
 > module BAPC2014.Armybase where
 
-> import Control.Applicative
+
 > import Control.Lens((^.))
-> import Data.Monoid
 > import Data.Ix
 > import Data.Ext
-
 > import Data.Geometry.Triangle
 > import Data.Geometry.Point
 > import Data.Geometry.Polygon
+> import Data.Geometry.Polygon.Convex
 > import Algorithms.Geometry.ConvexHull.GrahamScan
 > import qualified Data.Array   as A
 > import qualified Data.Foldable as F
@@ -90,10 +89,10 @@
 > maxBaseArea    :: PointSet -> Area
 > maxBaseArea [] = 0
 > maxBaseArea p  = case convexHull . NonEmpty.fromList $ map ext p of
->     ch@(ConvexHull h) -> case F.toList $ h ^. outerBoundary of
->                            [_,_]   -> 0
->                            [a,b,c] -> triangArea $ Triangle a b c
->                            _       -> maxAreaQuadrangle ch
+>     ch@(ConvexPolygon h) -> case F.toList $ h ^. outerBoundary of
+>                               [_,_]   -> 0
+>                               [a,b,c] -> triangArea $ Triangle a b c
+>                               _       -> maxAreaQuadrangle ch
 
 
 <div class="observation">
@@ -105,7 +104,7 @@
 subproblems, in both of which we have to find the largest triangle that has $p$
 and $q$ as vertices.
 
-> maxAreaQuadrangle :: ConvexHull () Int -> Area
+> maxAreaQuadrangle :: ConvexPolygon () Int -> Area
 > maxAreaQuadrangle = maximum' . map (uncurry3 maxAreaQuadrangleWith) . allChains
 >   where
 >     uncurry3 f (a,b,c) = f a b c
@@ -116,9 +115,9 @@
 
 > type Chain = Array Int (Point 2 Int)
 
-> allChains                 :: ConvexHull () Int
+> allChains                 :: ConvexPolygon () Int
 >                           -> [(Point 2 Int, Point 2 Int, (Chain,Chain))]
-> allChains (ConvexHull ch) =
+> allChains (ConvexPolygon ch) =
 >     [ (chA ! i, chA ! j, chains chA i j) | i <- [1..n-2], j <- rest i ]
 >   where
 >     n   = F.length $ ch^.outerBoundary
diff --git a/examples/BAPC2014/sample.in b/examples/BAPC2014/sample.in
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2014/sample.in
@@ -0,0 +1,25 @@
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+6
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+5
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+0 1
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+2 3
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diff --git a/examples/BAPC2014/sample.out b/examples/BAPC2014/sample.out
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2014/sample.out
@@ -0,0 +1,3 @@
+100
+12.5
+31
diff --git a/examples/BAPC2014/testdata.in b/examples/BAPC2014/testdata.in
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2014/testdata.in
@@ -0,0 +1,66406 @@
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diff --git a/examples/BAPC2014/testdata.out b/examples/BAPC2014/testdata.out
new file mode 100644
--- /dev/null
+++ b/examples/BAPC2014/testdata.out
@@ -0,0 +1,78 @@
+92210814
+88343466
+93671289
+87057782
+96232690
+94638324
+92852070
+92987572
+92744228
+97779736
+800
+200052736
+199995410
+0
+1280
+4096
+2304
+1024
+371513194
+374362054
+377188700
+376064444
+373665909
+376562872.5
+364442449
+365333194.5
+382802310.5
+371894230
+358267136
+368224075
+368543700
+377758295.5
+376702807.5
+356447037.5
+377131502
+377490682.5
+378118548
+372155990.5
+383333198.5
+352680168
+371294967
+376500650
+376274258
+368283202.5
+382473268
+368020279
+363707143.5
+373665106.5
+369035942
+366940652
+370708656
+371569688
+378140728
+373974300
+359873951.5
+360773601.5
+369896954
+351975481
+359847959.5
+367623183
+359879556
+379433260.5
+378837725
+364060059.5
+357774392
+370114715
+370832552
+357111020
+347979354.5
+323491260
+353187255.5
+381838262
+189877347.5
+361709734.5
+351329926
+318801902.5
+264706770
+334690037.5
diff --git a/examples/Demo/Delaunay.hs b/examples/Demo/Delaunay.hs
--- a/examples/Demo/Delaunay.hs
+++ b/examples/Demo/Delaunay.hs
@@ -1,9 +1,9 @@
 {-# LANGUAGE ScopedTypeVariables #-}
 module Demo.Delaunay where
 
-import           Algorithms.Geometry.EuclideanMST.EuclideanMST
 import           Algorithms.Geometry.DelaunayTriangulation.DivideAndConqueror
 import           Algorithms.Geometry.DelaunayTriangulation.Types
+import           Algorithms.Geometry.EuclideanMST.EuclideanMST
 import           Control.Applicative
 import           Control.Lens
 import           Data.Data
@@ -11,6 +11,7 @@
 import           Data.Geometry
 import           Data.Geometry.Ipe
 import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Semigroup
 import           Data.Traversable
 import           Options.Applicative
 
diff --git a/examples/Demo/DrawGPX.hs b/examples/Demo/DrawGPX.hs
--- a/examples/Demo/DrawGPX.hs
+++ b/examples/Demo/DrawGPX.hs
@@ -12,11 +12,12 @@
 import           Data.Ext
 import qualified Data.Foldable as F
 import           Data.Geometry
-import           Data.Geometry.PolyLine
 import           Data.Geometry.Ipe
+import           Data.Geometry.PolyLine
 import           Data.Geometry.Vector
 import           Data.List (isSuffixOf)
 import           Data.Maybe
+import           Data.Semigroup
 import qualified Data.Sequence as S
 import qualified Data.Text as T
 import           Data.Time.Calendar
diff --git a/examples/Demo/ExpectedPairwiseDistance.hs b/examples/Demo/ExpectedPairwiseDistance.hs
new file mode 100644
--- /dev/null
+++ b/examples/Demo/ExpectedPairwiseDistance.hs
@@ -0,0 +1,209 @@
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+module Demo.ExpectedPairwiseDistance where
+
+import           Algorithms.Geometry.Diameter
+import           Algorithms.Geometry.WellSeparatedPairDecomposition.Types
+import           Algorithms.Geometry.WellSeparatedPairDecomposition.WSPD
+import           Control.Lens
+import           Control.Monad ((<=<))
+import           Data.BinaryTree
+import qualified Data.ByteString as B
+import qualified Data.ByteString.Char8 as C
+import           Data.Char (isSpace)
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Maybe (mapMaybe)
+import           Data.Proxy
+import           Data.Data
+import           Data.Semigroup
+import qualified Data.Set as Set
+import           GHC.TypeLits (natVal,KnownNat)
+import           Options.Applicative hiding ((<>))
+
+
+import           Debug.Trace
+--------------------------------------------------------------------------------
+
+
+data Options = Options { _inPath    :: FilePath }
+               deriving Data
+
+options :: ParserInfo Options
+options = info (helper <*> parser)
+               (  progDesc "Compute expected pairwise distance of the points in the input file."
+               <> header   "Expected Pairwise Distance"
+               )
+  where
+    parser = Options
+          <$> strOption (help "Input file"
+                         <> short 'i'
+                        )
+
+--------------------------------------------------------------------------------
+
+-- | Evaluates the formula: $\sum_{p,q \in pts} \|pq\|*Prb[sample has size k
+-- and contains p and q]$ which solves to $\frac{{n-2 \choose k-2}}{{n \choose
+-- k}} \sum_{p,q} \|pq\|$
+--
+-- running time: $O(n^2)$, where $n$ is the number of points
+expectedPairwiseDistance       :: (Floating r, Arity d) => Int -> [Point d r :+ p] -> r
+expectedPairwiseDistance k pts = makeExpected k pts pairwiseDist
+
+-- | A $(1+\varepsilon)$-approximation of expectedPairwiseDistance
+--
+-- running time: $O(n(1/eps)^d + n\log n)$, where $n$ is the number of points
+approxExpectedPairwiseDistance          :: (Floating r, Ord r
+                                           , AlwaysTrueWSPD d, Index' 0 d
+                                         , Show r, Show p)
+                                         => r -> Int -> [Point d r :+ p] -> r
+approxExpectedPairwiseDistance eps k pts =
+  makeExpected k pts (approxPairwiseDistance eps)
+
+--------------------------------------------------------------------------------
+-- * Computing Distances
+
+-- | Sum of the pairwise distances
+pairwiseDist     :: (Floating r, Arity d) => [Point d r :+ p] -> r
+pairwiseDist pts = sum [ euclideanDist (p^.core) (q^.core) | p <- pts, q <- pts] / 2
+
+
+-- | $(1+\eps)$-approximation of the sum of the pairwise distances.
+--
+-- running time: $O(n(1/eps)^d + n\log n)$, where $n$ is the number of points
+approxPairwiseDistance         :: (Floating r, Ord r, AlwaysTrueWSPD d, Index' 0 d
+                                  , Show r, Show p)
+                               => r -> [Point d r :+ p] -> r
+approxPairwiseDistance _   []  = 0
+approxPairwiseDistance eps pts =
+    sum [ (size as)*(size bs)*euclideanDist (repr as) (repr bs) | (as,bs) <- pairs ]
+  where
+    t     = withSizes . fairSplitTree . NonEmpty.fromList $ pts
+    pairs = wellSeparatedPairs (4 / eps) t
+
+    size (access' -> (Sized (Size i) _))  = fromIntegral i
+    repr (access' -> (Sized _ (First p))) = p^.core
+
+
+-- wspPairs = fairSplitTree . NonEmpty.fromList
+
+--------------------------------------------------------------------------------
+-- * Helper stuff
+
+-- | Helper to turn the result of 'f k' into 'the expected 'f k', assuming that
+-- we select a set of k points.
+makeExpected         :: (Fractional r, Foldable t) => Int -> t a -> (t a -> r) -> r
+makeExpected k pts f = prb * f pts
+  where
+    n   = length pts
+    prb = ((n - 2) `choose` (k - 2)) / (n `choose` k)
+
+
+choose       :: (Integral a, Num b) => a -> a -> b
+n `choose` k = fromIntegral $ fac n' `div` (fac (n'-k') * fac k')
+  where
+    n' :: Integer
+    n' = fromIntegral n
+    k' :: Integer
+    k' = fromIntegral k
+
+    fac z = product [1..z]
+
+-- newtype WSPDMeasured a = WSPDMeasured a
+
+-- instance Measured (Sized (First a)) (WSPDMeasured a) where
+--   measure (WSPDMeasured p) = Sized 1 (First p)
+
+-- instance Measured v (WSPDMeasured (Point d r :+ p))
+--            => Measured v (SplitTree d p r v) where
+--   measure (Leaf p)      = measure $ WSPDMeasured p
+--   measure (Node _ nd _) = nd^.nodeData
+
+-- | Annotate the split tree with sizes
+withSizes :: SplitTree d p r a -> SplitTree d p r (Sized (First (Point d r :+ p)))
+withSizes = foldUp f Leaf
+  where
+    f l (NodeData j b _) r = let nd = (access' l) <> (access' r)
+                             in Node l (NodeData j b nd) r
+
+-- | Get the measurement for a given splittree
+access'               :: BinLeafTree (NodeData d r (Sized (First a))) a -> Sized (First a)
+access' (Leaf x)      = Sized 1 (First x)
+access' (Node _ nd _) = nd^.nodeData
+
+
+--
+-- | CVS file, in which every line consists of a name, followed by exactly d coordinates
+parseInput :: forall d r. (Arity d, KnownNat d, Read r)
+           => B.ByteString -> [Point d r :+ B.ByteString]
+parseInput = mapMaybe toPoint . drop 1 . C.lines
+  where
+    trim      = fst . C.spanEnd isSpace . C.dropWhile isSpace
+    fromList' = vectorFromList . take (fromInteger . natVal $ (Proxy :: Proxy d))
+
+    toPoint bs = let (n:rs) = map trim . C.split ',' $ bs
+                     p      = fmap Point . fromList' . map (read . C.unpack) $ rs
+                 in (:+ n) <$> p
+
+
+readInput :: (Arity d, KnownNat d, Read r) => FilePath -> IO [Point d r :+ C.ByteString]
+readInput = fmap parseInput . B.readFile
+
+test :: FilePath -> IO [Point 2 Double :+ C.ByteString]
+test = readInput
+
+testTree = fmap f .test
+  where
+    f pts = uncovered pts (4 / 0.05) (fairSplitTree $ NonEmpty.fromList pts)
+
+-- compareBoth     :: r -> FilePath -> IO (r, r, Bool)
+compareBoth eps = fmap f . test
+  where
+    f pts = let exact  = pairwiseDist pts
+                approx = approxPairwiseDistance eps pts
+            in (exact, approx, (1-eps)*exact <= approx && approx <= (1+eps)*exact)
+
+
+compareBoth1 eps pts = let exact  = pairwiseDist pts
+                           approx = approxPairwiseDistance eps pts
+                       in (exact, approx, (1-eps)*exact <= approx && approx <= (1+eps)*exact)
+
+
+mainWith (Options f) = compareBoth 0.05 f >>= print
+
+
+--------------------------------------------------------------------------------
+-- testing stuff
+
+
+-- | Computes all pairs of points that are uncovered by the WSPD with separation s
+uncovered         :: (Floating r, Ord r, AlwaysTrueWSPD d, Ord p)
+                  => [Point d r :+ p] -> r -> SplitTree d p r a -> [(Point d r :+ p, Point d r :+ p)]
+uncovered pts s t = Set.toList $ allPairs `Set.difference` covered
+  where
+    allPairs = Set.fromList [ (p,q) | p <- pts, q <- pts, p < q ]
+    covered  = Set.unions [ mkSet as bs | (as,bs) <- wellSeparatedPairs s t]
+
+mkSet as bs = Set.fromList [ (min a b,max a b) | a <- F.toList as, b <- F.toList bs]
+
+-- | Naively check if a WSP pair is actually well separated with respect to
+-- separation s. i.e. computes the maximum diameter of as and bs, and then
+-- tests by brute force if all pairs (a,b) from different sets are at distance
+-- at least s times the maximum diameter.
+isWellSeparated           :: (Floating r, Ord r, Arity d) => r -> WSP d p r a -> Bool
+isWellSeparated s (as,bs) =
+    and [ euclideanDist (a^.core) (b^.core) >= s*d | a <- F.toList as, b <- F.toList bs ]
+  where
+    d = (/2) . maximum . map (diameterNaive . F.toList) $ [as,bs]
+
+
+nonWellSeparated s = map (\(a,b,c) -> (a,b))
+                   . filter (\(a,b,c) -> not c)
+                   . map (\p@(a,b) -> (a,b,isWellSeparated s p))
+                   . wellSeparatedPairs s . fairSplitTree . NonEmpty.fromList
+
+
+points1 :: [Point 2 Double :+ ()]
+points1 = ext <$> [point2 0 0, point2 1 1, point2 2 10, point2 3 11, point2 5 5, point2 10 0]
diff --git a/examples/Demo/GPXParser.hs b/examples/Demo/GPXParser.hs
--- a/examples/Demo/GPXParser.hs
+++ b/examples/Demo/GPXParser.hs
@@ -6,19 +6,17 @@
 module Demo.GPXParser where
 
 
-import qualified Data.ByteString.Lazy as B
-
 import           Control.Applicative
+import           Control.Lens
 import           Control.Monad
+import qualified Data.ByteString.Lazy as B
+import           Data.Ext
+import           Data.Geometry.Point
 import           Data.Maybe
+import           Data.Semigroup
 import           Data.Time.Clock
 import           Data.Time.Format
 import           Text.XML.Expat.Tree
-
-import           Control.Lens
-
-import           Data.Geometry.Point
-import           Data.Ext
 
 import           Debug.Trace
 
diff --git a/examples/Demo/MinDisk.hs b/examples/Demo/MinDisk.hs
--- a/examples/Demo/MinDisk.hs
+++ b/examples/Demo/MinDisk.hs
@@ -3,29 +3,25 @@
 
 import           Algorithms.Geometry.SmallestEnclosingBall.RandomizedIncrementalConstruction
 import           Algorithms.Geometry.SmallestEnclosingBall.Types
-
 import           Control.Applicative
 import           Control.Lens
 import           Data.Data
 import           Data.Ext
 import qualified Data.Foldable as F
-import qualified Data.Traversable as Tr
 import           Data.Geometry
 import           Data.Geometry.Ball
-import           Data.Geometry.Line
-import           Data.Geometry.PolyLine
-import           Data.Monoid
-
---import           Data.Geometry.Ipe
-
 import           Data.Geometry.Ipe
 import           Data.Geometry.Ipe.Types
+import           Data.Geometry.Line
+import           Data.Geometry.PolyLine
 import           Data.Maybe
+import           Data.Semigroup
 import           Data.Seq2
+import qualified Data.Traversable as Tr
 import           Data.Vinyl
-import           System.Environment(getArgs)
-import           System.Random
 import           Options.Applicative
+import           System.Environment (getArgs)
+import           System.Random
 
 --------------------------------------------------------------------------------
 
diff --git a/examples/Main.hs b/examples/Main.hs
--- a/examples/Main.hs
+++ b/examples/Main.hs
@@ -11,36 +11,39 @@
 import qualified Demo.WriteEnsemble as EnsembleWriter
 import qualified Demo.MinDisk as MinDisk
 import qualified Demo.Delaunay as Delaunay
+import qualified Demo.ExpectedPairwiseDistance as ExpPWD
 
 
 
 
 --------------------------------------------------------------------------------
 
-data Options = BAPC           BAPCOptions
-             | DrawGPX        DrawGPX.Options
-             | EnsembleWriter EnsembleWriter.Options
-             | MinDisk        MinDisk.Options
-             | Delaunay       Delaunay.Options
+data Options = BAPC                      BAPCOptions
+             | DrawGPX                   DrawGPX.Options
+             | EnsembleWriter            EnsembleWriter.Options
+             | MinDisk                   MinDisk.Options
+             | Delaunay                  Delaunay.Options
+             | ExpectedPairwiseDistance  ExpPWD.Options
              deriving Data
 
 parser :: Parser Options
 parser = subparser (
-       command' DrawGPX        DrawGPX.options
-    <> command' EnsembleWriter EnsembleWriter.options
-    <> command' MinDisk        MinDisk.options
-    <> command' Delaunay       Delaunay.options
+       command' DrawGPX                        DrawGPX.options
+    <> command' EnsembleWriter                 EnsembleWriter.options
+    <> command' MinDisk                        MinDisk.options
+    <> command' Delaunay                       Delaunay.options
+    <> command' ExpectedPairwiseDistance       ExpPWD.options
     )
 
 
 mainWith       :: Options -> IO ()
 mainWith opts' = case opts' of
-  BAPC _              -> putStrLn "not yet"
-  DrawGPX opts        -> DrawGPX.mainWith opts
-  EnsembleWriter opts -> EnsembleWriter.mainWith opts
-  MinDisk opts        -> MinDisk.mainWith opts
-  Delaunay opts       -> Delaunay.mainWith opts
-
+  BAPC _                              -> putStrLn "not yet"
+  DrawGPX opts                        -> DrawGPX.mainWith opts
+  EnsembleWriter opts                 -> EnsembleWriter.mainWith opts
+  MinDisk opts                        -> MinDisk.mainWith opts
+  Delaunay opts                       -> Delaunay.mainWith opts
+  ExpectedPairwiseDistance opts       -> ExpPWD.mainWith opts
 
 --------------------------------------------------------------------------------
 
diff --git a/hgeometry.cabal b/hgeometry.cabal
--- a/hgeometry.cabal
+++ b/hgeometry.cabal
@@ -2,13 +2,13 @@
 -- documentation, see http://haskell.org/cabal/users-guide/
 
 name:                hgeometry
-version:             0.5.0.0
+version:             0.6.0.0
 synopsis:            Geometric Algorithms, Data structures, and Data types.
 description:
   HGeometry provides some basic geometry types, and geometric algorithms and
   data structures for them. The main two focusses are: (1) Strong type safety,
   and (2) implementations of geometric algorithms and data structures with good
-  asymptotic running time guarantees.
+  asymptotic running time guarantees. Note that HGeometry is still highly experimental, don't be surprised to find bugs.
 
 homepage:            https://fstaals.net/software/hgeometry
 license:             BSD3
@@ -17,7 +17,7 @@
 maintainer:          frank@fstaals.net
 -- copyright:
 
-tested-with:         GHC >= 7.8.4
+tested-with:         GHC >= 7.10.2
 
 category:            Geometry
 build-type:          Simple
@@ -27,7 +27,17 @@
                      test/Data/Geometry/pointInPolygon.ipe
                      test/Data/Geometry/Polygon/Convex/convexTests.ipe
                      test/Algorithms/Geometry/SmallestEnclosingDisk/manual.ipe
+                     test/Algorithms/Geometry/LineSegmentIntersection/manual.ipe
+                     examples/BAPC2014/sample.in
+                     examples/BAPC2014/sample.out
+                     examples/BAPC2014/testdata.in
+                     examples/BAPC2014/testdata.out
+                     examples/BAPC2012/G.in
+                     examples/BAPC2012/G.out
+                     examples/BAPC2012/sampleG.in
+                     examples/BAPC2012/sampleG.out
 
+
 cabal-version:       >=1.10
 source-repository head
   type:     git
@@ -40,17 +50,22 @@
   manual:      True
 
 library
-  ghc-options: -Wall -fno-warn-unticked-promoted-constructors
+  ghc-options: -Wall -fno-warn-unticked-promoted-constructors -fno-warn-type-defaults
 
-  exposed-modules:  Data.Geometry
+  exposed-modules:
+                    -- * Generic Geometry
+                    Data.Geometry
                     Data.Geometry.Properties
-                    Data.Geometry.Vector
-                    -- Data.Geometry.Vector.Vinyl
                     Data.Geometry.Transformation
-                    Data.Geometry.Vector.VectorFixed
-                    Data.Geometry.Interval
                     Data.Geometry.Boundary
+                    Data.Geometry.Duality
 
+                    -- * Basic Geometry Types
+                    Data.Geometry.Vector
+                    Data.Geometry.Vector.VectorFixed
+                    -- Data.Geometry.Vector.Vinyl
+                    Data.Geometry.Interval
+                    Data.Geometry.Interval.Util
                     Data.Geometry.Point
                     Data.Geometry.Line
                     Data.Geometry.Line.Internal
@@ -58,10 +73,7 @@
                     Data.Geometry.SubLine
                     Data.Geometry.HalfLine
                     Data.Geometry.PolyLine
-
                     Data.Geometry.Triangle
-
-
                     -- Data.Geometry.Plane
                     Data.Geometry.Slab
                     Data.Geometry.Box
@@ -70,12 +82,19 @@
                     Data.Geometry.Polygon
                     Data.Geometry.Polygon.Convex
 
-                    Data.Geometry.Duality
+                    -- * Geometric Data Structures
+                    Data.Geometry.IntervalTree
+                    Data.Geometry.SegmentTree
+                    Data.Geometry.SegmentTree.Generic
 
+                    Data.Geometry.KDTree
+                    Data.Geometry.PlanarSubdivision
+
+                    -- * Algorithms
                     Algorithms.Util
 
+                    -- * Geometric Algorithms
                     Algorithms.Geometry.ConvexHull.GrahamScan
-                    Algorithms.Geometry.ConvexHull.Types
                     Algorithms.Geometry.ConvexHull.DivideAndConqueror
 
                     Algorithms.Geometry.SmallestEnclosingBall.Types
@@ -90,15 +109,20 @@
 
                     Algorithms.Geometry.EuclideanMST.EuclideanMST
 
-
-                    Algorithms.Graph.DFS
-                    Algorithms.Graph.MST
+                    Algorithms.Geometry.WellSeparatedPairDecomposition.WSPD
+                    Algorithms.Geometry.WellSeparatedPairDecomposition.Types
 
+                    Algorithms.Geometry.Diameter
 
-                    Data.UnBounded
-                    Data.Range
+                    Algorithms.Geometry.LineSegmentIntersection.Naive
+                    Algorithms.Geometry.LineSegmentIntersection.BentleyOttmann
+                    Algorithms.Geometry.LineSegmentIntersection.Types
 
+                    -- * Graph Algorithms
+                    Algorithms.Graph.DFS
+                    Algorithms.Graph.MST
 
+                    -- * Ipe Types
                     Data.Geometry.Ipe
                     Data.Geometry.Ipe.Literal
                     Data.Geometry.Ipe.Attributes
@@ -109,24 +133,32 @@
                     Data.Geometry.Ipe.IpeOut
                     Data.Geometry.Ipe.FromIpe
 
-
-
+                    -- * General Data Types
+                    Data.UnBounded
+                    Data.Range
                     Data.Ext
                     Data.Seq2
+                    Data.Seq
                     Data.CircularSeq
                     Data.Sequence.Util
                     Data.BinaryTree
+                    Data.BinaryTree.Zipper
+
                     Data.CircularList.Util
+                    Data.BalBST
+                    Data.Util
 
+                    -- * Planar Graphs
                     Data.Permutation
                     Data.PlanarGraph
                     Data.PlaneGraph
 
-
+                    -- * Other
                     System.Random.Shuffle
                     Control.Monad.State.Persistent
 
 
+
   other-modules:
                    Data.Geometry.Ipe.ParserPrimitives
 
@@ -134,27 +166,29 @@
 
 
   -- other-extensions:
-  build-depends: base             >= 4.7       &&     < 5
-               , containers       >= 0.5.5
-               , vinyl            >= 0.5       &&     < 0.6
-               , linear           >= 1.10
-               , lens             >= 4.2
-               , singletons       >= 1.0       &&     < 2.0
-               , bifunctors       >= 5
-               , semigroupoids    >= 5
-               , Frames           >= 0.1.0.0
+  build-depends:
 
-               , text             >= 1.1.1.0
-               , bytestring       >= 0.10
+                  Frames           >= 0.1.3.0
+                , base             >= 4.8       &&     < 5
+                , bifunctors       >= 4.1
+                , bytestring       >= 0.10
+                , containers       >= 0.5.5
+                , contravariant    >= 1.4
+                , lens             >= 4.2
+                , linear           >= 1.10
+                , semigroupoids    >= 5
+                , semigroups       >= 0.18
+                , singletons       >= 2.0
+                , text             >= 1.1.1.0
+                , vinyl            >= 0.5       &&     < 0.6
+                , deepseq          >= 1.1
 
-               , semigroups       >= 0.15
-               , bifunctors       >= 4.1
                -- , validation       >= 0.4
 
                , parsec           >= 3
                -- , tranformers      > 0.3
 
-               , vector           >= 0.10
+               , vector           >= 0.11
                , fixed-vector     >= 0.6.4.0
                , data-clist       >= 0.0.7.2
 
@@ -193,9 +227,10 @@
                     , DeriveFunctor
                     , DeriveFoldable
                     , DeriveTraversable
-
+                    , DeriveGeneric
                     , AutoDeriveTypeable
 
+
                     , FlexibleInstances
                     , FlexibleContexts
                     , MultiParamTypeClasses
@@ -213,7 +248,7 @@
                  , vinyl
                  , Frames
                  , semigroups
-                 , optparse-applicative
+                 , optparse-applicative   >= 0.13.0.0
                  , text
                  , hexpat
                  , bytestring
@@ -228,6 +263,7 @@
                  Demo.WriteEnsemble
                  Demo.MinDisk
                  Demo.Delaunay
+                 Demo.ExpectedPairwiseDistance
 
                  Demo.GPXParser
 
@@ -273,21 +309,32 @@
   default-language:     Haskell2010
   hs-source-dirs:       test
   main-is:              Spec.hs
+  ghc-options:   -O2 -fno-warn-unticked-promoted-constructors
 
   other-modules: Data.RangeSpec
+                 Data.EdgeOracleSpec
+                 Data.PlanarGraphSpec
                  Data.Geometry.Ipe.ReaderSpec
                  Data.Geometry.PolygonSpec
                  Data.Geometry.PointSpec
                  Data.Geometry.Polygon.Convex.ConvexSpec
+                 Data.Geometry.KDTreeSpec
+                 Data.Geometry.IntervalSpec
+                 Data.Geometry.BoxSpec
 
+
                  Algorithms.Geometry.SmallestEnclosingDisk.RISpec
                  Algorithms.Geometry.DelaunayTriangulation.DTSpec
+                 Algorithms.Geometry.WellSeparatedPairDecomposition.WSPDSpec
+                 Algorithms.Geometry.LineSegmentIntersection.BentleyOttmannSpec
 
+                 QuickCheck.Instances
                  Util
 
 
   build-depends:        base
-                      , hspec >= 2.1
+                      , hspec        >= 2.1
+                      , QuickCheck   >= 2.5
                       , hgeometry
                       , Frames
                       , lens
@@ -312,6 +359,7 @@
                     , PatternSynonyms
                     , ViewPatterns
                     , LambdaCase
+                    , TupleSections
 
 
                     , StandaloneDeriving
@@ -329,7 +377,7 @@
 
 test-suite bapc_examples
   type:          exitcode-stdio-1.0
-  ghc-options:   -O2
+  ghc-options:   -O2 -fno-warn-unticked-promoted-constructors
   main-is:       bapc_examples.hs
   hs-source-dirs: examples
   build-depends: base
@@ -354,3 +402,55 @@
                     , PolyKinds
                     , PatternSynonyms
                     , ViewPatterns
+
+benchmark benchmarks
+
+  hs-source-dirs: benchmark test
+
+  main-is: Benchmarks.hs
+  type: exitcode-stdio-1.0
+
+  other-modules:
+
+  build-depends:
+                base
+              , criterion >= 1.1.4.0 && < 1.2
+              , semigroups
+              , deepseq
+              , deepseq-generics
+              , hgeometry
+              , Frames
+              , lens
+              , QuickCheck
+
+  ghc-options: -Wall -O2 -rtsopts -fno-warn-unticked-promoted-constructors
+
+  default-language:    Haskell2010
+
+  default-extensions: TypeFamilies
+                    , GADTs
+                    , KindSignatures
+                    , DataKinds
+                    , TypeOperators
+                    , ConstraintKinds
+                    , PolyKinds
+                    , RankNTypes
+
+                    , PatternSynonyms
+                    , ViewPatterns
+                    , LambdaCase
+                    , TupleSections
+
+
+                    , StandaloneDeriving
+                    , GeneralizedNewtypeDeriving
+                    , DeriveFunctor
+                    , DeriveFoldable
+                    , DeriveTraversable
+
+                    , AutoDeriveTypeable
+
+                    , FlexibleInstances
+                    , FlexibleContexts
+                    , MultiParamTypeClasses
+                    , OverloadedStrings
diff --git a/src/Algorithms/Geometry/ConvexHull/DivideAndConqueror.hs b/src/Algorithms/Geometry/ConvexHull/DivideAndConqueror.hs
--- a/src/Algorithms/Geometry/ConvexHull/DivideAndConqueror.hs
+++ b/src/Algorithms/Geometry/ConvexHull/DivideAndConqueror.hs
@@ -1,30 +1,26 @@
 module Algorithms.Geometry.ConvexHull.DivideAndConqueror( convexHull
-                                                        , module Types
                                                         ) where
 
-import           Data.Semigroup.Foldable
-import           Data.Semigroup
-import           Algorithms.Geometry.ConvexHull.Types as Types
-import           Control.Lens((^.))
+import           Control.Lens ((^.))
 import           Data.BinaryTree
 import           Data.Ext
-import           Data.Function(on)
+import           Data.Function (on)
 import           Data.Geometry.Point
 import           Data.Geometry.Polygon
-import qualified Data.Geometry.Polygon.Convex as Convex
+import           Data.Geometry.Polygon.Convex
 import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Semigroup
+import           Data.Semigroup.Foldable
 
--- | O(n log n) time ConvexHull using divide and conqueror.
+-- | \(O(n \log n)\) time ConvexHull using divide and conqueror.
 convexHull :: (Ord r, Num r)
-           => NonEmpty.NonEmpty (Point 2 r :+ p) -> ConvexHull p r
+           => NonEmpty.NonEmpty (Point 2 r :+ p) -> ConvexPolygon p r
 convexHull = unMerge
-           . foldMap1 (Merge . ConvexHull . fromPoints . (:[]) . _unElem)
+           . foldMap1 (Merge . ConvexPolygon . fromPoints . (:[]) . _unElem)
            . asBalancedBinLeafTree
            . NonEmpty.sortBy (compare `on` (^.core))
 
-newtype Merge r p = Merge { unMerge :: ConvexHull p r }
+newtype Merge r p = Merge { unMerge :: ConvexPolygon p r }
 
 instance (Num r, Ord r) => Semigroup (Merge r p) where
-  (Merge lp) <> (Merge rp) = Merge . ConvexHull $ ch
-    where
-      (ch,_,_) = Convex.merge (lp^.extractHull) (rp^.extractHull)
+  (Merge lp) <> (Merge rp) = let (ch,_,_) = merge lp rp in Merge ch
diff --git a/src/Algorithms/Geometry/ConvexHull/GrahamScan.hs b/src/Algorithms/Geometry/ConvexHull/GrahamScan.hs
--- a/src/Algorithms/Geometry/ConvexHull/GrahamScan.hs
+++ b/src/Algorithms/Geometry/ConvexHull/GrahamScan.hs
@@ -1,28 +1,27 @@
 module Algorithms.Geometry.ConvexHull.GrahamScan( convexHull
                                                 , upperHull
                                                 , lowerHull
-                                                , module Types
                                                 ) where
 
-import           Algorithms.Geometry.ConvexHull.Types as Types
-import           Control.Lens((^.))
+import           Control.Lens ((^.))
 import           Data.Ext
 import           Data.Geometry.Point
 import           Data.Geometry.Polygon
+import           Data.Geometry.Polygon.Convex (ConvexPolygon(..))
 import qualified Data.List.NonEmpty as NonEmpty
+import           Data.List.NonEmpty (NonEmpty(..))
 import           Data.Monoid
-import           Data.List.NonEmpty(NonEmpty(..))
 
 
--- | O(n log n) time ConvexHull using Graham-Scan. The resulting polygon is
+-- | \(O(n \log n)\) time ConvexHull using Graham-Scan. The resulting polygon is
 -- given in clockwise order.
 convexHull            :: (Ord r, Num r)
-                      => NonEmpty (Point 2 r :+ p) -> ConvexHull p r
-convexHull (p :| []) = ConvexHull . fromPoints $ [p]
+                      => NonEmpty (Point 2 r :+ p) -> ConvexPolygon p r
+convexHull (p :| []) = ConvexPolygon . fromPoints $ [p]
 convexHull ps        = let ps' = NonEmpty.toList . NonEmpty.sortBy incXdecY $ ps
                            uh  = NonEmpty.tail . hull' $         ps'
                            lh  = NonEmpty.tail . hull' $ reverse ps'
-                       in ConvexHull . fromPoints . reverse $ lh ++ uh
+                       in ConvexPolygon . fromPoints . reverse $ lh ++ uh
 
 upperHull  :: (Ord r, Num r) => NonEmpty (Point 2 r :+ p) -> NonEmpty (Point 2 r :+ p)
 upperHull = hull id
@@ -37,7 +36,7 @@
 hull               :: (Ord r, Num r)
                    => ([Point 2 r :+ p] -> [Point 2 r :+ p])
                    -> NonEmpty (Point 2 r :+ p) -> NonEmpty (Point 2 r :+ p)
-hull f h@(_ :| []) = h
+hull _ h@(_ :| []) = h
 hull f pts         = hull' .  f
                    . NonEmpty.toList . NonEmpty.sortBy incXdecY $ pts
 
@@ -53,11 +52,11 @@
     hull'' h  []    = h
     hull'' h (p:ps) = hull'' (cleanMiddle (p:h)) ps
 
-    cleanMiddle [b,a]                           = [b,a]
-    cleanMiddle h@(c:b:a:rest)
-      | rightTurn (a^.core) (b^.core) (c^.core) = h
-      | otherwise                               = cleanMiddle (c:a:rest)
-
+    cleanMiddle h@[_,_]                         = h
+    cleanMiddle h@(z:y:x:rest)
+      | rightTurn (x^.core) (y^.core) (z^.core) = h
+      | otherwise                               = cleanMiddle (z:x:rest)
+    cleanMiddle _                               = error "cleanMiddle: too few points"
 
 rightTurn       :: (Ord r, Num r) => Point 2 r -> Point 2 r -> Point 2 r -> Bool
 rightTurn a b c = ccw a b c == CW
diff --git a/src/Algorithms/Geometry/ConvexHull/Types.hs b/src/Algorithms/Geometry/ConvexHull/Types.hs
deleted file mode 100644
--- a/src/Algorithms/Geometry/ConvexHull/Types.hs
+++ /dev/null
@@ -1,14 +0,0 @@
-{-# LANGUAGE TemplateHaskell #-}
-module Algorithms.Geometry.ConvexHull.Types where
-
-import           Control.Lens
-import           Data.Ext
-import           Data.Geometry.Point
-import           Data.Geometry.Polygon
-
-
-
--- | Two dimensional convex hulls
-newtype ConvexHull p r = ConvexHull { _extractHull :: (SimplePolygon p r) }
-                       deriving (Show,Eq)
-makeLenses ''ConvexHull
diff --git a/src/Algorithms/Geometry/DelaunayTriangulation/DivideAndConqueror.hs b/src/Algorithms/Geometry/DelaunayTriangulation/DivideAndConqueror.hs
--- a/src/Algorithms/Geometry/DelaunayTriangulation/DivideAndConqueror.hs
+++ b/src/Algorithms/Geometry/DelaunayTriangulation/DivideAndConqueror.hs
@@ -3,7 +3,6 @@
 
 import           Algorithms.Geometry.ConvexHull.GrahamScan as GS
 import           Algorithms.Geometry.DelaunayTriangulation.Types
-import           Control.Applicative
 import           Control.Lens
 import           Control.Monad.Reader
 import           Control.Monad.State
@@ -16,10 +15,9 @@
 import           Data.Function (on)
 import           Data.Geometry
 import           Data.Geometry.Ball (disk, insideBall)
-import           Data.Geometry.Interval
 import           Data.Geometry.Polygon
 import qualified Data.Geometry.Polygon.Convex as Convex
-import           Data.Geometry.Polygon.Convex (ConvexPolygon)
+import           Data.Geometry.Polygon.Convex (ConvexPolygon(..), simplePolygon)
 import qualified Data.IntMap.Strict as IM
 import qualified Data.List as L
 import qualified Data.List.NonEmpty as NonEmpty
@@ -46,8 +44,8 @@
 
 -- | Computes the delaunay triangulation of a set of points.
 --
--- Running time: $O(n \log n)$
--- (note: We use an IntMap in the implementation. So maybe actually $O(n \log^2 n)$)
+-- Running time: \(O(n \log n)\)
+-- (note: We use an IntMap in the implementation. So maybe actually \(O(n \log^2 n)\))
 --
 -- pre: the input is a *SET*, i.e. contains no duplicate points. (If the
 -- input does contain duplicate points, the implementation throws them away)
@@ -74,11 +72,11 @@
 delaunayTriangulation' pts mapping'@(vtxMap,_)
   | size' pts == 1 = let (Leaf p) = pts
                          i        = lookup' vtxMap (p^.core)
-                     in (IM.singleton i CL.empty, fromPoints [withID p i])
-  | size' pts <= 3 = let pts'            = NonEmpty.fromList
-                                         . map (\p -> withID p (lookup' vtxMap (p^.core)))
-                                         . F.toList $ pts
-                         (ConvexHull ch) = GS.convexHull pts'
+                     in (IM.singleton i CL.empty, ConvexPolygon $ fromPoints [withID p i])
+  | size' pts <= 3 = let pts'  = NonEmpty.fromList
+                               . map (\p -> withID p (lookup' vtxMap (p^.core)))
+                               . F.toList $ pts
+                         ch    = GS.convexHull pts'
                      in (fromHull mapping' ch, ch)
   | otherwise      = let (Node lt _ rt) = pts
                          (ld,lch)       = delaunayTriangulation' lt mapping'
@@ -91,16 +89,17 @@
 
 -- | Mapping that says for each vtx in the convex hull what the first entry in
 -- the adj. list should be. The input polygon is given in Clockwise order
-firsts :: SimplePolygon (p :+ VertexID) r -> IM.IntMap VertexID
+firsts :: ConvexPolygon (p :+ VertexID) r -> IM.IntMap VertexID
 firsts = IM.fromList . map (\s -> (s^.end.extra.extra, s^.start.extra.extra))
-       . F.toList . outerBoundaryEdges
+       . F.toList . outerBoundaryEdges . _simplePolygon
 
 
 -- | Given a polygon; construct the adjacency list representation
 -- pre: at least two elements
-fromHull              :: Ord r => Mapping p r -> SimplePolygon (p :+ q) r -> Adj
+fromHull              :: Ord r => Mapping p r -> ConvexPolygon (p :+ q) r -> Adj
 fromHull (vtxMap,_) p = let vs@(u:v:vs') = map (lookup' vtxMap . (^.core))
-                                         . F.toList . CS.rightElements $ p^.outerBoundary
+                                         . F.toList . CS.rightElements
+                                         $ p^.simplePolygon.outerBoundary
                             es           = zipWith3 f vs (tail vs ++ [u]) (vs' ++ [u,v])
                             f prv c nxt  = (c,CL.fromList . L.nub $ [prv, nxt])
                         in IM.fromList es
@@ -108,7 +107,7 @@
 
 -- | Merge the two delaunay triangulations.
 --
--- running time: $O(n)$ (although we cheat a bit by using a IntMap)
+-- running time: \(O(n)\) (although we cheat a bit by using a IntMap)
 merge                            :: (Ord r, Fractional r)
                                  => Adj
                                  -> Adj
diff --git a/src/Algorithms/Geometry/DelaunayTriangulation/Naive.hs b/src/Algorithms/Geometry/DelaunayTriangulation/Naive.hs
--- a/src/Algorithms/Geometry/DelaunayTriangulation/Naive.hs
+++ b/src/Algorithms/Geometry/DelaunayTriangulation/Naive.hs
@@ -1,22 +1,21 @@
 module Algorithms.Geometry.DelaunayTriangulation.Naive where
 
-import Algorithms.Geometry.DelaunayTriangulation.Types
-
-import Control.Applicative
-import Control.Monad(forM_)
-import Control.Lens
-import Data.Function(on)
+import           Algorithms.Geometry.DelaunayTriangulation.Types
+import           Control.Lens
+import           Control.Monad (forM_)
+import qualified Data.CircularList as C
+import           Data.Ext
 import qualified Data.Foldable as F
-import qualified Data.Vector as V
-import qualified Data.Vector.Mutable as MV
+import           Data.Function (on)
+import           Data.Geometry
+import           Data.Geometry.Ball (disk, insideBall)
+import qualified Data.List as L
 import qualified Data.List.NonEmpty as NonEmpty
 import qualified Data.Map as M
-import qualified Data.CircularList as C
-import Data.Ext
-import Data.Geometry
-import Data.Geometry.Ball(disk, insideBall)
-import qualified Data.List as L
+import qualified Data.Vector as V
+import qualified Data.Vector.Mutable as MV
 
+--------------------------------------------------------------------------------
 
 -- | Naive O(n^4) time implementation of the delaunay triangulation. Simply
 -- tries each triple (p,q,r) and tests if it is delaunay, i.e. if there are no
@@ -24,7 +23,7 @@
 --
 -- pre: the input is a *SET*, i.e. contains no duplicate points. (If the
 -- input does contain duplicate points, the implementation throws them away)
-delaunayTriangulation     :: (Ord r, Fractional r,       Show r, Show p)
+delaunayTriangulation     :: (Ord r, Fractional r)
                           => NonEmpty.NonEmpty (Point 2 r :+ p) -> Triangulation p r
 delaunayTriangulation pts = Triangulation ptIds ptsV adjV
   where
diff --git a/src/Algorithms/Geometry/DelaunayTriangulation/Types.hs b/src/Algorithms/Geometry/DelaunayTriangulation/Types.hs
--- a/src/Algorithms/Geometry/DelaunayTriangulation/Types.hs
+++ b/src/Algorithms/Geometry/DelaunayTriangulation/Types.hs
@@ -7,16 +7,13 @@
 import           Data.Ext
 import           Data.Geometry
 import           Data.Geometry.Ipe
+import           Data.Geometry.PlanarSubdivision
 import qualified Data.IntMap.Strict as IM
 import qualified Data.Map as M
 import qualified Data.Map.Strict as SM
-import           Data.Monoid (mempty)
-import qualified Data.Permutation as P
 import           Data.PlaneGraph
 import qualified Data.Vector as V
 
-import           Debug.Trace
-
 --------------------------------------------------------------------------------
 
 -- We store all adjacency lists in clockwise order
@@ -80,30 +77,50 @@
 -- - x: the data value we are interested in computing
 type ST' a = ST (SM.Map (VertexID,VertexID) ArcID) ArcID a
 
+
+-- | convert the triangulation into a planarsubdivision
+--
+-- running time: \(O(n\log n)\).
+toPlanarSubdivision :: proxy s -> Triangulation p r -> PlanarSubdivision s p () () r
+toPlanarSubdivision px tr = PlanarSubdivision g
+  where
+    g = toPlaneGraph px tr & vertexData.traverse  %~ (\(v :+ e) -> VertexData v e)
+                           & dartData.traverse._2 %~ EdgeData Visible
+                           & faceData.traverse    %~ FaceData []
+
+-- | convert the triangulation into a plane graph
+--
+-- running time: \(O(n\log n)\).
 toPlaneGraph    :: forall proxy s p r.
                    proxy s -> Triangulation p r -> PlaneGraph s Primal_ p () () r
-toPlaneGraph _ tr = (planarGraph . P.toCycleRep n $ perm)&vertexData .~ tr^.positions
+toPlaneGraph _ tr = g & vertexData .~ tr^.positions
   where
-    neighs    = C.rightElements <$> tr^.neighbours
-    n         = sum . fmap length $ neighs
+    g       = fromAdjacencyLists . V.toList . V.imap f $ tr^.neighbours
+    f i adj = (VertexId i, VertexId <$> adj)
 
-    vtxIDs = [0..]
-    perm = trd' . foldr toOrbit (ST mempty 0 mempty) $ zip vtxIDs (V.toList neighs)
 
-    -- | Given a vertex with its adjacent vertices (u,vs) (in CCW order) convert this
-    -- vertex with its adjacent vertices into an Orbit
-    toOrbit                     :: (VertexID,[VertexID]) -> ST' [[Dart s]]
-                                -> ST' [[Dart s]]
-    toOrbit (u,vs) (ST m a dss) =
-      let (ST m' a' ds') = foldr (toDart . (u,)) (ST m a mempty) vs
-      in ST m' a' (ds':dss)
+  -- (planarGraph' . P.toCycleRep n $ perm)&vertexData .~ tr^.positions
+  -- where
+  --   neighs    = C.rightElements <$> tr^.neighbours
+  --   n         = sum . fmap length $ neighs
 
+  --   vtxIDs = [0..]
+  --   perm = trd' . foldr toOrbit (ST mempty 0 mempty) $ zip vtxIDs (V.toList neighs)
 
-    -- | Given an edge (u,v) and a triplet (m,a,ds) we construct a new dart
-    -- representing this edge.
-    toDart                   :: (VertexID,VertexID) -> ST' [Dart s] -> ST' [Dart s]
-    toDart (u,v) (ST m a ds) = let dir = if u < v then Positive else Negative
-                                   t'  = (min u v, max u v)
-                               in case M.lookup t' m of
-      Just a' -> ST m                  a     (Dart (Arc a') dir : ds)
-      Nothing -> ST (SM.insert t' a m) (a+1) (Dart (Arc a)  dir : ds)
+  --   -- | Given a vertex with its adjacent vertices (u,vs) (in CCW order) convert this
+  --   -- vertex with its adjacent vertices into an Orbit
+  --   toOrbit                     :: (VertexID,[VertexID]) -> ST' [[Dart s]]
+  --                               -> ST' [[Dart s]]
+  --   toOrbit (u,vs) (ST m a dss) =
+  --     let (ST m' a' ds') = foldr (toDart . (u,)) (ST m a mempty) vs
+  --     in ST m' a' (ds':dss)
+
+
+  --   -- | Given an edge (u,v) and a triplet (m,a,ds) we construct a new dart
+  --   -- representing this edge.
+  --   toDart                   :: (VertexID,VertexID) -> ST' [Dart s] -> ST' [Dart s]
+  --   toDart (u,v) (ST m a ds) = let dir = if u < v then Positive else Negative
+  --                                  t'  = (min u v, max u v)
+  --                              in case M.lookup t' m of
+  --     Just a' -> ST m                  a     (Dart (Arc a') dir : ds)
+  --     Nothing -> ST (SM.insert t' a m) (a+1) (Dart (Arc a)  dir : ds)
diff --git a/src/Algorithms/Geometry/Diameter.hs b/src/Algorithms/Geometry/Diameter.hs
new file mode 100644
--- /dev/null
+++ b/src/Algorithms/Geometry/Diameter.hs
@@ -0,0 +1,35 @@
+module Algorithms.Geometry.Diameter where
+
+import Control.Lens
+import Data.Ext
+import Data.Geometry
+import Data.List(maximumBy)
+
+--------------------------------------------------------------------------------
+
+
+diameterNaive :: (Ord r, Floating r, Arity d) => [Point d r :+ p] -> r
+diameterNaive = maybe 0 (\(p,q) -> euclideanDist (p^.core) (q^.core))
+              . diametralPairNaive
+
+-- | Computes the Euclidean diametral pair by naively trying all pairs.
+--
+-- running time: \(O(n^2)\)
+diametralPairNaive :: (Ord r, Num r, Arity d)
+                   => [Point d r :+ p] -> Maybe (Point d r :+ p, Point d r :+ p)
+diametralPairNaive = diametralPairWithNaive squaredEuclideanDist
+
+
+-- | Given a distance function and a list of points pts, computes the diametral
+-- pair by naively trying all pairs.
+--
+-- running time: \(O(n^2)\)
+diametralPairWithNaive               :: Ord r
+                                     => (Point d r -> Point d r -> r)
+                                     -> [Point d r :+ p]
+                                     -> Maybe (Point d r :+ p, Point d r :+ p)
+diametralPairWithNaive f pts@(_:_:_) = Just $ maximumBy cmp [ (p,q) | p <- pts, q <- pts ]
+  where
+    f' (p,q) = f (p^.core) (q^.core)
+    tp `cmp` tq = f' tp `compare` f' tq
+diametralPairWithNaive _ _           = Nothing
diff --git a/src/Algorithms/Geometry/EuclideanMST/EuclideanMST.hs b/src/Algorithms/Geometry/EuclideanMST/EuclideanMST.hs
--- a/src/Algorithms/Geometry/EuclideanMST/EuclideanMST.hs
+++ b/src/Algorithms/Geometry/EuclideanMST/EuclideanMST.hs
@@ -22,7 +22,7 @@
 -- pre: the input is a *SET*, i.e. contains no duplicate points. (If the input
 -- does contain duplicate points, the implementation throws them away)
 --
--- running time: $O(n \log n)$
+-- running time: \(O(n \log n)\)
 euclideanMST     :: (Ord r, Fractional r)
                  => NonEmpty.NonEmpty (Point 2 r :+ p) -> Tree (Point 2 r :+ p)
 euclideanMST pts = (\v -> g^.vDataOf v) <$> t
diff --git a/src/Algorithms/Geometry/LineSegmentIntersection/BentleyOttmann.hs b/src/Algorithms/Geometry/LineSegmentIntersection/BentleyOttmann.hs
new file mode 100644
--- /dev/null
+++ b/src/Algorithms/Geometry/LineSegmentIntersection/BentleyOttmann.hs
@@ -0,0 +1,213 @@
+module Algorithms.Geometry.LineSegmentIntersection.BentleyOttmann where
+
+import           Algorithms.Geometry.LineSegmentIntersection.Types
+import           Control.Lens hiding (contains)
+import qualified Data.BalBST as SS -- status struct
+import           Data.Ext
+import           Data.Function (on)
+import           Data.Geometry.Interval
+import           Data.Geometry.Line
+import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
+import           Data.Geometry.Properties
+import qualified Data.List as L
+import           Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.List.NonEmpty as NonEmpty
+import qualified Data.Map as M
+import           Data.Maybe
+import           Data.Ord (Down(..), comparing)
+import           Data.Semigroup
+import qualified Data.Set as EQ -- event queue
+import           Data.Vinyl
+import           Frames.CoRec
+
+import           Debug.Trace
+
+--------------------------------------------------------------------------------
+
+-- | Compute all intersections
+--
+-- \(O((n+k)\log n)\), where \(k\) is the number of intersections.
+intersections    :: (Ord r, Fractional r)
+                 => [LineSegment 2 p r] -> Intersections p r
+intersections ss = merge $ sweep pts (SS.empty $ ordAtNav undefined)
+  where
+    pts = EQ.fromAscList . groupStarts . L.sort . concatMap f $ ss
+    f s = let [p,q] = L.sortBy ordPoints [s^.start.core,s^.end.core]
+          in [Event p (Start $ s :| []), Event q (End s)]
+
+-- | Group the segments with the intersection points
+merge :: Ord r =>  [IntersectionPoint p r] -> Intersections p r
+merge = foldr (\(IntersectionPoint p a) -> M.insertWith (<>) p a) M.empty
+
+-- | Group the startpoints such that segments with the same start point
+-- correspond to one event.
+groupStarts                          :: Eq r => [Event p r] -> [Event p r]
+groupStarts []                       = []
+groupStarts (Event p (Start s) : es) = Event p (Start ss) : groupStarts rest
+  where
+    (ss',rest) = L.span sameStart es
+
+    -- sort the segs on lower endpoint
+    ss         = let (x:|xs) = s in x :| (xs ++ concatMap startSegs ss')
+
+    sameStart (Event q (Start _)) = p == q
+    sameStart _                   = False
+groupStarts (e : es)           = e : groupStarts es
+
+
+--------------------------------------------------------------------------------
+-- * Data type for Events
+
+-- | Type of segment
+data EventType s = Start !(NonEmpty s)| Intersection | End !s deriving (Show)
+
+instance Eq (EventType s) where
+  a == b = a `compare` b == EQ
+
+instance Ord (EventType s) where
+  (Start _)    `compare` (Start _)    = EQ
+  (Start _)    `compare` _            = LT
+  Intersection `compare` (Start _)    = GT
+  Intersection `compare` Intersection = EQ
+  Intersection `compare` (End _)      = LT
+  (End _)      `compare` (End _)      = EQ
+  (End _)      `compare` _            = GT
+
+-- | The actual event consists of a point and its type
+data Event p r = Event { eventPoint :: !(Point 2 r)
+                       , eventType  :: !(EventType (LineSegment 2 p r))
+                       } deriving (Show,Eq)
+
+instance Ord r => Ord (Event p r) where
+  -- decreasing on the y-coord, then increasing on x-coord, and increasing on event-type
+  (Event p s) `compare` (Event q t) = case ordPoints p q of
+                                        EQ -> s `compare` t
+                                        x  -> x
+
+-- | An ordering that is decreasing on y, increasing on x
+ordPoints     :: Ord r => Point 2 r -> Point 2 r -> Ordering
+ordPoints a b = let f p = (Down $ p^.yCoord, p^.xCoord) in comparing f a b
+
+
+-- | Get the segments that start at the given event point
+startSegs   :: Event p r -> [LineSegment 2 p r]
+startSegs e = case eventType e of
+                Start ss -> NonEmpty.toList ss
+                _        -> []
+
+--------------------------------------------------------------------------------
+
+-- | The navigator that we use that orders the segments that intersect at a
+-- horizontal line (from left to right)
+ordAtNav   :: (Ord r, Fractional r) => r -> SS.TreeNavigator r (LineSegment 2 p r)
+ordAtNav y = SS.Nav (\s x -> h s <= x) (min `on` h)
+  where
+    h s = match (s `intersect` horizontalLine y) $
+         (H $ \NoIntersection -> error "ordAtNav: No intersection")
+      :& (H $ \p              -> p^.xCoord)
+      :& (H $ \_              -> rightEndpoint s) -- the intersection is s itself
+      :& RNil
+
+
+--------------------------------------------------------------------------------
+-- * The Main Sweep
+
+type EventQueue p r = EQ.Set (Event p r)
+
+type StatusStructure p r = SS.BalBST r (LineSegment 2 p r)
+
+-- | Run the sweep handling all events
+sweep       :: (Ord r, Fractional r)
+            => EventQueue p r -> StatusStructure p r -> [IntersectionPoint p r]
+sweep eq ss = case EQ.minView eq of
+    Nothing      -> []
+    Just (e,eq') -> handle e eq' ss
+
+-- | Handle an event point
+handle                           :: (Ord r, Fractional r)
+                                 => Event p r -> EventQueue p r -> StatusStructure p r
+                                 -> [IntersectionPoint p r]
+handle e@(eventPoint -> p) eq ss = toReport <> sweep eq' ss'
+  where
+    starts                   = startSegs e
+    (before,contains',after) = extractContains p ss
+    (ends,contains)          = L.partition (endsAt p) contains'
+
+
+    toReport = case starts ++ contains' of
+                 (_:_:_) -> [IntersectionPoint p $ associated (starts <> ends) contains]
+                 _       -> []
+
+    -- new status structure
+    ss' = before `SS.join` newSegs `SS.join` after
+
+    newSegs = toStatusStruct p $ starts ++ contains
+
+    -- the new eeventqueue
+    eq' = foldr EQ.insert eq es
+
+    -- the new events:
+    es | SS.null newSegs = maybeToList $ app (findNewEvent p) sl sr
+       | otherwise       = let s'  = fst <$> SS.minView newSegs
+                               s'' = fst <$> SS.maxView newSegs
+                           in catMaybes [ app (findNewEvent p) sl  s'
+                                        , app (findNewEvent p) s'' sr
+                                        ]
+    sl = fst <$> SS.maxView before
+    sr = fst <$> SS.minView after
+
+    app f x y = do { x' <- x ; y' <- y ; f x' y'}
+
+-- | split the status structure, extracting the segments that contain p.
+-- the result is (before,contains,after)
+extractContains      :: (Fractional r, Ord r)
+                     => Point 2 r -> StatusStructure p r
+                     -> (StatusStructure p r, [LineSegment 2 p r], StatusStructure p r)
+extractContains p ss = (before, mid1 ++ mid2, after)
+  where
+    n = ordAtNav (p^.yCoord)
+    SS.Split before (mid1,mid2) after = SS.splitExtract pred' sel $ ss { SS.nav = n}
+
+    pred' s = not $ SS.goLeft n s (p^.xCoord)
+    sel   s = p `onSegment` s
+
+
+-- | Given a point and the linesegements that contain it. Create a piece of
+-- status structure for it.
+toStatusStruct      :: (Fractional r, Ord r)
+                    => Point 2 r -> [LineSegment 2 p r] -> StatusStructure p r
+toStatusStruct p xs = ss { SS.nav = ordAtNav $ p^.yCoord } `SS.join` hors
+  where
+    (hors',rest) = L.partition isHorizontal xs
+    ss           = SS.fromList (ordAtNav $ maxY xs) rest
+    hors         = SS.fromList (SS.ordNavBy rightEndpoint) hors'
+
+    isHorizontal s  = s^.start.core.yCoord == s^.end.core.yCoord
+
+    -- find the y coord of the first interesting thing below the sweep at y
+    maxY = maximum . filter (< p^.yCoord)
+         . concatMap (\s -> [s^.start.core.yCoord,s^.end.core.yCoord])
+
+-- | Get the right endpoint of a segment
+rightEndpoint   :: Ord r => LineSegment 2 p r -> r
+rightEndpoint s = (s^.start.core.xCoord) `max` (s^.end.core.xCoord)
+
+-- | Test if a segment ends at p
+endsAt                      :: Ord r => Point 2 r -> LineSegment 2 p r -> Bool
+endsAt p (LineSegment' a b) = all (\q -> ordPoints (q^.core) p /= GT) [a,b]
+
+--------------------------------------------------------------------------------
+-- * Finding New events
+
+-- | Find all events
+findNewEvent       :: (Ord r, Fractional r)
+                   => Point 2 r -> LineSegment 2 p r -> LineSegment 2 p r
+                   -> Maybe (Event p r)
+findNewEvent p l r = match (l `intersect` r) $
+     (H $ \NoIntersection -> Nothing)
+  :& (H $ \q              -> if ordPoints q p == GT then Just (Event q Intersection)
+                                      else Nothing)
+  :& (H $ \_              -> Nothing) -- full segment intersectsions are handled
+                                      -- at insertion time
+  :& RNil
diff --git a/src/Algorithms/Geometry/LineSegmentIntersection/Naive.hs b/src/Algorithms/Geometry/LineSegmentIntersection/Naive.hs
new file mode 100644
--- /dev/null
+++ b/src/Algorithms/Geometry/LineSegmentIntersection/Naive.hs
@@ -0,0 +1,53 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+module Algorithms.Geometry.LineSegmentIntersection.Naive where
+
+import           Algorithms.Geometry.LineSegmentIntersection.Types
+import           Control.Lens
+import           Data.Ext
+import           Data.Geometry.Interval
+import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
+import           Data.Geometry.Properties
+import qualified Data.Map as M
+import           Data.Semigroup
+import           Data.Vinyl
+import           Frames.CoRec
+
+
+-- | Compute all intersections (naively)
+--
+-- \(O(n^2)\)
+intersections :: forall r p. (Ord r, Fractional r)
+              => [LineSegment 2 p r] -> Intersections p r
+intersections = foldr collect mempty . pairs
+
+-- | Test if the two segments intersect, and if so add the segment to the map
+collect          :: (Ord r, Fractional r)
+                 => (LineSegment 2 p r, LineSegment 2 p r)
+                 -> Intersections p r -> Intersections p r
+collect (s,s') m = match (s `intersect` s') $
+     (H $ \NoIntersection -> m)
+  :& (H $ \p              -> handlePoint s s' p $ m)
+  :& (H $ \s''            -> foldr (handlePoint s s') m [s''^.start.core, s''^.end.core])
+  :& RNil
+
+-- | Add s and s' to the map with key p
+handlePoint        :: Ord r
+                   => LineSegment 2 p r -> LineSegment 2 p r -> Point 2 r
+                   -> Intersections p r -> Intersections p r
+handlePoint s s' p = addTo p s . addTo p s'
+
+-- | figure out which map to add the point to
+addTo                  :: Ord r => Point 2 r -> LineSegment 2 p r
+                       -> Intersections p r -> Intersections p r
+addTo p s
+  | p `isEndPointOf` s = M.insertWith (<>) p (associated [s] [])
+  | otherwise          = M.insertWith (<>) p (associated [] [s])
+
+isEndPointOf       :: Eq r => Point 2 r -> LineSegment 2 p r -> Bool
+p `isEndPointOf` s = p == s^.start.core || p == s^.end.core
+
+
+pairs        :: [a] -> [(a, a)]
+pairs []     = []
+pairs (x:xs) = map (x,) xs ++ pairs xs
diff --git a/src/Algorithms/Geometry/LineSegmentIntersection/Types.hs b/src/Algorithms/Geometry/LineSegmentIntersection/Types.hs
new file mode 100644
--- /dev/null
+++ b/src/Algorithms/Geometry/LineSegmentIntersection/Types.hs
@@ -0,0 +1,70 @@
+{-# LANGUAGE TemplateHaskell #-}
+module Algorithms.Geometry.LineSegmentIntersection.Types where
+
+import           Control.Lens
+import           Data.Ext
+import           Data.Geometry.Interval
+import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
+import           Data.Geometry.Properties
+import           Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.List.NonEmpty as NonEmpty
+import qualified Data.List as L
+import qualified Data.Map as Map
+import           Data.Semigroup
+
+
+-- get the endpoints of a line segment
+endPoints'   :: (HasEnd s, HasStart s) => s -> (StartCore s, EndCore s)
+endPoints' s = (s^.start.core,s^.end.core)
+
+
+type Set' l =
+  Map.Map (Point (Dimension l) (NumType l), Point (Dimension l) (NumType l)) (NonEmpty l)
+
+data Associated p r = Associated { _endPointOf        :: Set' (LineSegment 2 p r)
+                                 , _interiorTo        :: Set' (LineSegment 2 p r)
+                                 } deriving (Show)
+
+instance (Eq p, Eq r) => Eq (Associated p r) where
+  (Associated es is) == (Associated es' is') = f es es' && f is is'
+    where
+      f xs ys = and $ zipWith (\(p,pa) (q,qa) -> p == q && pa `sameElements` qa)
+                        (Map.toAscList xs) (Map.toAscList ys)
+
+      g = L.nub . NonEmpty.toList
+      sameElements (g -> xs) (g -> ys) = L.null $ (xs L.\\ ys) ++ (ys L.\\ xs)
+
+
+associated       :: Ord r
+                 => [LineSegment 2 p r] -> [LineSegment 2 p r] -> Associated p r
+associated es is = Associated (f es) (f is)
+  where
+    f = foldr (\s -> Map.insertWith (<>) (endPoints' s) (s :| [])) mempty
+
+
+endPointOf :: Associated p r -> [LineSegment 2 p r]
+endPointOf = concatMap NonEmpty.toList . Map.elems . _endPointOf
+
+interiorTo :: Associated p r -> [LineSegment 2 p r]
+interiorTo = concatMap NonEmpty.toList . Map.elems . _interiorTo
+
+
+instance Ord r => Semigroup (Associated p r) where
+  (Associated es is) <> (Associated es' is') = Associated (es <> es') (is <> is')
+
+instance Ord r => Monoid (Associated p r) where
+  mempty = Associated mempty mempty
+  mappend = (<>)
+
+type Intersections p r = Map.Map (Point 2 r) (Associated p r)
+
+data IntersectionPoint p r =
+  IntersectionPoint { _intersectionPoint :: !(Point 2 r)
+                    , _associatedSegs    :: !(Associated p r)
+                    } deriving (Show,Eq)
+makeLenses ''IntersectionPoint
+
+
+
+-- newtype E a b = E (a -> b)
diff --git a/src/Algorithms/Geometry/SmallestEnclosingBall/RandomizedIncrementalConstruction.hs b/src/Algorithms/Geometry/SmallestEnclosingBall/RandomizedIncrementalConstruction.hs
--- a/src/Algorithms/Geometry/SmallestEnclosingBall/RandomizedIncrementalConstruction.hs
+++ b/src/Algorithms/Geometry/SmallestEnclosingBall/RandomizedIncrementalConstruction.hs
@@ -3,19 +3,15 @@
 module Algorithms.Geometry.SmallestEnclosingBall.RandomizedIncrementalConstruction where
 
 import           Algorithms.Geometry.SmallestEnclosingBall.Types
-
 import           Control.Lens
 import           Data.Ext
-import qualified Data.Foldable as F
 import           Data.Geometry
 import           Data.Geometry.Ball
 import qualified Data.List as L
 import           Data.List.NonEmpty
-import           Data.Maybe(fromMaybe)
-import           Data.Monoid
-
+import           Data.Maybe (fromMaybe)
 import           System.Random
-import           System.Random.Shuffle(shuffle)
+import           System.Random.Shuffle (shuffle)
 
 
 
@@ -29,7 +25,7 @@
 
 smallestEnclosingDisk g pts@(_:_:_) = let (p:q:pts') = shuffle g pts
                                       in smallestEnclosingDisk' p q pts'
-smallestEnclosingDisk g _           = error "smallestEnclosingDisk: Too few points"
+smallestEnclosingDisk _ _           = error "smallestEnclosingDisk: Too few points"
 
 -- | Smallest enclosing disk.
 smallestEnclosingDisk'     :: (Ord r, Fractional r)
@@ -50,10 +46,10 @@
                                             -> DiskResult p r
 smallestEnclosingDiskWithPoint p (a :| pts) = foldr addPoint (initial p a) $ L.tails pts
   where
-    addPoint []      br   = br
-    addPoint (q:pts) br@(DiskResult d _)
+    addPoint []       br   = br
+    addPoint (q:pts') br@(DiskResult d _)
       | (q^.core) `inClosedBall` d = br
-      | otherwise                  = smallestEnclosingDiskWithPoints p q (a:pts)
+      | otherwise                  = smallestEnclosingDiskWithPoints p q (a:pts')
 
 
 
diff --git a/src/Algorithms/Geometry/WellSeparatedPairDecomposition/Types.hs b/src/Algorithms/Geometry/WellSeparatedPairDecomposition/Types.hs
new file mode 100644
--- /dev/null
+++ b/src/Algorithms/Geometry/WellSeparatedPairDecomposition/Types.hs
@@ -0,0 +1,73 @@
+{-# LANGUAGE TemplateHaskell #-}
+module Algorithms.Geometry.WellSeparatedPairDecomposition.Types where
+
+import           Control.Lens hiding (Level)
+import           Data.BinaryTree
+import           Data.Ext
+import           Data.Geometry.Box
+import           Data.Geometry.Point
+import           Data.Geometry.Vector
+import           Data.Semigroup
+import qualified Data.Seq2 as S2
+import qualified Data.Sequence as S
+import qualified Data.Traversable as Tr
+
+--------------------------------------------------------------------------------
+
+type SplitTree d p r a = BinLeafTree (NodeData d r a) (Point d r :+ p)
+
+type PointSet d p r a = SplitTree d p r a
+
+type WSP d p r a = (PointSet d p r a, PointSet d p r a)
+
+-- | Data that we store in the split tree
+data NodeData d r a = NodeData { _splitDim :: !Int
+                               , _bBox     :: !(Box d () r)
+                               , _nodeData :: !a
+                               }
+deriving instance (Arity d, Show r, Show a) => Show (NodeData d r a)
+deriving instance (Arity d, Eq r,   Eq a)   => Eq   (NodeData d r a)
+
+makeLenses ''NodeData
+
+instance Semigroup v => Measured v (NodeData d r v) where
+  measure = _nodeData
+
+instance Functor (NodeData d r) where
+  fmap = Tr.fmapDefault
+
+instance Foldable (NodeData d r) where
+  foldMap = Tr.foldMapDefault
+
+instance Traversable (NodeData d r) where
+  traverse f (NodeData d b x) = NodeData d b <$> f x
+
+--------------------------------------------------------------------------------
+-- * Implementation types
+
+type PointSeq d p r = S2.ViewL1 (Point d r :+ p)
+
+
+data Level = Level { _unLevel   :: Int
+                   , _widestDim :: Maybe Int
+                   } deriving (Show,Eq,Ord)
+makeLenses ''Level
+
+nextLevel             :: Level -> Level
+nextLevel (Level i _) = Level (i+1) Nothing
+
+
+
+type Idx = Int
+
+
+data ShortSide = L | R deriving (Show,Eq)
+
+data FindAndCompact d r p = FAC { _leftPart  :: !(S.Seq (Point d r :+ p))
+                                , _rightPart :: !(S.Seq (Point d r :+ p))
+                                , _shortSide :: !ShortSide
+                                }
+deriving instance (Arity d, Show r, Show p) => Show (FindAndCompact d r p)
+deriving instance (Arity d, Eq r,   Eq p)   => Eq   (FindAndCompact d r p)
+
+makeLenses ''FindAndCompact
diff --git a/src/Algorithms/Geometry/WellSeparatedPairDecomposition/WSPD.hs b/src/Algorithms/Geometry/WellSeparatedPairDecomposition/WSPD.hs
new file mode 100644
--- /dev/null
+++ b/src/Algorithms/Geometry/WellSeparatedPairDecomposition/WSPD.hs
@@ -0,0 +1,463 @@
+{-# LANGUAGE TemplateHaskell  #-}
+{-# LANGUAGE LambdaCase  #-}
+module Algorithms.Geometry.WellSeparatedPairDecomposition.WSPD where
+
+import           Algorithms.Geometry.WellSeparatedPairDecomposition.Types
+import           Control.Lens hiding (Level, levels)
+import           Control.Monad.Reader
+import           Control.Monad.ST (ST,runST)
+import           Data.BinaryTree
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry.Box
+import           Data.Geometry.Transformation
+import           Data.Geometry.Properties
+import           Data.Geometry.Point
+import           Data.Geometry.Vector
+import qualified Data.Geometry.Vector as GV
+import qualified Data.List as L
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Maybe
+import           Data.Ord (comparing)
+import           Data.Range
+import qualified Data.Range as Range
+import           Data.Semigroup
+import qualified Data.Seq2 as S2
+import qualified Data.Sequence as S
+import qualified Data.Vector as V
+import qualified Data.Vector.Mutable as MV
+import           GHC.TypeLits
+import qualified Data.IntMap.Strict as IntMap
+
+import Debug.Trace
+
+--------------------------------------------------------------------------------
+
+-- | Construct a split tree
+--
+-- running time: \(O(n \log n)\)
+fairSplitTree     :: (Fractional r, Ord r, Arity d, Index' 0 d,
+                      KnownNat d
+                        , Show r, Show p
+
+                     )
+                  => NonEmpty.NonEmpty (Point d r :+ p) -> SplitTree d p r ()
+fairSplitTree pts = foldUp node' Leaf $ fairSplitTree' n pts'
+  where
+    pts' = GV.imap sortOn . pure . g $ pts
+    n    = length $ pts'^.GV.element (C :: C 0)
+
+    sortOn' i = NonEmpty.sortWith (^.core.unsafeCoord i)
+    sortOn  i = S2.viewL1FromNonEmpty . sortOn' (i + 1)
+    -- sorts the points on the first coordinate, and then associates each point
+    -- with an index,; its rank in terms of this first coordinate.
+    g = NonEmpty.zipWith (\i (p :+ e) -> p :+ (i :+ e)) (NonEmpty.fromList [0..])
+      . sortOn' 1
+
+    -- node' :: b -> a -> b -> b
+    -- node'       :: SplitTree d p r () -> Int -> SplitTree d p r () -> SplitTree d p r ()
+    node' l j r = Node l (NodeData j (bbOf l <> bbOf r) ()) r
+
+
+-- | Given a split tree, generate the Well separated pairs
+--
+-- running time: \(O(s^d n)\)
+wellSeparatedPairs   :: (Floating r, Ord r, AlwaysTrueWSPD d)
+                     => r -> SplitTree d p r a -> [WSP d p r a]
+wellSeparatedPairs s = f
+  where
+    f (Leaf _)     = []
+    f (Node l _ r) = findPairs s l r ++ f l ++ f r
+
+
+
+-- -- | Given a split tree, generate the well separated pairs such that one set is
+-- -- a singleton.
+-- -- running time: \(O(s^d n\log n)\)
+-- wellSeparatedPairSingletons   :: (Fractional r, Ord r, AlwaysTrueWSPD d)
+--                               => r -> SplitTree d p r a -> [(Point d r :+ p, PointSet d p r (Sized a))]
+-- wellSeparatedPairSingletons s t = concatMap split $ wellSeparatedPairs s t'
+--   where
+--     split (l,r) = undefined
+--       -- | measure l <= measure r = map (,r) $ F.toList l
+--       -- | otherwise              = map (,l) $ F.toList r
+--     t' = foldUpData (\l nd r -> )
+
+--     t
+
+
+--------------------------------------------------------------------------------
+-- * Building the split tree
+
+-- | Given the points, sorted in every dimension, recursively build a split tree
+--
+-- The algorithm works in rounds. Each round takes O(n) time, and halves the
+-- number of points. Thus, the total running time is O(n log n).
+--
+-- The algorithm essentially builds a path in the split tree; at every node on
+-- the path that we construct, we split the point set into two sets (L,R)
+-- according to the longest side of the bounding box.
+--
+-- The smaller set is "assigned" to the current node and set asside. We
+-- continue to build the path with the larger set until the total number of
+-- items remaining is less than n/2.
+--
+-- To start the next round, each node on the path needs to have the points
+-- assigned to that node, sorted in each dimension (i.e. the Vector
+-- (PointSeq))'s. Since we have the level assignment, we can compute these
+-- lists by traversing each original input list (i.e. one for every dimension)
+-- once, and partition the points based on their level assignment.
+fairSplitTree'       :: (Fractional r, Ord r, Arity d, Index' 0 d, KnownNat d
+                        , Show r, Show p
+                        )
+                     => Int -> GV.Vector d (PointSeq d (Idx :+ p) r)
+                     -> BinLeafTree Int (Point d r :+ p)
+fairSplitTree' n pts
+    | n <= 1    = let (p S2.:< _) = pts^.GV.element (C :: C 0) in Leaf (dropIdx p)
+    | otherwise = foldr node' (V.last path) $ V.zip nodeLevels (V.init path)
+  where
+    -- note that points may also be assigned level 'Nothing'.
+    (levels, nodeLevels'@(maxLvl NonEmpty.:| _)) = runST $ do
+        lvls  <- MV.replicate n Nothing
+        ls    <- runReaderT (assignLevels (n `div` 2) 0 pts (Level 0 Nothing) []) lvls
+        lvls' <- V.unsafeFreeze lvls
+        pure (lvls',ls)
+
+    -- TODO: We also need to report the levels in the order in which they are
+    -- assigned to nodes
+
+    nodeLevels = V.fromList . L.reverse . NonEmpty.toList $ nodeLevels'
+
+    -- levels = traceShow ("Levels",levels',maxLvl) levels'
+
+    -- path = traceShow ("path", path',nodeLevels) path'
+    distrPts = distributePoints (1 + maxLvl^.unLevel) levels pts
+
+    path = recurse <$> distrPts -- (traceShow ("distributed pts",distrPts) distrPts)
+
+    -- node' (lvl,lc) rc | traceShow ("node' ",lvl,lc,rc) False = undefined
+    node' (lvl,lc) rc = case lvl^?widestDim._Just of
+                          Nothing -> error "Unknown widest dimension"
+                          Just j  -> Node lc j rc
+    recurse pts' = fairSplitTree' (length $ pts'^.GV.element (C :: C 0))
+                                  (reIndexPoints pts')
+
+-- | Assign the points to their the correct class. The 'Nothing' class is
+-- considered the last class
+distributePoints          :: (Arity d , Show r, Show p)
+                          => Int -> V.Vector (Maybe Level)
+                          -> GV.Vector d (PointSeq d (Idx :+ p) r)
+                          -> V.Vector (GV.Vector d (PointSeq d (Idx :+ p) r))
+distributePoints k levels = transpose . fmap (distributePoints' k levels)
+
+transpose :: Arity d => GV.Vector d (V.Vector a) -> V.Vector (GV.Vector d a)
+transpose = V.fromList . map GV.vectorFromListUnsafe . L.transpose
+          . map V.toList . F.toList
+
+-- | Assign the points to their the correct class. The 'Nothing' class is
+-- considered the last class
+distributePoints'              :: Int                      -- ^ number of classes
+                               -> V.Vector (Maybe Level)   -- ^ level assignment
+                               -> PointSeq d (Idx :+ p) r  -- ^ input points
+                               -> V.Vector (PointSeq d (Idx :+ p) r)
+distributePoints' k levels pts
+  | otherwise
+  = fmap fromSeqUnsafe $ V.create $ do
+    v <- MV.replicate k mempty
+    forM_ pts $ \p ->
+      append v (level p) p
+    pure v
+  where
+    level p = maybe (k-1) _unLevel $ levels V.! (p^.extra.core)
+    append v i p = MV.read v i >>= MV.write v i . (S.|> p)
+
+
+
+-- | Given a sequence of points, whose index is increasing in the first
+-- dimension, i.e. if idx p < idx q, then p[0] < q[0].
+-- Reindex the points so that they again have an index
+-- in the range [0,..,n'], where n' is the new number of points.
+--
+-- running time: O(n' * d) (more or less; we are actually using an intmap for
+-- the lookups)
+--
+-- alternatively: I can unsafe freeze and thaw an existing vector to pass it
+-- along to use as mapping. Except then I would have to force the evaluation
+-- order, i.e. we cannot be in 'reIndexPoints' for two of the nodes at the same
+-- time.
+--
+-- so, basically, run reIndex points in ST as well.
+reIndexPoints      :: (Arity d, Index' 0 d)
+                   => GV.Vector d (PointSeq d (Idx :+ p) r)
+                   -> GV.Vector d (PointSeq d (Idx :+ p) r)
+reIndexPoints ptsV = fmap reIndex ptsV
+  where
+    pts = ptsV^.GV.element (C :: C 0)
+
+    reIndex = fmap (\p -> p&extra.core %~ fromJust . flip IntMap.lookup mapping')
+    mapping' = IntMap.fromAscList $ zip (map (^.extra.core) . F.toList $ pts) [0..]
+
+-- | ST monad with access to the vector storign the level of the points.
+type RST s = ReaderT (MV.MVector s (Maybe Level)) (ST s)
+
+-- | Assigns the points to a level. Returns the list of levels used. The first
+-- level in the list is the level assigned to the rest of the nodes. Their
+-- level is actually still set to Nothing in the underlying array.
+assignLevels                  :: (Fractional r, Ord r, Arity d, KnownNat d
+                                 , Show r, Show p
+                                 )
+                              => Int -- ^ Number of items we need to collect
+                              -> Int -- ^ Number of items we collected so far
+                              -> GV.Vector d (PointSeq d (Idx :+ p) r)
+                              -> Level -- ^ next level to use
+                              -> [Level] -- ^ Levels used so far
+                              -> RST s (NonEmpty.NonEmpty Level)
+assignLevels h m pts l prevLvls
+  | m >= h    = pure (l NonEmpty.:| prevLvls)
+  | otherwise = do
+    pts' <- compactEnds pts
+    -- find the widest dimension j = i+1
+    let j    = widestDimension pts'
+        i    = j - 1 -- traceShow  ("i",j,pts') j - 1
+        extJ = (extends pts')^.ix' i
+        mid  = midPoint extJ
+
+    -- find the set of points that we have to delete, by looking at the sorted
+    -- list L_j. As a side effect, this will remove previously assigned points
+    -- from L_j.
+    (lvlJPts,deletePts) <- findAndCompact j (pts'^.ix' i) mid
+    let pts''     = pts'&ix' i .~ lvlJPts
+        l'        = l&widestDim .~ Just j
+    forM_ deletePts $ \p ->
+      assignLevel p l'
+    assignLevels h (m + length deletePts) pts'' (nextLevel l) (l' : prevLvls)
+
+-- | Remove already assigned pts from the ends of all vectors.
+compactEnds        :: Arity d
+                   => GV.Vector d (PointSeq d (Idx :+ p) r)
+                   -> RST s (GV.Vector d (PointSeq d (Idx :+ p) r))
+compactEnds = traverse compactEnds'
+
+-- | Assign level l to point p
+assignLevel     :: (c :+ (Idx :+ p)) -> Level -> RST s ()
+assignLevel p l = ask >>= \levels -> lift $ MV.write levels (p^.extra.core) (Just l)
+
+-- | Get the level of a point
+levelOf   :: (c :+ (Idx :+ p)) -> RST s (Maybe Level)
+levelOf p = ask >>= \levels -> lift $ MV.read levels (p^.extra.core)
+
+-- | Test if the point already has a level assigned to it.
+hasLevel :: c :+ (Idx :+ p) -> RST s Bool
+hasLevel = fmap isJust . levelOf
+
+-- | Remove allready assigned points from the sequence
+--
+-- pre: there are points remaining
+compactEnds'               :: PointSeq d (Idx :+ p) r
+                           -> RST s (PointSeq d (Idx :+ p) r)
+compactEnds' (l0 S2.:< s0) = fmap fromSeqUnsafe . goL $ l0 S.<| s0
+  where
+    goL s@(S.viewl -> l S.:< s') = hasLevel l >>= \case
+                                     False -> goR s
+                                     True  -> goL s'
+    goR s@(S.viewr -> s' S.:> r) = hasLevel r >>= \case
+                                     False -> pure s
+                                     True  -> goR s'
+
+
+-- | Given the points, ordered by their j^th coordinate, split the point set
+-- into a "left" and a "right" half, i.e. the points whose j^th coordinate is
+-- at most the given mid point m, and the points whose j^th coordinate is
+-- larger than m.
+--
+-- We return a pair (Largest set, Smallest set)
+--
+--
+--fi ndAndCompact works by simultaneously traversing the points from left to
+-- right, and from right to left. As soon as we find a point crossing the mid
+-- point we stop and return. Thus, in principle this takes only O(|Smallest
+-- set|) time.
+--
+-- running time: O(|Smallest set|) + R, where R is the number of *old* points
+-- (i.e. points that should have been removed) in the list.
+findAndCompact                   :: (Ord r, Arity d
+                                    , Show r, Show p
+                                    )
+                                 => Int
+                                    -- ^ the dimension we are in, i.e. so that we know
+                                    -- which coordinate of the point to compare
+                                 -> PointSeq d (Idx :+ p) r
+                                 -> r -- ^ the mid point
+                                 -> RST s ( PointSeq d (Idx :+ p) r
+                                          , PointSeq d (Idx :+ p) r
+                                          )
+findAndCompact j (l0 S2.:< s0) m = fmap select . stepL $ l0 S.<| s0
+  where
+    -- stepL and stepR together build a data structure (FAC l r S) that
+    -- contains the left part of the list, i.e. the points before midpoint, and
+    -- the right part of the list., and a value S that indicates which part is
+    -- the short side.
+
+    -- stepL takes a step on the left side of the list; if the left point l
+    -- already has been assigned, we continue waling along (and "ignore" the
+    -- point). If it has not been assigned, and is before the mid point, we
+    -- take a step from the right, and add l onto the left part. If it is
+    -- larger than the mid point, we have found our split.
+    -- stepL :: S.Seq (Point d r :+ (Idx :+ p)) -> ST s (FindAndCompact d r (Idx :+ p))
+    stepL s = case S.viewl s of
+      S.EmptyL  -> pure $ FAC mempty mempty L
+      l S.:< s' -> hasLevel l >>= \case
+                     False -> if l^.core.unsafeCoord j <= m
+                                 then addL l <$> stepR s'
+                                 else pure $ FAC mempty s L
+                     True  -> stepL s' -- delete, continue left
+
+    -- stepR :: S.Seq (Point d r :+ (Idx :+ p)) -> ST s (FindAndCompact d r (Idx :+ p))
+    stepR s = case S.viewr s of
+      S.EmptyR  -> pure $ FAC mempty mempty R
+      s' S.:> r -> hasLevel r >>= \case
+                     False -> if r^.core.unsafeCoord j >= m
+                                 then addR r <$> stepL s'
+                                 else pure $ FAC s mempty R
+                     True  -> stepR s'
+
+
+    addL l x = x&leftPart  %~ (l S.<|)
+    addR r x = x&rightPart %~ (S.|> r)
+
+    select = over both fromSeqUnsafe . select'
+
+    -- select' f | traceShow ("select'", f) False = undefined
+    select' (FAC l r L) = (r, l)
+    select' (FAC l r R) = (l, r)
+
+
+-- | Find the widest dimension of the point set
+--
+-- pre: points are sorted according to their dimension
+widestDimension :: (Num r, Ord r, Arity d) => GV.Vector d (PointSeq d p r) -> Int
+widestDimension = fst . L.maximumBy (comparing snd) . zip [1..] . F.toList . widths
+
+widths :: (Num r, Arity d) => GV.Vector d (PointSeq d p r) -> GV.Vector d r
+widths = fmap Range.width . extends
+
+
+
+-- | get the extends of the set of points in every dimension, i.e. the left and
+-- right boundaries.
+--
+-- pre: points are sorted according to their dimension
+extends :: Arity d => GV.Vector d (PointSeq d p r) -> GV.Vector d (Range r)
+extends = GV.imap (\i pts@(l S2.:< _) ->
+                     let (_ S2.:> r) = S2.viewL1toR1 pts
+                     in ClosedRange (l^.core.unsafeCoord (i + 1))
+                                    (r^.core.unsafeCoord (i + 1)))
+
+
+--------------------------------------------------------------------------------
+-- * Finding Well Separated Pairs
+
+type AlwaysTrueWSPD d = ( Arity d, KnownNat d
+                        , AlwaysTruePFT d, AlwaysTrueTransformation d)
+
+findPairs                     :: (Floating r, Ord r, AlwaysTrueWSPD d)
+                              => r -> SplitTree d p r a -> SplitTree d p r a
+                              -> [WSP d p r a]
+findPairs s l r
+  | areWellSeparated' s l r   = [(l,r)]
+  | maxWidth l <=  maxWidth r = concatMap (findPairs s l) $ children' r
+  | otherwise                 = concatMap (findPairs s r) $ children' l
+
+
+-- | Test if the two sets are well separated with param s
+areWellSeparated                     :: ( AlwaysTrueWSPD d, Fractional r, Ord r)
+                                     => r -- ^ separation factor
+                                     -> SplitTree d p r a
+                                     -> SplitTree d p r a -> Bool
+areWellSeparated _ (Leaf _) (Leaf _) = True
+areWellSeparated s l        r        = boxBox s (bbOf l)   (bbOf r)
+
+
+-- areWellSeparated s (Leaf p)      (Node _ nd _) = pointBox s (p^.core) (nd^.bBox)
+-- areWellSeparated s (Node _ nd _) (Leaf p)      = pointBox s (p^.core) (nd^.bBox)
+-- areWellSeparated s (Node _ ld _) (Node _ rd _) = boxBox   s (ld^.bBox) (rd^.bBox)
+
+
+-- -- | Test if the point and the box are far enough appart
+-- pointBox       :: (Fractional r, Ord r, AlwaysTruePFT d, AlwaysTrueTransformation d)
+--                => r -> Point d r -> Box d p r -> Bool
+-- pointBox s p b = not $ p `inBox` b'
+--   where
+--     v  = (centerPoint b)^.vector
+--     b' = translateBy v . scaleUniformlyBy s . translateBy ((-1) *^ v) $ b
+
+-- | Test if the two boxes are sufficiently far appart
+boxBox         :: (Fractional r, Ord r, AlwaysTruePFT d, AlwaysTrueTransformation d)
+               => r -> Box d p r -> Box d p r -> Bool
+boxBox s lb rb = boxBox' lb rb && boxBox' rb lb
+  where
+    boxBox' b' b = not $ b' `intersects` bOut
+      where
+        v    = (centerPoint b)^.vector
+        bOut = translateBy v . scaleUniformlyBy s . translateBy ((-1) *^ v) $ b
+
+--------------------------------------------------------------------------------
+-- * Alternative def if wellSeparated that uses fractional
+
+
+areWellSeparated'                     :: (Floating r, Ord r, Arity d)
+                                      => r
+                                      -> SplitTree d p r a
+                                      -> SplitTree d p r a
+                                      -> Bool
+areWellSeparated' _ (Leaf _) (Leaf _) = True
+areWellSeparated' s l        r        = boxBox1 s (bbOf l) (bbOf r)
+
+-- (Leaf p)      (Node _ nd _) = pointBox' s (p^.core) (nd^.bBox)
+-- areWellSeparated' s (Node _ nd _) (Leaf p)      = pointBox' s (p^.core) (nd^.bBox)
+-- areWellSeparated' s (Node _ ld _) (Node _ rd _) = boxBox'   s (ld^.bBox) (rd^.bBox)
+
+boxBox1         :: (Floating r, Ord r, Arity d) => r -> Box d p r -> Box d p r -> Bool
+boxBox1 s lb rb = euclideanDist (centerPoint lb) (centerPoint rb) >= (s+1)*d
+  where
+    diam b = euclideanDist (b^.minP.core.cwMin) (b^.maxP.core.cwMax)
+    d      = max (diam lb) (diam rb)
+
+
+
+
+--------------------------------------------------------------------------------
+-- * Helper stuff
+
+
+-- | Computes the maximum width of a splitTree
+maxWidth                             :: (Arity d, KnownNat d, Num r)
+                                     => SplitTree d p r a -> r
+maxWidth (Leaf _)                    = 0
+maxWidth (Node _ (NodeData i b _) _) = fromJust $ widthIn' i b
+
+-- | 'Computes' the bounding box of a split tree
+bbOf                             :: Ord r => SplitTree d p r a -> Box d () r
+bbOf (Leaf p)                    = boundingBox $ p^.core
+bbOf (Node _ (NodeData _ b _) _) = b
+
+
+children'              :: BinLeafTree v a -> [BinLeafTree v a]
+children' (Leaf _)     = []
+children' (Node l _ r) = [l,r]
+
+
+fromSeqUnsafe                         :: S.Seq a -> S2.ViewL1 a
+fromSeqUnsafe (S.viewl -> (l S.:< s)) = l S2.:< s
+fromSeqUnsafe _                       = error "fromSeqUnsafe: Empty seq"
+
+
+-- | Turn a traversal into lens
+ix'   :: (Arity d, KnownNat d) => Int -> Lens' (GV.Vector d a) a
+ix' i = singular (GV.element' i)
+
+
+dropIdx                 :: core :+ (t :+ extra) -> core :+ extra
+dropIdx (p :+ (_ :+ e)) = p :+ e
+
+--------------------------------------------------------------------------------
diff --git a/src/Algorithms/Graph/DFS.hs b/src/Algorithms/Graph/DFS.hs
--- a/src/Algorithms/Graph/DFS.hs
+++ b/src/Algorithms/Graph/DFS.hs
@@ -12,7 +12,7 @@
 
 -- | DFS on a planar graph.
 --
--- Running time: $O(n)$
+-- Running time: \(O(n)\)
 --
 -- Note that since our planar graphs are always connected there is no need need
 -- for dfs to take a list of start vertices.
@@ -30,7 +30,7 @@
 
 -- | DFS, from a given vertex, on a graph in AdjacencyLists representation.
 --
--- Running time: $O(n)$
+-- Running time: \(O(n)\)
 dfs'          :: forall s w. AdjacencyLists s w -> VertexId s w -> Tree (VertexId s w)
 dfs' g start = runST $ do
                  bv     <- UMV.replicate n False -- bit vector of marks
diff --git a/src/Algorithms/Graph/MST.hs b/src/Algorithms/Graph/MST.hs
--- a/src/Algorithms/Graph/MST.hs
+++ b/src/Algorithms/Graph/MST.hs
@@ -23,14 +23,14 @@
 --
 -- The algorithm used is Kruskal's.
 --
--- running time: $O(n \log n)$
+-- running time: \(O(n \log n)\)
 mst   :: Ord e => PlanarGraph s w v e f -> Tree (VertexId s w)
 mst g = makeTree g $ mstEdges g
   -- TODO: Add edges/darts to the output somehow.
 
 -- | Computes the set of edges in the Minimum spanning tree
 --
--- running time: $O(n \log n)$
+-- running time: \(O(n \log n)\)
 mstEdges   :: Ord e => PlanarGraph s w v e f -> [Dart s]
 mstEdges g = runST $ do
           uf <- new (numVertices g)
@@ -67,7 +67,7 @@
 -- | Union find DS
 newtype UF s a = UF { _unUF :: UMV.MVector s (Int,Int) }
 
-new   :: Enum a => Int -> ST s (UF s a)
+new   :: Int -> ST s (UF s a)
 new n = do
           v <- UMV.new n
           forM_ [0..n-1] $ \i ->
diff --git a/src/Control/Monad/State/Persistent.hs b/src/Control/Monad/State/Persistent.hs
--- a/src/Control/Monad/State/Persistent.hs
+++ b/src/Control/Monad/State/Persistent.hs
@@ -5,13 +5,9 @@
                                      , runPersistentState
                                      ) where
 
-
-import Control.Applicative
 import Control.Monad.State
 import Control.Monad.Identity(Identity(..))
-
 import Data.List.NonEmpty(NonEmpty(..),(<|),toList)
-
 
 --------------------------------------------------------------------------------
 
diff --git a/src/Data/BalBST.hs b/src/Data/BalBST.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/BalBST.hs
@@ -0,0 +1,355 @@
+{-# LANGUAGE RecordWildCards #-}
+module Data.BalBST where
+
+import           Data.Bifunctor
+import           Data.Function (on)
+import           Data.Functor.Contravariant
+import qualified Data.List as L
+import           Data.Maybe
+import qualified Data.Tree as T
+import           Prelude hiding (lookup,null)
+
+--------------------------------------------------------------------------------
+
+-- | Describes how to search in a tree
+data TreeNavigator k a = Nav { goLeft     :: a -> k -> Bool
+                             , extractKey :: a -> a -> k
+                             }
+
+instance Contravariant (TreeNavigator k) where
+  contramap f (Nav gL eK) = Nav (\a k -> gL (f a) k) (\x y -> eK (f x) (f y))
+
+
+ordNav :: Ord a => TreeNavigator a a
+ordNav = Nav (<=) min
+
+
+ordNavBy   :: Ord b => (a -> b) ->  TreeNavigator b a
+ordNavBy f = Nav (\x k -> f x <= k) (min `on` f)
+
+
+-- instance Functor (TreeNavigator k) where
+--   fmap f Nav{..} = Nav (\b k -> )
+
+
+
+-- | A balanced binary search tree
+data BalBST k a = BalBST { nav    :: !(TreeNavigator k a)
+                         , toTree :: !(Tree k a)
+                         }
+
+instance (Show k, Show a) => Show (BalBST k a) where
+  show (BalBST _ t) = "BalBST (" ++ show t ++ ")"
+
+
+data Color = Red | Black deriving (Show,Read,Eq,Ord)
+
+type Height = Int
+
+-- Red-Black tree with values in the leaves
+data Tree k a = Empty
+              | Leaf !a
+              | Node !Color !Height (Tree k a) !k (Tree k a) deriving (Show,Eq)
+
+--------------------------------------------------------------------------------
+
+-- | Creates an empty BST
+empty   :: TreeNavigator k a -> BalBST k a
+empty n = BalBST n Empty
+
+
+-- | \(O(n\log n)\)
+fromList :: TreeNavigator k a -> [a] -> BalBST k a
+fromList n = foldr insert (empty n)
+
+fromList' :: Ord a => [a] -> BalBST a a
+fromList' = fromList ordNav
+
+
+-- -- | \(O(n)\)
+-- fromAscList :: TreeNavigator k a -> [a] -> BalBST k a
+-- fromAscList = undefined
+
+
+--------------------------------------------------------------------------------
+
+-- | Check if the tree is empty
+null                  :: BalBST k a -> Bool
+null (BalBST _ Empty) = True
+null _                = False
+
+-- | Test if an element occurs in the BST.
+-- \(O(\log n)\)
+lookup :: Eq a => a -> BalBST k a -> Maybe a
+lookup x (BalBST Nav{..} t) = lookup' t
+  where
+    lookup' Empty            = Nothing
+    lookup' (Leaf y)         = if x == y then Just y else Nothing
+    lookup' (Node _ _ l k r)
+      | goLeft x k           = lookup' l
+      | otherwise            = lookup' r
+
+-- | \(O(\log n)\)
+member   :: Eq a => a -> BalBST k a -> Bool
+member x = isJust . lookup x
+
+
+
+
+
+-- | Insert an element in the BST.
+--
+-- \(O(\log n)\)
+insert :: a -> BalBST k a -> BalBST k a
+insert x (BalBST n@Nav{..} t) = BalBST n (blacken $ insert' t)
+  where
+    insert' Empty    = Leaf x
+    insert' (Leaf y) = let k     = extractKey x y
+                           (l,r) = if goLeft x k then (x,y) else (y,x)
+                       in red 2 (Leaf l) k (Leaf r)
+    insert' (Node c h l k r)
+      | goLeft  x k  = balance c h (insert' l) k r
+      | otherwise    = balance c h l           k (insert' r)
+
+
+
+-- delete = undefined
+
+-- delete                        :: Eq a => a -> BalBST k a -> BalBST k a
+-- delete x (BalBST n@Nav{..} t) = delete' t
+--   where
+--     delete' Empty      = Empty
+--     delete' l@(Leaf y) = if x == y then Empty else l
+--     delete' (Node c h l k r)
+--       | goLeft x k     =
+
+
+--------------------------------------------------------------------------------
+
+
+-- | Extract the minimum from the tree
+-- \(O(\log n)\)
+minView              :: BalBST k a -> Maybe (a, Tree k a)
+minView (BalBST n t) = minView' t
+  where
+    minView' Empty            = Nothing
+    minView' (Leaf x)         = Just (x,Empty)
+    minView' (Node _ _ l _ r) = fmap (flip (joinWith n) r) <$> minView' l
+
+-- | Extract the maximum from the tree
+-- \(O(\log n)\)
+maxView              :: BalBST k a -> Maybe (a, Tree k a)
+maxView (BalBST n t) = maxView' t
+  where
+    maxView' Empty            = Nothing
+    maxView' (Leaf x)         = Just (x,Empty)
+    maxView' (Node _ _ l _ r) = fmap (joinWith n l) <$> maxView' r
+
+-- | Joins two BSTs. Assumes that the ranges are disjoint. It takes the left Tree nav
+--
+-- \(O(\log n)\)
+join                           :: BalBST k a -> BalBST k a -> BalBST k a
+join (BalBST n l) (BalBST _ r) = BalBST n $ joinWith n l r
+
+-- | Joins two BSTs' with a specific Tree Navigator
+--
+-- \(O(\log n)\)
+joinWith               :: TreeNavigator k a -> Tree k a -> Tree k a -> Tree k a
+joinWith Nav{..} tl tr
+    | lh >= rh         = blacken $ joinL tl tr
+    | otherwise        = blacken $ joinR tl tr
+  where
+    rh = height tr
+    lh = height tl
+
+    joinL Empty      _           = Empty
+    joinL l          Empty       = l
+    joinL l@(Leaf x) r@(Leaf y)  = red 2 l (extractKey x y) r
+    joinL l@(Node c h ll k lr) r
+      | h == rh                  = let lm = unsafeMax lr
+                                       rm = unsafeMin r
+                                   in balance Red (h+1) l (extractKey lm rm) r
+      | otherwise                = balance c h ll k (joinL lr r)
+        -- lh >= rh
+    joinL _ _ = error "joinL. absurd"
+
+
+    joinR _          Empty       = Empty
+    joinR Empty      r           = r
+
+    joinR l@(Leaf x) r@(Leaf y)  = red 2 l (extractKey x y) r
+    joinR l r@(Node c h rl k rr)
+      | h == lh                  = let lm = unsafeMax l
+                                       rm = unsafeMin rl
+                                   in balance Red (h+1) l (extractKey lm rm) r
+      | otherwise                = balance c h (joinR l rl) k rr
+        -- lh >= rh
+    joinR _ _ = error "joinR absurd"
+
+
+--------------------------------------------------------------------------------
+-- | Splitting and extracting
+
+-- | A pair that is strict in its first argument and lazy in the second.
+data Pair a b = Pair { fst' :: !a
+                     , snd' :: b
+                     } deriving (Show,Eq,Functor,Foldable,Traversable)
+
+
+collect        :: b -> [Pair a b] -> Pair [a] b
+collect def [] = Pair [] def
+collect _   xs = Pair (map fst' xs) (snd' $ last xs)
+
+
+-- | Extract a prefix from the tree, i.e. a repeated 'minView'
+--
+-- \(O(\log n +k)\), where \(k\) is the size of the extracted part
+extractPrefix                      :: BalBST k a -> [Pair a (Tree k a)]
+extractPrefix (BalBST n@Nav{..} t) = extractPrefix' t
+  where
+    extractPrefix' Empty            = []
+    extractPrefix' (Leaf x)         = [Pair x Empty]
+    extractPrefix' (Node _ _ l _ r) = ls ++ extractPrefix' r
+      where
+        ls = map (fmap $ flip (joinWith n) r) $ extractPrefix' l
+
+-- | Extract a suffix from the tree, i.e. a repeated 'minView'
+--
+-- \(O(\log n +k)\), where \(k\) is the size of the extracted part
+extractSuffix                      :: BalBST k a -> [Pair a (Tree k a)]
+extractSuffix (BalBST n@Nav{..} t) = extract t
+  where
+    extract Empty            = []
+    extract (Leaf x)         = [Pair x Empty]
+    extract (Node _ _ l _ r) = rs ++ extract l
+      where
+        rs = map (fmap $ joinWith n l) $ extract r
+
+-- | Result of splititng a tree
+data Split a b = Split a !b a deriving (Show,Eq)
+
+-- | Splits the tree at x. Note that if x occurs more often, no guarantees are
+-- given which one is found.
+--
+-- \(O(\log n)\)
+split                        :: Eq a => a -> BalBST k a -> Split (Tree k a) (Maybe a)
+split x (BalBST n@Nav{..} t) = split' t
+  where
+    split' Empty                  = Split Empty Nothing Empty
+    split' l@(Leaf y)
+      | x == y                    = Split Empty (Just y) Empty
+      | goLeft x (extractKey x y) = Split l     Nothing  Empty
+      | otherwise                 = Split Empty Nothing  l
+    split' (Node _ _ l k r)
+      | goLeft x k                = let Split l' mx r' = split' l
+                                    in Split l' mx (joinWith n r' r)
+      | otherwise                 = let Split l' mx r' = split' r
+                                    in Split (joinWith n l l') mx r'
+
+-- | split based on a monotonic predicate
+--
+-- \(O(\log n)\)
+splitMonotone                        :: (a -> Bool) -> BalBST k a
+                                     -> (BalBST k a, BalBST k a)
+splitMonotone p (BalBST n@Nav{..} t) = bimap (BalBST n) (BalBST n) $ split' t
+  where
+    split' Empty        = (Empty,Empty)
+    split' l@(Leaf y)
+      | p y             = (Empty,l)
+      | otherwise       = (l,Empty)
+    split' (Node _ _ l _ r)
+      | p (unsafeMin r) = let (l',m) = split' l in (l',joinWith n m r)
+      | otherwise       = let (m,r') = split' r in (joinWith n l m, r')
+
+
+-- | Splits at a given monotone predicate p, and then selects everything that
+-- satisfies the predicate sel.
+splitExtract           :: (a -> Bool) -> (a -> Bool) -> BalBST k a
+                       -> Split (BalBST k a) ([a],[a])
+splitExtract p sel bst = Split (BalBST n before) (reverse mid1,mid2) (BalBST n after)
+  where
+    n                = nav bst
+    (before',after') = splitMonotone p bst
+
+    extract def = collect def . L.takeWhile (sel . fst')
+
+    Pair mid1 before = extract (toTree before') $ extractSuffix before'
+    Pair mid2 after  = extract (toTree after')  $ extractPrefix after'
+
+
+--------------------------------------------------------------------------------
+
+
+data T k a = Internal !Color !Height !k | Val !a deriving (Show,Eq,Ord)
+
+toRoseTree :: Tree k a -> Maybe (T.Tree (T k a))
+toRoseTree Empty            = Nothing
+toRoseTree (Leaf x)         = Just $ T.Node (Val x) []
+toRoseTree (Node c h l k r) = Just $ T.Node (Internal c h k) (mapMaybe toRoseTree [l,r])
+
+
+showTree :: (Show k, Show a) => BalBST k a -> String
+showTree = maybe "Empty" T.drawTree . fmap (fmap show) . toRoseTree . toTree
+
+-- | Get the minimum in the tree. Errors when the tree is empty
+--
+-- \(O(\log n)\)
+unsafeMin                  :: Tree k a -> a
+unsafeMin (Leaf x)         = x
+unsafeMin (Node _ _ l _ _) = unsafeMin l
+unsafeMin _                = error "unsafeMin: Empty"
+
+-- | Get the maximum in the tree. Errors when the tree is empty
+--
+-- \(O(\log n)\)
+unsafeMax                  :: Tree k a -> a
+unsafeMax (Leaf x)         = x
+unsafeMax (Node _ _ _ _ r) = unsafeMax r
+unsafeMax _                = error "unsafeMax: Empty"
+
+-- | Extract all elements in the tree
+--
+-- \(O(n)\)
+toList :: BalBST k a -> [a]
+toList = toList' . toTree
+
+-- | Extract all elements in the tree
+--
+-- \(O(n)\)
+toList'                  :: Tree k a -> [a]
+toList' Empty            = []
+toList' (Leaf x)         = [x]
+toList' (Node _ _ l _ r) = toList' l ++ toList' r
+
+
+--------------------------------------------------------------------------------
+-- * Helper stuff
+
+black :: Height -> Tree k a -> k -> Tree k a -> Tree k a
+black = Node Black
+
+red :: Height -> Tree k a -> k -> Tree k a -> Tree k a
+red = Node Red
+
+
+blacken                    :: Tree k a -> Tree k a
+blacken (Node Red h l k r) = Node Black h l k r
+blacken t                  = t
+
+-- | rebalance the tree
+balance  :: Color -> Height -> Tree k a -> k -> Tree k a -> Tree k a
+balance Black h (Node Red _ (Node Red _ a x b) y c) z d = mkNode h a x b y c z d
+balance Black h (Node Red _ a x (Node Red _ b y c)) z d = mkNode h a x b y c z d
+balance Black h a x (Node Red _ (Node Red _ b y c) z d) = mkNode h a x b y c z d
+balance Black h a x (Node Red _ b y (Node Red _ c z d)) = mkNode h a x b y c z d
+balance co h a x b                                      = Node co h a x b
+
+mkNode                 :: Height
+                       -> Tree k a -> k -> Tree k a -> k -> Tree k a  -> k -> Tree k a
+                       -> Tree k a
+mkNode h a x b y c z d = red h (black h a x b) y (black h c z d)
+
+height                  :: Tree k a -> Height
+height Empty            = 0
+height (Leaf _)         = 1
+height (Node _ h _ _ _) = h
diff --git a/src/Data/BinaryTree.hs b/src/Data/BinaryTree.hs
--- a/src/Data/BinaryTree.hs
+++ b/src/Data/BinaryTree.hs
@@ -1,19 +1,25 @@
-{-# Language DeriveFunctor #-}
+{-# Language DeriveFunctor#-}
 {-# Language FunctionalDependencies #-}
 module Data.BinaryTree where
 
-import Control.Applicative
-import Data.Foldable
-import Data.List.NonEmpty(NonEmpty)
-import Data.Semigroup
-import Data.Traversable
-import Data.Semigroup.Foldable
+import           Control.DeepSeq
+import           Data.List.NonEmpty (NonEmpty(..),(<|))
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Maybe (mapMaybe)
+import           Data.Semigroup
+import           Data.Semigroup.Foldable
+import qualified Data.Tree as Tree
+import qualified Data.Vector as V
+import           GHC.Generics (Generic)
 
-data BinLeafTree v a = Leaf a
-                     | Node (BinLeafTree v a) v (BinLeafTree v a)
-                     deriving (Show,Read,Eq,Ord,Functor)
+--------------------------------------------------------------------------------
 
+data BinLeafTree v a = Leaf !a
+                     | Node (BinLeafTree v a) !v (BinLeafTree v a)
+                     deriving (Show,Read,Eq,Ord,Functor,Generic)
 
+instance (NFData v, NFData a) => NFData (BinLeafTree v a)
+
 class Semigroup v => Measured v a | a -> v where
   measure :: a -> v
 
@@ -40,24 +46,143 @@
 instance Measured v a => Semigroup (BinLeafTree v a) where
   l <> r = node l r
 
+-- | Create a balanced tree, i.e. a tree of height \(O(\log n)\) with the
+-- elements in the leaves.
+--
+-- \(O(n)\) time.
+asBalancedBinLeafTree :: NonEmpty a -> BinLeafTree Size (Elem a)
+asBalancedBinLeafTree = repeatedly merge . fmap (Leaf . Elem)
+  where
+    repeatedly _ (t :| []) = t
+    repeatedly f ts        = repeatedly f $ f ts
 
-asBalancedBinLeafTree    :: NonEmpty a -> BinLeafTree Size (Elem a)
-asBalancedBinLeafTree ys = asBLT (length ys') ys'
+    merge ts@(_ :| [])  = ts
+    merge (l :| r : []) = node l r :| []
+    merge (l :| r : ts) = node l r <| (merge $ NonEmpty.fromList ts)
+-- -- the implementation below produces slightly less high trees, but runs in
+-- -- \(O(n \log n)\) time, as on every level it traverses the list passed down.
+-- asBalancedBinLeafTree ys = asBLT (length ys') ys' where ys' = toList ys
+
+--     asBLT _ [x] = Leaf (Elem x)
+--     asBLT n xs  = let h       = n `div` 2
+--                       (ls,rs) = splitAt h xs
+--                   in node (asBLT h ls) (asBLT (n-h) rs)
+
+-- | Given a function to combine internal nodes into b's and leafs into b's,
+-- traverse the tree bottom up, and combine everything into one b.
+foldUp                  :: (b -> v -> b -> b) -> (a -> b) -> BinLeafTree v a -> b
+foldUp _ g (Leaf x)     = g x
+foldUp f g (Node l x r) = f (foldUp f g l) x (foldUp f g r)
+
+
+-- | Traverses the tree bottom up, recomputing the assocated values.
+foldUpData     :: (w -> v -> w -> w) -> (a -> w) -> BinLeafTree v a -> BinLeafTree w a
+foldUpData f g = foldUp f' Leaf
   where
-    ys' = toList ys
+    f' l v r = Node l (f (access' l) v (access' r)) r
 
-    asBLT _ [x] = Leaf (Elem x)
-    asBLT n xs  = let h       = n `div` 2
-                      (ls,rs) = splitAt h xs
-                  in node (asBLT h ls) (asBLT (n-h) rs)
+    access' (Leaf x)     = g x
+    access' (Node _ v _) = v
 
-newtype Size = Size Int deriving (Show,Read,Eq,Num,Integral,Enum,Real,Ord)
+-- | Takes two trees, that have the same structure, and uses the provided
+-- functions to "zip" them together
+zipExactWith                                  :: (u -> v -> w)
+                                              -> (a -> b -> c)
+                                              -> BinLeafTree u a
+                                              -> BinLeafTree v b
+                                              -> BinLeafTree w c
+zipExactWith _ g (Leaf x)     (Leaf y)        = Leaf (x `g` y)
+zipExactWith f g (Node l m r) (Node l' m' r') = Node (zipExactWith f g l l')
+                                                     (m `f` m')
+                                                     (zipExactWith f g r r')
+zipExactWith _ _ _            _               =
+    error "zipExactWith: tree structures not the same "
 
+newtype Size = Size Int deriving (Show,Read,Eq,Num,Integral,Enum,Real,Ord,Generic,NFData)
+
 instance Semigroup Size where
   x <> y = x + y
 
+instance Monoid Size where
+  mempty = Size 0
+  mappend = (<>)
+
 newtype Elem a = Elem { _unElem :: a }
                deriving (Show,Read,Eq,Ord,Functor,Foldable,Traversable)
 
 instance Measured Size (Elem a) where
   measure _ = 1
+
+
+data Sized a = Sized !Size a
+             deriving (Show,Eq,Ord,Functor,Foldable,Traversable,Generic)
+instance NFData a => NFData (Sized a)
+
+instance Semigroup a => Semigroup (Sized a) where
+  (Sized i a) <> (Sized j b) = Sized (i <> j) (a <> b)
+
+instance Monoid a => Monoid (Sized a) where
+  mempty = Sized mempty mempty
+  (Sized i a) `mappend` (Sized j b) = Sized (i <> j) (a `mappend` b)
+
+-- instance Semigroup a => Measured Size (Sized a) where
+--   measure (Sized i _) = i
+
+
+--------------------------------------------------------------------------------
+-- * Converting into a Data.Tree
+
+data RoseElem v a = InternalNode v | LeafNode a deriving (Show,Eq,Functor)
+
+toRoseTree              :: BinLeafTree v a -> Tree.Tree (RoseElem v a)
+toRoseTree (Leaf x)     = Tree.Node (LeafNode x) []
+toRoseTree (Node l v r) = Tree.Node (InternalNode v) (map toRoseTree [l,r])
+
+
+drawTree :: (Show v, Show a) => BinLeafTree v a -> String
+drawTree = Tree.drawTree . fmap show . toRoseTree
+
+
+--------------------------------------------------------------------------------
+-- * Internal Node Tree
+
+
+data BinaryTree a = Nil
+                  | Internal (BinaryTree a) !a (BinaryTree a)
+                  deriving (Show,Read,Eq,Ord,Functor,Foldable,Traversable,Generic)
+instance NFData a => NFData (BinaryTree a)
+
+-- | Get the element stored at the root, if it exists
+access                  :: BinaryTree a -> Maybe a
+access Nil              = Nothing
+access (Internal _ x _) = Just x
+
+-- | Create a balanced binary tree
+--
+-- \(O(n)\)
+asBalancedBinTree :: [a] -> BinaryTree a
+asBalancedBinTree = mkTree . V.fromList
+  where
+    mkTree v = let n = V.length v
+                   h = n `div` 2
+                   x = v V.! h
+               in if n == 0 then Nil
+                            else Internal (mkTree $ V.slice 0 h v) x
+                                          (mkTree $ V.slice (h+1) (n - h -1) v)
+
+
+foldBinaryUp                      :: b -> (a -> b -> b -> b)
+                                  -> BinaryTree a -> BinaryTree (a,b)
+foldBinaryUp _ _ Nil              = Nil
+foldBinaryUp e f (Internal l x r) = let l' = foldBinaryUp e f l
+                                        r' = foldBinaryUp e f r
+                                        g  = maybe e snd . access
+                                        b  = f x (g l') (g r')
+                                    in Internal l' (x,b) r'
+
+toRoseTree'                  :: BinaryTree a -> Maybe (Tree.Tree a)
+toRoseTree' Nil              = Nothing
+toRoseTree' (Internal l v r) = Just $ Tree.Node v $ mapMaybe toRoseTree' [l,r]
+
+drawTree' :: Show a => BinaryTree a -> String
+drawTree' = maybe "Nil" (Tree.drawTree . fmap show) . toRoseTree'
diff --git a/src/Data/BinaryTree/Zipper.hs b/src/Data/BinaryTree/Zipper.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/BinaryTree/Zipper.hs
@@ -0,0 +1,65 @@
+module Data.BinaryTree.Zipper where
+
+import Data.BinaryTree
+import Data.Semigroup
+
+--------------------------------------------------------------------------------
+
+data Ctx a = Top | L (Ctx a) a (BinaryTree a) | R (BinaryTree a) a (Ctx a)
+           deriving (Show,Read,Eq,Ord,Functor,Foldable,Traversable)
+
+data BinaryTreeZipper a = Loc (BinaryTree a) (Ctx a)
+           deriving (Show,Read,Eq,Ord,Functor,Foldable,Traversable)
+
+-- | Focus on the root
+top   :: BinaryTree a -> BinaryTreeZipper a
+top t = Loc t Top
+
+-- | Go to the left child
+left                            :: BinaryTreeZipper a -> Maybe (BinaryTreeZipper a)
+left (Loc (Internal l x r) ctx) = Just $ Loc l (L ctx x r)
+left (Loc Nil _)                = Nothing
+
+-- | Go to the right child
+right                            :: BinaryTreeZipper a -> Maybe (BinaryTreeZipper a)
+right (Loc (Internal l x r) ctx) = Just $ Loc r (R l x ctx)
+right (Loc Nil _)                = Nothing
+
+-- | Move to the parent
+up                     :: BinaryTreeZipper a -> Maybe (BinaryTreeZipper a)
+up (Loc _ Top)         = Nothing
+up (Loc l (L ctx x r)) = Just $ Loc (Internal l x r) ctx
+up (Loc r (R l x ctx)) = Just $ Loc (Internal l x r) ctx
+
+-- | Navigate to the root
+toRoot   :: BinaryTreeZipper a -> BinaryTreeZipper a
+toRoot z = toRoot' z (Just z)
+  where
+    toRoot' z' Nothing   = z'
+    toRoot' _  (Just z') = toRoot' z' (up z')
+
+
+-- | Returns a list of zippers; one focussed on each node in the tree
+visitAll   :: BinaryTree a -> [BinaryTreeZipper a]
+visitAll t = visitAll' (top t)
+  where
+    f           = maybe [] visitAll'
+    visitAll' z = z : f (left z) <> f (right z)
+
+-- | Get the value stored at the current node
+accessZ           :: BinaryTreeZipper a -> Maybe a
+accessZ (Loc t _) = access t
+
+
+-- | Returns all subtrees; i.e. every node with all its decendents
+subTrees :: BinaryTree a -> [BinaryTree a]
+subTrees t = Nil : subTrees' t
+  where
+    subTrees' Nil                 = []
+    subTrees' tt@(Internal l _ r) = tt : subTrees' l <> subTrees' r
+
+
+-- | Splits the tree here, returns a pair (innerTree,outerTree)
+splitTree             :: BinaryTreeZipper a -> (BinaryTree a, BinaryTree a)
+splitTree (Loc t ctx) = let (Loc r _) = toRoot $ Loc Nil ctx
+                        in (t, r)
diff --git a/src/Data/CircularList/Util.hs b/src/Data/CircularList/Util.hs
--- a/src/Data/CircularList/Util.hs
+++ b/src/Data/CircularList/Util.hs
@@ -1,14 +1,13 @@
-{-# LANGUAGE CPP #-}
 module Data.CircularList.Util where
 
 import           Control.Lens
 import           Data.Tuple
 import qualified Data.CircularList as C
 import qualified Data.List as L
-import qualified Data.Traversable as T
 
-import Debug.Trace
 
+--------------------------------------------------------------------------------
+
 -- $setup
 -- >>> let ordList = C.fromList [5,6,10,20,30,1,2,3]
 
@@ -66,17 +65,3 @@
 xs `isShiftOf` ys = let rest = tail . C.leftElements
                     in maybe False (\xs' -> rest xs' == rest ys) $
                          C.focus ys >>= flip C.rotateTo xs
-
-
--- minimumBy     :: (a -> a -> Ordering) -> C.CList a -> a
--- minimumBy cmp = L.minimumBy cmp . C.rightElements
-
-#if !MIN_VERSION_data_clist(0,1,0)
-
-instance Foldable C.CList where
-  foldMap = T.foldMapDefault
-
-instance T.Traversable C.CList where
-  traverse f = fmap C.fromList . T.traverse f . C.rightElements
-
-#endif
diff --git a/src/Data/CircularSeq.hs b/src/Data/CircularSeq.hs
--- a/src/Data/CircularSeq.hs
+++ b/src/Data/CircularSeq.hs
@@ -4,6 +4,8 @@
                        , fromNonEmpty
                        , fromList
 
+                       , toNonEmpty
+
                        , focus
                        , index, adjust
                        , item
@@ -21,11 +23,18 @@
 
                        , findRotateTo
                        , rotateTo
+
+                       , zipLWith, zipL
+                       , zip3LWith
+
+
+                       , insertOrd, insertOrdBy
+                       , isShiftOf
                        ) where
 
-import           Control.Applicative
-import           Control.Lens(lens, Lens')
+import           Control.Lens (lens, Lens', bimap)
 import qualified Data.Foldable as F
+import qualified Data.List as L
 import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Maybe (listToMaybe)
 import           Data.Semigroup
@@ -37,11 +46,17 @@
 
 --------------------------------------------------------------------------------
 
+-- $setup
+-- >>> let ordList = fromList [5,6,10,20,30,1,2,3]
+
+
 -- | Nonempty circular sequence
 data CSeq a = CSeq !(Seq a) !a !(Seq a)
-            deriving (Eq)
                      -- we keep the seq balanced, i.e. size left >= size right
 
+instance Eq a => Eq (CSeq a) where
+  a == b = asSeq a == asSeq b
+
 instance Show a => Show (CSeq a) where
   showsPrec d s = showParen (d > app_prec) $
                     showString (("CSeq " <>) . show . F.toList . rightElements $ s)
@@ -79,7 +94,7 @@
 
 -- | Access the i^th item  (w.r.t the focus) in the CSeq (indices modulo n).
 --
--- running time: $O(\log (i \mod n))$
+-- running time: \(O(\log (i \mod n))\)
 --
 -- >>> index (fromList [0..5]) 1
 -- 1
@@ -104,7 +119,7 @@
 
 -- | Adjusts the i^th element w.r.t the focus in the CSeq
 --
--- running time: $O(\log (i \mod n))$
+-- running time: \(O(\log (i \mod n))\)
 --
 -- >>> adjust (const 1000) 2 (fromList [0..5])
 -- CSeq [0,1,1000,3,4,5]
@@ -201,11 +216,14 @@
 fromList (x:xs) = withFocus x $ S.fromList xs
 fromList []     = error "fromList: Empty list"
 
+toNonEmpty :: CSeq a -> NonEmpty.NonEmpty a
+toNonEmpty = NonEmpty.fromList . F.toList
+
 -- | Rotates i elements to the right.
 --
 -- pre: 0 <= i < n
 --
--- running time: $O(\log i)$ amortized
+-- running time: \(O(\log i)\) amortized
 --
 -- >>> rotateNR 0 $ fromList [1..5]
 -- CSeq [1,2,3,4,5]
@@ -223,7 +241,7 @@
 --
 -- pre: 0 <= i < n
 --
--- running time: $O(\log i)$ amoritzed
+-- running time: \(O(\log i)\) amoritzed
 --
 -- >>> rotateNL 0 $ fromList [1..5]
 -- CSeq [1,2,3,4,5]
@@ -244,7 +262,7 @@
 
 -- | Reversres the direction of the CSeq
 --
--- running time: $O(n)$
+-- running time: \(O(n)\)
 --
 -- >>> reverseDirection $ fromList [1..5]
 -- CSeq [1,5,4,3,2]
@@ -279,3 +297,76 @@
 
 allRotations'   :: CSeq a -> [CSeq a]
 allRotations' s = take (length s) . iterate rotateR $ s
+
+-- | "Left zip": zip the two CLists, pairing up every element in the *left*
+-- list with its corresponding element in the right list. If there are more
+-- items in the right clist they are discarded.
+zipLWith         :: (a -> b -> c) -> CSeq a -> CSeq b -> CSeq c
+zipLWith f as bs = fromList $ zipWith f (F.toList as) (F.toList bs)
+
+-- | see 'zipLWith
+zipL :: CSeq a -> CSeq b -> CSeq (a, b)
+zipL = zipLWith (,)
+
+
+-- | same as zipLWith but with three items
+zip3LWith            :: (a -> b -> c -> d) -> CSeq a -> CSeq b -> CSeq c -> CSeq d
+zip3LWith f as bs cs = fromList $ zipWith3 f (F.toList as) (F.toList bs) (F.toList cs)
+
+
+
+
+-- | Given a circular seq, whose elements are in increasing order, insert the
+-- new element into the Circular seq in its sorted order.
+--
+-- >>> insertOrd 1 $ fromList [2]
+-- CSeq [2,1]
+-- >>> insertOrd 2 $ fromList [1,3]
+-- CSeq [1,2,3]
+-- >>> insertOrd 31 ordList
+-- CSeq [5,6,10,20,30,31,1,2,3]
+-- >>> insertOrd 1 ordList
+-- CSeq [5,6,10,20,30,1,1,2,3]
+-- >>> insertOrd 4 ordList
+-- CSeq [5,6,10,20,30,1,2,3,4]
+-- >>> insertOrd 11 ordList
+-- CSeq [5,6,10,11,20,30,1,2,3]
+--
+-- running time: \(O(n)\)
+insertOrd :: Ord a => a -> CSeq a -> CSeq a
+insertOrd = insertOrdBy compare
+
+-- | Insert an element into an increasingly ordered circular list, with
+-- specified compare operator.
+--
+-- running time: \(O(n)\)
+insertOrdBy       :: (a -> a -> Ordering) -> a -> CSeq a -> CSeq a
+insertOrdBy cmp x = fromList . insertOrdBy' cmp x . F.toList . rightElements
+
+-- | List version of insertOrdBy; i.e. the list contains the elements in
+-- cirulcar order. Again produces a list that has the items in circular order.
+insertOrdBy'         :: (a -> a -> Ordering) -> a -> [a] -> [a]
+insertOrdBy' cmp x xs = case (rest, x `cmp` head rest) of
+    ([],  _)   -> L.insertBy cmp x pref
+    (z:zs, GT) -> (z : L.insertBy cmp x zs) ++ pref
+    (_:_,  EQ) -> (x : xs) -- == x : rest ++ pref
+    (_:_,  LT) -> rest ++ L.insertBy cmp x pref
+  where
+    -- split the list at its maximum.
+    (pref,rest) = splitIncr cmp xs
+
+-- given a list of elements that is supposedly a a cyclic-shift of a list of
+-- increasing items, find the splitting point. I.e. returns a pair of lists
+-- (ys,zs) such that xs = zs ++ ys, and ys ++ zs is (supposedly) in sorted
+-- order.
+splitIncr              :: (a -> a -> Ordering) -> [a] -> ([a],[a])
+splitIncr _   []       = ([],[])
+splitIncr cmp xs@(x:_) = swap . bimap (map snd) (map snd)
+                      . L.break (\(a,b) -> (a `cmp` b) == GT) $ zip (x:xs) xs
+
+-- | Test if the circular list is a cyclic shift of the second list.
+-- Running time: O(n), where n is the size of the smallest list
+isShiftOf         :: Eq a => CSeq a -> CSeq a -> Bool
+xs `isShiftOf` ys = let rest = tail . F.toList . leftElements
+                    in maybe False (\xs' -> rest xs' == rest ys) $
+                       rotateTo (focus ys) xs
diff --git a/src/Data/Ext.hs b/src/Data/Ext.hs
--- a/src/Data/Ext.hs
+++ b/src/Data/Ext.hs
@@ -1,19 +1,21 @@
+{-# LANGUAGE DeriveAnyClass  #-}
 module Data.Ext where
 
-import Control.Applicative
 import Control.Lens
-import Data.Semigroup
 import Data.Biapplicative
-import Data.Bifunctor.Apply
 import Data.Bifoldable
+import Data.Bifunctor.Apply
 import Data.Bitraversable
+import Data.Functor.Apply (liftF2)
+import Data.Semigroup
 import Data.Semigroup.Bifoldable
 import Data.Semigroup.Bitraversable
-import Data.Functor.Apply(liftF2)
+import GHC.Generics (Generic)
+import Control.DeepSeq
 
 --------------------------------------------------------------------------------
 
-data core :+ extra = core :+ extra deriving (Show,Read,Eq,Ord,Bounded)
+data core :+ extra = core :+ extra deriving (Show,Read,Eq,Ord,Bounded,Generic,NFData)
 infixr 1 :+
 
 
diff --git a/src/Data/Geometry.hs b/src/Data/Geometry.hs
--- a/src/Data/Geometry.hs
+++ b/src/Data/Geometry.hs
@@ -25,5 +25,5 @@
 import Data.Geometry.Properties
 import Data.Geometry.Transformation
 import Data.Geometry.Vector
-import Linear.Affine hiding (Point, origin)
+import Linear.Affine hiding (Point, Vector, origin)
 import Linear.Vector
diff --git a/src/Data/Geometry/Ball.hs b/src/Data/Geometry/Ball.hs
--- a/src/Data/Geometry/Ball.hs
+++ b/src/Data/Geometry/Ball.hs
@@ -2,6 +2,7 @@
 {-# LANGUAGE UndecidableInstances #-}
 module Data.Geometry.Ball where
 
+import           Control.DeepSeq
 import           Control.Lens
 import           Data.Bifunctor
 import           Data.Ext
@@ -16,22 +17,28 @@
 import qualified Data.Traversable as T
 import           Data.Vinyl
 import           Frames.CoRec
-import           GHC.TypeLits
-import           Linear.Affine(qdA, (.-.), (.+^))
-import           Linear.Vector((^/),(*^),(^+^))
+import           GHC.Generics (Generic)
+import           Linear.Matrix
+import           Linear.V3 (V3(..))
 
 --------------------------------------------------------------------------------
 -- * A d-dimensional ball
 
 -- | A d-dimensional ball.
-data Ball d p r = Ball { _center        :: Point d r :+ p
-                       , _squaredRadius :: r
-                       }
+data Ball d p r = Ball { _center        :: !(Point d r :+ p)
+                       , _squaredRadius :: !r
+                       } deriving Generic
 makeLenses ''Ball
 
-deriving instance (Show r, Show p, Arity d) => Show (Ball d p r)
-deriving instance (Eq r, Eq p, Arity d)     => Eq (Ball d p r)
+-- | A lens to get/set the radius of a Ball
+radius :: Floating r => Lens' (Ball d p r) r
+radius = lens (sqrt . _squaredRadius) (\(Ball c _) r -> Ball c (r^2))
 
+
+deriving instance (Show r, Show p, Arity d)     => Show (Ball d p r)
+-- deriving instance (NFData p, NFData r, Arity d) => NFData (Ball d p r)
+deriving instance (Eq r, Eq p, Arity d)         => Eq (Ball d p r)
+
 type instance NumType   (Ball d p r) = r
 type instance Dimension (Ball d p r) = d
 
@@ -67,7 +74,7 @@
 
 -- | Test if a point lies strictly inside a ball
 --
--- >>> (point2 0.5 0) `insideBall` unitBall
+-- >>> (point2 0.5 0.0) `insideBall` unitBall
 -- True
 -- >>> (point2 1 0) `insideBall` unitBall
 -- False
@@ -101,6 +108,8 @@
 
 type Sphere d p r = Boundary (Ball d p r)
 
+
+pattern Sphere     :: Point d r :+ p -> r -> Sphere d p r
 pattern Sphere c r = Boundary (Ball c r)
 
 
@@ -111,11 +120,13 @@
 
 type Disk p r = Ball 2 p r
 
+pattern Disk     :: Point 2 r :+ p -> r -> Disk p r
 pattern Disk c r = Ball c r
 
 
 type Circle p r = Sphere 2 p r
 
+pattern Circle     :: Point 2 r :+ p ->  r -> Circle p r
 pattern Circle c r = Sphere c r
 
 -- | Given three points, get the disk through the three points. If the three
@@ -125,7 +136,7 @@
 -- Just (Ball {_center = Point2 [0.0,0.0] :+ (), _squaredRadius = 100.0})
 disk       :: (Eq r, Fractional r)
            => Point 2 r -> Point 2 r -> Point 2 r -> Maybe (Disk () r)
-disk p q r = match ((f p) `intersect` (f q)) $
+disk p q r = match (f p `intersect` f q) $
        (H $ \NoIntersection -> Nothing)
     :& (H $ \c@(Point _)    -> Just $ Ball (ext c) (qdA c p))
     :& (H $ \_              -> Nothing)
@@ -139,7 +150,24 @@
                midPoint = p' .+^ (v ^/ 2)
            in perpendicularTo (Line midPoint v)
 
+-- | Creates a circle from three points on the boundary
+from3Points :: Fractional r
+            => Point 2 r :+ p -> Point 2 r :+ q -> Point 2 r :+ s -> Circle () r
+from3Points (p@(Point2 px py) :+ _) (Point2 qx qy :+ _) (Point2 sx sy :+ _) =
+    Circle (ext c) (squaredEuclideanDist c p)
+  where
+    f  x y = x^2 + y^2
+    fx x y = V3 (f x y) y       1
+    fy x y = V3 x       (f x y) 1
 
+    xnom   = det33 $ V3 (fx px py) (fx qx qy) (fx sx sy)
+    ynom   = det33 $ V3 (fy px py) (fy qx qy) (fy sx sy)
+
+    denom  = (2 *) . det33 $ V3 (V3 px py 1) (V3 qx qy 1) (V3 sx sy 1)
+    c      = point2 (xnom / denom) (ynom / denom)
+
+
+
 newtype Touching p = Touching p deriving (Show,Eq,Ord,Functor,F.Foldable,T.Traversable)
 
 -- | No intersection, one touching point, or two points
@@ -154,10 +182,10 @@
   nonEmptyIntersection = defaultNonEmptyIntersection
 
   (Line p' v) `intersect` (Circle (c :+ _) r) = case discr `compare` 0 of
-                                                LT -> coRec $ NoIntersection
+                                                LT -> coRec NoIntersection
                                                 EQ -> coRec . Touching $ q' (lambda (+))
                                                 GT -> let [l1,l2] = L.sort [lambda (-), lambda (+)]
-                                                      in coRec $ (q' l1, q' l2)
+                                                      in coRec (q' l1, q' l2)
     where
       (Vector2 vx vy)   = v
       -- (px, py) is the vector/point after translating the circle s.t. it is centered at the
diff --git a/src/Data/Geometry/Boundary.hs b/src/Data/Geometry/Boundary.hs
--- a/src/Data/Geometry/Boundary.hs
+++ b/src/Data/Geometry/Boundary.hs
@@ -1,9 +1,6 @@
 module Data.Geometry.Boundary where
 
-
-import Data.Geometry.Properties
-import qualified Data.Foldable as F
-import qualified Data.Traversable as T
+import           Data.Geometry.Properties
 import           Data.Geometry.Transformation
 
 --------------------------------------------------------------------------------
diff --git a/src/Data/Geometry/Box.hs b/src/Data/Geometry/Box.hs
--- a/src/Data/Geometry/Box.hs
+++ b/src/Data/Geometry/Box.hs
@@ -1,19 +1,21 @@
 {-# LANGUAGE TemplateHaskell  #-}
-{-# LANGUAGE MultiParamTypeClasses  #-}
 {-# LANGUAGE ScopedTypeVariables  #-}
-{-# LANGUAGE DeriveFunctor  #-}
+{-# LANGUAGE DeriveAnyClass  #-}
+{-# OPTIONS_GHC -fno-warn-orphans #-}
 module Data.Geometry.Box( module Data.Geometry.Box.Internal
                         , topSide, leftSide, bottomSide, rightSide
                         , sides, sides'
                         ) where
 
-
-import           Control.Applicative
-import qualified Data.Traversable as Tr
-import           Data.Geometry.Box.Internal
-import           Data.Geometry.LineSegment
+import Control.DeepSeq
+import Data.Geometry.Box.Internal
+import Data.Geometry.LineSegment
+import Data.Geometry.Vector
 
 --------------------------------------------------------------------------------
+
+deriving instance (NFData p, NFData r, Arity d) => NFData (Box d p r)
+
 
 topSide :: Num r => Rectangle p r -> LineSegment 2 p r
 topSide = (\(l,r,_,_) -> ClosedLineSegment l r) . corners
diff --git a/src/Data/Geometry/Box/Internal.hs b/src/Data/Geometry/Box/Internal.hs
--- a/src/Data/Geometry/Box/Internal.hs
+++ b/src/Data/Geometry/Box/Internal.hs
@@ -1,41 +1,73 @@
 {-# LANGUAGE TemplateHaskell  #-}
-{-# LANGUAGE MultiParamTypeClasses  #-}
 {-# LANGUAGE ScopedTypeVariables  #-}
-{-# LANGUAGE DeriveFunctor  #-}
 module Data.Geometry.Box.Internal where
 
-import           Control.Applicative
+import           Control.DeepSeq
 import           Control.Lens
 import           Data.Bifunctor
 import           Data.Ext
-import qualified Data.Semigroup.Foldable as F
-import qualified Data.Range as R
+import           Frames.CoRec (asA)
 import           Data.Geometry.Point
 import           Data.Geometry.Properties
 import           Data.Geometry.Transformation
+import           Data.Geometry.Vector (Vector, Arity, Index',C(..))
 import qualified Data.Geometry.Vector as V
 import qualified Data.List.NonEmpty as NE
-import           Data.Geometry.Vector(Vector, Arity, Index',C(..))
+import           Data.Proxy
+import qualified Data.Range as R
 import           Data.Semigroup
+import qualified Data.Semigroup.Foldable as F
+import qualified Data.Vector.Fixed as FV
+import           GHC.TypeLits
+import           GHC.Generics (Generic)
 
-import qualified Data.Vector.Fixed                as FV
 
-import           GHC.TypeLits
+--------------------------------------------------------------------------------
 
+-- | Coordinate wize minimum
+newtype CWMin a = CWMin { _cwMin :: a }
+                deriving (Show,Eq,Ord,Functor,Foldable,Traversable,Generic,NFData)
+makeLenses ''CWMin
+
+instance (Arity d, Ord r) => Semigroup (CWMin (Point d r)) where
+  (CWMin p) <> (CWMin q) = CWMin . Point $ FV.zipWith min (p^.vector) (q^.vector)
+
+-- | Coordinate wize maximum
+newtype CWMax a = CWMax { _cwMax :: a }
+                deriving (Show,Eq,Ord,Functor,Foldable,Traversable,Generic,NFData)
+makeLenses ''CWMax
+
+instance (Arity d, Ord r) => Semigroup (CWMax (Point d r)) where
+  (CWMax p) <> (CWMax q) = CWMax . Point $ FV.zipWith max (p^.vector) (q^.vector)
+
+
 --------------------------------------------------------------------------------
 -- * d-dimensional boxes
 
-data Box d p r = Box { _minP :: Min (Point d r) :+ p
-                     , _maxP :: Max (Point d r) :+ p
-                     }
+
+data Box d p r = Box { _minP :: !(CWMin (Point d r) :+ p)
+                     , _maxP :: !(CWMax (Point d r) :+ p)
+                     } deriving Generic
 makeLenses ''Box
 
 -- | Given the point with the lowest coordinates and the point with highest
 -- coordinates, create a box.
-fromCornerPoints          :: Point d r :+ p -> Point d r :+ p -> Box d p r
-fromCornerPoints low high = Box (low&core %~ Min) (high&core %~ Max)
+box          :: Point d r :+ p -> Point d r :+ p -> Box d p r
+box low high = Box (low&core %~ CWMin) (high&core %~ CWMax)
 
+-- | Build a d dimensional Box given d ranges.
+fromExtent    :: Arity d => Vector d (R.Range r) -> Box d () r
+fromExtent rs = Box (CWMin (Point $ fmap (^.R.lower.R.unEndPoint) rs) :+ mempty)
+                    (CWMax (Point $ fmap (^.R.upper.R.unEndPoint) rs) :+ mempty)
 
+
+-- | Center of the box
+centerPoint   :: (Arity d, Fractional r) => Box d p r -> Point d r
+centerPoint b = Point $ w V.^/ 2
+  where w = b^.minP.core.cwMin.vector V.^+^ b^.maxP.core.cwMax.vector
+
+
+
 deriving instance (Show r, Show p, Arity d) => Show (Box d p r)
 deriving instance (Eq r, Eq p, Arity d)     => Eq   (Box d p r)
 deriving instance (Ord r, Ord p, Arity d)   => Ord  (Box d p r)
@@ -45,25 +77,44 @@
 
 type instance IntersectionOf (Box d p r) (Box d q r) = '[ NoIntersection, Box d () r]
 
--- In principle this should also just work for Boxes in higher dimensions. It is just
--- that we need a better way to compute their corners
-instance (Num r, Ord r) => (Rectangle p r) `IsIntersectableWith` (Rectangle p r) where
-
+instance (Ord r, Arity d) => (Box d p r) `IsIntersectableWith` (Box d q r) where
   nonEmptyIntersection = defaultNonEmptyIntersection
 
-  box@(Box a b) `intersect` box'@(Box c d)
-      |    box  `containsACornerOf` box'
-        || box' `containsACornerOf` box = coRec $ Box (mi :+ ()) (ma :+ ())
-      | otherwise                       = coRec NoIntersection
+  bx `intersect` bx' = f . sequence $ FV.zipWith intersect' (extent bx) (extent bx')
     where
+      f = maybe (coRec NoIntersection) (coRec . fromExtent)
+      r `intersect'` s = asA (Proxy :: Proxy (R.Range r)) $ r `intersect` s
 
-      mi = (a^.core) `max` (c^.core)
-      ma = (b^.core) `min` (d^.core)
 
-      bx `containsACornerOf` bx' = let (a',b',c',d') = corners bx'
-                                   in any (\(p :+ _) -> p `inBox` bx) [a',b',c',d']
 
+-- -- In principle this should also just work for Boxes in higher dimensions. It is just
+-- -- that we need a better way to compute their corners
+-- instance (Num r, Ord r) => (Rectangle p r) `IsIntersectableWith` (Rectangle p r) where
 
+--   nonEmptyIntersection = defaultNonEmptyIntersection
+
+--   box@(Box a b) `intersect` box'@(Box c d)
+--       |    box  `containsACornerOf` box'
+--         || box' `containsACornerOf` box = coRec $ Box (mi :+ ()) (ma :+ ())
+--       | otherwise                       = coRec NoIntersection
+--     where
+
+--       mi = (a^.core) `max` (c^.core)
+--       ma = (b^.core) `min` (d^.core)
+
+--       bx `containsACornerOf` bx' = let (a',b',c',d') = corners bx'
+--                                    in any (\(p :+ _) -> p `inBox` bx) [a',b',c',d']
+
+
+type instance IntersectionOf (Point d r) (Box d p r) = '[ NoIntersection, Point d r]
+
+instance (Arity d, Ord r) => (Point d r) `IsIntersectableWith` (Box d p r) where
+  nonEmptyIntersection = defaultNonEmptyIntersection
+  p `intersect` b
+    | not $ p `inBox` b = coRec NoIntersection
+    | otherwise         = coRec p
+
+
 instance PointFunctor (Box d p) where
   pmap f (Box mi ma) = Box (first (fmap f) mi) (first (fmap f) ma)
 
@@ -82,10 +133,10 @@
 -- * Functions on d-dimensonal boxes
 
 minPoint :: Box d p r -> Point d r :+ p
-minPoint b = let (Min p :+ e) = b^.minP in p :+ e
+minPoint b = let (CWMin p :+ e) = b^.minP in p :+ e
 
 maxPoint :: Box d p r -> Point d r :+ p
-maxPoint b = let (Max p :+ e) = b^.maxP in p :+ e
+maxPoint b = let (CWMax p :+ e) = b^.maxP in p :+ e
 
 -- | Check if a point lies a box
 --
@@ -101,10 +152,10 @@
 -- starting at zero.
 --
 -- >>> extent (boundingBoxList' [point3 1 2 3, point3 10 20 30] :: Box 3 () Int)
--- Vector3 [Range {_lower = Closed 1, _upper = Closed 10},Range {_lower = Closed 2, _upper = Closed 20},Range {_lower = Closed 3, _upper = Closed 30}]
-extent                                 :: (Arity d)
+-- Vector3 [Range (Closed 1) (Closed 10),Range (Closed 2) (Closed 20),Range (Closed 3) (Closed 30)]
+extent                                 :: Arity d
                                        => Box d p r -> Vector d (R.Range r)
-extent (Box (Min a :+ _) (Max b :+ _)) = FV.zipWith R.ClosedRange (toVec a) (toVec b)
+extent (Box (CWMin a :+ _) (CWMax b :+ _)) = FV.zipWith R.ClosedRange (toVec a) (toVec b)
 
 -- | Get the size of the box (in all dimensions). Note that the resulting vector is 0 indexed
 -- whereas one would normally count dimensions starting at zero.
@@ -166,8 +217,8 @@
                                      , Point 2 r :+ p
                                      )
 corners r     = let w = width r
-                    p = (_maxP r)&core %~ getMax
-                    q = (_minP r)&core %~ getMin
+                    p = (_maxP r)&core %~ _cwMax
+                    q = (_minP r)&core %~ _cwMin
                 in ( p&core.xCoord %~ (subtract w)
                    , p
                    , q&core.xCoord %~ (+ w)
@@ -178,26 +229,23 @@
 -- * Constructing bounding boxes
 
 class IsBoxable g where
-  boundingBox :: (Monoid p, Semigroup p, Ord (NumType g))
-              => g -> Box (Dimension g) p (NumType g)
-
-type IsAlwaysTrueBoundingBox g p = (Semigroup p, Arity (Dimension g))
-
+  boundingBox :: Ord (NumType g) => g -> Box (Dimension g) () (NumType g)
 
 
-boundingBoxList :: (IsBoxable g, Monoid p, F.Foldable1 c, Ord (NumType g)
-                    , IsAlwaysTrueBoundingBox g p
-                    ) => c g -> Box (Dimension g) p (NumType g)
+boundingBoxList :: (IsBoxable g, F.Foldable1 c, Ord (NumType g), Arity (Dimension g))
+                => c g -> Box (Dimension g) () (NumType g)
 boundingBoxList = F.foldMap1 boundingBox
 
 
 -- | Unsafe version of boundingBoxList, that does not check if the list is non-empty
-boundingBoxList' :: (IsBoxable g, Monoid p, Ord (NumType g)
-                    , IsAlwaysTrueBoundingBox g p
-                    ) => [g] -> Box (Dimension g) p (NumType g)
+boundingBoxList' :: (IsBoxable g, Ord (NumType g), Arity (Dimension g))
+                 => [g] -> Box (Dimension g) () (NumType g)
 boundingBoxList' = boundingBoxList . NE.fromList
 
 ----------------------------------------
 
 instance IsBoxable (Point d r) where
-  boundingBox p = Box (Min p :+ mempty) (Max p :+ mempty)
+  boundingBox p = Box (ext $ CWMin p) (ext $ CWMax p)
+
+instance IsBoxable (Box d p r) where
+  boundingBox (Box m m') = Box (m&extra .~ ()) (m'&extra .~ ())
diff --git a/src/Data/Geometry/Duality.hs b/src/Data/Geometry/Duality.hs
--- a/src/Data/Geometry/Duality.hs
+++ b/src/Data/Geometry/Duality.hs
@@ -1,6 +1,5 @@
 module Data.Geometry.Duality where
 
-import Control.Applicative
 import Data.Geometry.Line
 import Data.Geometry.Point
 import Data.Maybe(fromJust)
diff --git a/src/Data/Geometry/HalfLine.hs b/src/Data/Geometry/HalfLine.hs
--- a/src/Data/Geometry/HalfLine.hs
+++ b/src/Data/Geometry/HalfLine.hs
@@ -1,8 +1,9 @@
 {-# LANGUAGE TemplateHaskell  #-}
 {-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE DeriveAnyClass #-}
 module Data.Geometry.HalfLine where
 
-import           Control.Applicative
+
 import           Control.Lens
 import           Data.Ext
 import qualified Data.Foldable as F
@@ -14,13 +15,10 @@
 import           Data.Geometry.SubLine
 import           Data.Geometry.Transformation
 import           Data.Geometry.Vector
-import           Data.Range
 import qualified Data.Traversable as T
 import           Data.UnBounded
-import           Frames.CoRec
-import           Linear.Affine(Affine(..),distanceA)
-import           Linear.Vector((*^))
-
+import           GHC.Generics (Generic)
+import           Control.DeepSeq
 
 --------------------------------------------------------------------------------
 -- * d-dimensional Half-Lines
@@ -28,11 +26,13 @@
 -- | d-dimensional Half-Lines
 data HalfLine d r = HalfLine { _startPoint        :: Point  d r
                              , _halfLineDirection :: Vector d r
-                             }
+                             } deriving Generic
 makeLenses ''HalfLine
 
-deriving instance (Show r, Arity d) => Show    (HalfLine d r)
-deriving instance (Eq r, Arity d)   => Eq      (HalfLine d r)
+deriving instance (Show r, Arity d)   => Show    (HalfLine d r)
+deriving instance (Eq r, Arity d)     => Eq      (HalfLine d r)
+deriving instance (NFData r, Arity d) => NFData  (HalfLine d r)
+
 deriving instance Arity d           => Functor (HalfLine d)
 deriving instance Arity d           => F.Foldable    (HalfLine d)
 deriving instance Arity d           => T.Traversable (HalfLine d)
diff --git a/src/Data/Geometry/Interval.hs b/src/Data/Geometry/Interval.hs
--- a/src/Data/Geometry/Interval.hs
+++ b/src/Data/Geometry/Interval.hs
@@ -1,21 +1,24 @@
 {-# LANGUAGE TemplateHaskell  #-}
-{-# LANGUAGE MultiParamTypeClasses #-}
-{-# LANGUAGE DeriveFunctor #-}
-module Data.Geometry.Interval-- (
-                             -- -- * 1 dimensional Intervals
-                             --   Interval(..)
-                             -- , Intersection(..)
+module Data.Geometry.Interval(
+                             -- * 1 dimensional Intervals
+                               Interval(..)
+                             , pattern OpenInterval
+                             , pattern ClosedInterval
+                             , pattern Interval
 
-                             -- -- * querying the start and end of intervals
-                             -- , HasStart(..), HasEnd(..)
-                             -- -- * Working with intervals
-                             -- , width
-                             -- , inInterval
-                             -- )
+                             -- * querying the start and end of intervals
+                             , HasStart(..), HasEnd(..)
+                             -- * Working with intervals
+                             , inInterval
+                             , shiftLeft'
+
+                             , module Data.Range
+                             )
        where
 
-import           Control.Applicative
-import           Control.Lens(makeLenses, (^.),(%~),(&), Lens')
+import           Control.DeepSeq
+import           Control.Lens (makeLenses, (^.),(%~),(&), Lens')
+import           Data.Bifunctor
 import           Data.Bitraversable
 import           Data.Ext
 import qualified Data.Foldable as F
@@ -25,14 +28,16 @@
 import qualified Data.Traversable as T
 import           Data.Vinyl
 import           Frames.CoRec
-import           Data.Bifunctor
+import           GHC.Generics (Generic)
 
 --------------------------------------------------------------------------------
 
 -- | An Interval is essentially a 'Data.Range' but with possible payload
 newtype Interval a r = GInterval { _unInterval :: Range (r :+ a) }
-                     deriving (Eq)
+                     deriving (Eq,Generic)
 makeLenses ''Interval
+
+deriving instance (NFData a, NFData r) => NFData (Interval a r)
 
 instance (Show a, Show r) => Show (Interval a r) where
   show ~(Interval l u) = concat [ "Interval (", show l, ") (", show u,")"]
diff --git a/src/Data/Geometry/Interval/Util.hs b/src/Data/Geometry/Interval/Util.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Geometry/Interval/Util.hs
@@ -0,0 +1,21 @@
+{-# LANGUAGE TemplateHaskell #-}
+module Data.Geometry.Interval.Util where
+
+import Control.DeepSeq
+import Control.Lens
+import Data.Range
+import GHC.Generics (Generic)
+
+
+-- | Open on left endpoint; so Closed before open
+newtype L r = L { _unL :: EndPoint r } deriving (Show,Eq,Generic,NFData)
+makeLenses ''L
+instance Ord r => Ord (L r) where
+  a `compare` b = f ( _unL a) `compare` f (_unL b)
+    where
+      f (Open x)   = (x,True)
+      f (Closed x) = (x,False)
+
+-- | Order on right endpoint; so Open before Closed
+newtype R r = R { _unR :: EndPoint r } deriving (Show,Eq,Ord,Generic,NFData)
+makeLenses ''R
diff --git a/src/Data/Geometry/IntervalTree.hs b/src/Data/Geometry/IntervalTree.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Geometry/IntervalTree.hs
@@ -0,0 +1,160 @@
+{-# LANGUAGE TemplateHaskell #-}
+{-# LANGUAGE DeriveAnyClass #-}
+module Data.Geometry.IntervalTree( NodeData(..)
+                                 , splitPoint, intervalsLeft, intervalsRight
+                                 , IntervalTree(..), unIntervalTree
+                                 , IntervalLike(..)
+                                 , createTree, fromIntervals
+                                 , insert, delete
+                                 , stab, search
+                                 , toList
+                                 ) where
+
+
+import           Control.DeepSeq
+import           Control.Lens
+import           Data.BinaryTree
+import           Data.Ext
+import           Data.Geometry.Interval
+import           Data.Geometry.Interval.Util
+import           Data.Geometry.Properties
+import qualified Data.List as List
+import qualified Data.Map as M
+import           GHC.Generics (Generic)
+
+--------------------------------------------------------------------------------
+
+-- | Information stored in a node of the Interval Tree
+data NodeData i r = NodeData { _splitPoint     :: !r
+                             , _intervalsLeft  :: !(M.Map (L r) [i])
+                             , _intervalsRight :: !(M.Map (R r) [i])
+                             } deriving (Show,Eq,Ord,Generic,NFData)
+makeLenses ''NodeData
+
+
+
+-- | IntervalTree type, storing intervals of type i
+newtype IntervalTree i r =
+  IntervalTree { _unIntervalTree :: BinaryTree (NodeData i r) }
+  deriving (Show,Eq,Generic,NFData)
+makeLenses ''IntervalTree
+
+-- | Given an ordered list of points, create an interval tree
+--
+-- \(O(n)\)
+createTree     :: Ord r => [r] -> IntervalTree i r
+createTree pts = IntervalTree . asBalancedBinTree
+               . map (\m -> NodeData m mempty mempty) $ pts
+
+
+-- | Build an interval tree
+--
+-- \(O(n \log n)\)
+fromIntervals    :: (Ord r, IntervalLike i, NumType i ~ r)
+                 => [i] -> IntervalTree i r
+fromIntervals is = foldr insert (createTree pts) is
+  where
+    endPoints (toRange -> Range' a b) = [a,b]
+    pts = List.sort . concatMap endPoints $ is
+
+-- | Lists the intervals. We don't guarantee anything about the order
+--
+-- running time: \(O(n)\).
+toList :: IntervalTree i r -> [i]
+toList = toList' . _unIntervalTree
+  where
+    toList' Nil              = []
+    toList' (Internal l v r) =
+      concat [concat $ v^..intervalsLeft.traverse, toList' l, toList' r]
+
+--------------------------------------------------------------------------------
+
+-- | Find all intervals that stab x
+--
+-- \(O(\log n + k)\), where k is the output size
+search :: Ord r => r -> IntervalTree i r -> [i]
+search = stab
+
+-- | Find all intervals that stab x
+--
+-- \(O(\log n + k)\), where k is the output size
+stab                    :: Ord r => r -> IntervalTree i r -> [i]
+stab x (IntervalTree t) = stab' t
+  where
+    stab' Nil = []
+    stab' (Internal l (NodeData m ll rr) r)
+      | x <= m    = let is = f (<= L (Closed x)) . M.toAscList $ ll
+                    in is ++ stab' l
+      | otherwise = let is = f (>= R (Closed x)) . M.toDescList $ rr
+                    in is ++ stab' r
+    f p = concatMap snd . List.takeWhile (p . fst)
+
+--------------------------------------------------------------------------------
+
+-- | Insert :
+-- pre: the interval intersects some midpoint in the tree
+--
+-- \(O(\log n)\)
+insert                    :: (Ord r, IntervalLike i, NumType i ~ r)
+                          => i -> IntervalTree i r -> IntervalTree i r
+insert i (IntervalTree t) = IntervalTree $ insert' t
+  where
+    ri@(Range a b) = toRange i
+
+    insert' Nil = Nil
+    insert' (Internal l nd@(_splitPoint -> m) r)
+      | m `inRange` ri = Internal l (insert'' nd) r
+      | b <= Closed m  = Internal (insert' l) nd r
+      | otherwise      = Internal l nd (insert' r)
+
+    insert'' (NodeData m l r) = NodeData m (M.insertWith (++) (L a) [i] l)
+                                           (M.insertWith (++) (R b) [i] r)
+
+
+-- | Delete an interval from the Tree
+--
+-- \(O(\log n)\) (under some general position assumption)
+delete :: (Ord r, IntervalLike i, NumType i ~ r, Eq i)
+          => i -> IntervalTree i r -> IntervalTree i r
+delete i (IntervalTree t) = IntervalTree $ delete' t
+  where
+    ri@(Range a b) = toRange i
+
+    delete' Nil = Nil
+    delete' (Internal l nd@(_splitPoint -> m) r)
+      | m `inRange` ri = Internal l (delete'' nd) r
+      | b <= Closed m  = Internal (delete' l) nd r
+      | otherwise      = Internal l nd (delete' r)
+
+    delete'' (NodeData m l r) = NodeData m (M.update f (L a) l) (M.update f (R b) r)
+    f is = let is' = List.delete i is in if null is' then Nothing else Just is'
+
+
+
+--------------------------------------------------------------------------------
+
+
+-- | Anything that looks like an interval
+class IntervalLike i where
+  toRange :: i -> Range (NumType i)
+
+instance IntervalLike (Range r) where
+  toRange = id
+
+instance IntervalLike (Interval p r) where
+  toRange = fmap (^.core) . _unInterval
+
+
+
+--------------------------------------------------------------------------------
+
+test'' = fromIntervals test
+test  = [Interval (Open (97 :+ ())) (Closed (228 :+ ())) ,Interval (Open (18 :+ ())) (Open (79 :+ ())),Interval (Closed (126 :+ ())) (Open (167 :+ ())),Interval (Closed (105 :+ ())) (Closed (158 :+ ())),Interval (Closed (126 :+ ())) (Closed (211 :+ ())),Interval (Closed (111 :+ ())) (Open (194 :+ ())),Interval (Closed (120 :+ ())) (Open (302 :+ ())),Interval (Closed (92 :+ ())) (Closed (140 :+ ()))]
+
+-- test = fromIntervals [ closedInterval 0 10
+--                      , closedInterval 5 15
+--                      , closedInterval 1 4
+--                      , closedInterval 3 9
+--                      ]
+
+-- closedInterval a b = ClosedInterval (ext a) (ext b)
diff --git a/src/Data/Geometry/Ipe/Attributes.hs b/src/Data/Geometry/Ipe/Attributes.hs
--- a/src/Data/Geometry/Ipe/Attributes.hs
+++ b/src/Data/Geometry/Ipe/Attributes.hs
@@ -5,15 +5,11 @@
 {-# LANGUAGE UndecidableInstances #-}
 module Data.Geometry.Ipe.Attributes where
 
-import           Control.Applicative hiding (Const)
 import           Control.Lens hiding (rmap, Const)
-import qualified Data.Foldable as F
-import qualified Data.Geometry.Transformation as Transf
 import           Data.Semigroup
 import           Data.Singletons
 import           Data.Singletons.TH
 import           Data.Text(Text)
-import qualified Data.Traversable as T
 import           Data.Vinyl
 import           Data.Vinyl.Functor
 import           Data.Vinyl.TypeLevel
@@ -64,8 +60,12 @@
 newtype Attr (f :: TyFun u * -> *) -- Symbol repr. the Type family mapping
                                    -- Labels in universe u to concrete types
              (label :: u) = GAttr { _getAttr :: Maybe (Apply f label) }
-                          deriving (Show,Read,Eq,Ord)
 
+deriving instance Show (Apply f label) => Show (Attr f label)
+deriving instance Read (Apply f label) => Read (Attr f label)
+deriving instance Eq   (Apply f label) => Eq   (Attr f label)
+deriving instance Ord  (Apply f label) => Ord  (Attr f label)
+
 makeLenses ''Attr
 
 pattern Attr x = GAttr (Just x)
@@ -74,7 +74,7 @@
 -- | Give pref. to the *RIGHT*
 instance Monoid (Attr f l) where
   mempty                 = NoAttr
-  _ `mappend` b@(Attr x) = b
+  _ `mappend` b@(Attr _) = b
   a `mappend` _          = a
 
 
@@ -106,7 +106,7 @@
 
 zipRecsWith                       :: (forall a. f a -> g a -> h a)
                                   -> Rec f as -> Rec g as -> Rec h as
-zipRecsWith f RNil      _         = RNil
+zipRecsWith _ RNil      _         = RNil
 zipRecsWith f (r :& rs) (s :& ss) = f r s :& zipRecsWith f rs ss
 
 attrLens   :: (at ∈ ats) => proxy at -> Lens' (Attributes f ats) (Maybe (Apply f at))
@@ -117,7 +117,7 @@
 
 setAttr               :: forall proxy at ats f. (at ∈ ats)
                       => proxy at -> Apply f at -> Attributes f ats -> Attributes f ats
-setAttr p a (Attrs r) = Attrs $ rput (Attr a :: Attr f at) r
+setAttr _ a (Attrs r) = Attrs $ rput (Attr a :: Attr f at) r
 
 
 -- | gets and removes the attribute from Attributes
@@ -242,6 +242,8 @@
                            } deriving (Show,Eq)
 makeLenses ''IpeArrow
 
+normalArrow :: IpeArrow r
+normalArrow = IpeArrow "normal" (IpeSize $ Named "normal/normal")
 
 -- -- | and their types
 -- type family PathAttrElf (r :: *) (s :: PathAttributeUniverse) :: * where
diff --git a/src/Data/Geometry/Ipe/FromIpe.hs b/src/Data/Geometry/Ipe/FromIpe.hs
--- a/src/Data/Geometry/Ipe/FromIpe.hs
+++ b/src/Data/Geometry/Ipe/FromIpe.hs
@@ -1,15 +1,14 @@
 module Data.Geometry.Ipe.FromIpe where
 
-import Control.Applicative
-import qualified Data.Traversable as Tr
-import Data.Ext
-import Data.Geometry.Ipe.Types
-import Data.Geometry.Line
-import Data.Geometry.LineSegment
+import           Control.Lens
+import           Data.Ext
+import           Data.Geometry.Ipe.Types
+import           Data.Geometry.Line
+import           Data.Geometry.LineSegment
 import qualified Data.Geometry.PolyLine as PolyLine
-import Data.Geometry.Polygon
-import Control.Lens
+import           Data.Geometry.Polygon
 import qualified Data.Seq2 as S2
+import qualified Data.Traversable as Tr
 
 
 -- | Try to convert a path into a line segment, fails if the path is not a line
diff --git a/src/Data/Geometry/Ipe/IpeOut.hs b/src/Data/Geometry/Ipe/IpeOut.hs
--- a/src/Data/Geometry/Ipe/IpeOut.hs
+++ b/src/Data/Geometry/Ipe/IpeOut.hs
@@ -1,29 +1,24 @@
 {-# LANGUAGE OverloadedStrings #-}
 module Data.Geometry.Ipe.IpeOut where
 
-import           Control.Applicative
 import           Control.Lens
 import           Data.Bifunctor
 import           Data.Ext
-import qualified Data.Foldable as F
 import           Data.Geometry.Ball
 import           Data.Geometry.Boundary
+import           Data.Geometry.Box
 import           Data.Geometry.Ipe.Attributes
 import           Data.Geometry.Ipe.Types
 import           Data.Geometry.LineSegment
 import           Data.Geometry.Point
-import           Data.Geometry.Box
-import           Data.Geometry.Polygon
 import           Data.Geometry.PolyLine
+import           Data.Geometry.Polygon
+import           Data.Geometry.Polygon.Convex
 import           Data.Geometry.Properties
 import           Data.Geometry.Transformation
-import qualified Data.List.NonEmpty as NE
 import           Data.Semigroup
-import           Data.Proxy
 import qualified Data.Seq2 as S2
-import           Data.Text(Text)
-import qualified Data.Traversable as Tr
-import           Data.Vinyl
+import           Data.Text (Text)
 
 --------------------------------------------------------------------------------
 
@@ -31,6 +26,7 @@
 -- 'g' into an ipe object of type 'i'.
 newtype IpeOut g i = IpeOut { asIpe :: g -> i } deriving (Functor)
 
+
 -- | Given an geometry object, and a record with its attributes, construct an ipe
 -- Object representing it using the default conversion.
 asIpeObject :: (HasDefaultIpeOut g, DefaultIpeOut g ~ i, NumType g ~ r)
@@ -107,6 +103,10 @@
   type DefaultIpeOut (SimplePolygon p r) = Path
   defaultIpeOut = flip addAttributes ipeSimplePolygon $
                     mempty <> attr SFill (IpeColor "red")
+
+instance HasDefaultIpeOut (ConvexPolygon p r) where
+  type DefaultIpeOut (ConvexPolygon p r) = Path
+  defaultIpeOut = IpeOut $ asIpe defaultIpeOut . view simplePolygon
 
 --------------------------------------------------------------------------------
 -- * Point Converters
diff --git a/src/Data/Geometry/Ipe/ParserPrimitives.hs b/src/Data/Geometry/Ipe/ParserPrimitives.hs
--- a/src/Data/Geometry/Ipe/ParserPrimitives.hs
+++ b/src/Data/Geometry/Ipe/ParserPrimitives.hs
@@ -10,12 +10,11 @@
                                          , pNotFollowedBy
                                          ) where
 
-import           Control.Applicative hiding (many,(<|>))
+
 import           Text.Parsec(try)
 import           Text.Parsec(ParsecT, Stream)
 import           Text.Parsec.Text
 import           Text.ParserCombinators.Parsec hiding (Parser,try)
-
 import qualified Data.Text as T
 
 
@@ -127,10 +126,10 @@
   return $ f x
 
 -- | the variants with missing brackets
-(***>) :: (Functor m, Monad m) => m a -> m b -> m b
+(***>) :: Monad m => m a -> m b -> m b
 p ***> q = (\_ s -> s) <$> p <***> q
 
-(<***) :: (Functor m, Monad m) => m b -> m t -> m b
+(<***) :: Monad m => m b -> m t -> m b
 p <*** q = (\s _ -> s) <$> p <***> q
 
 class Reversable s where
diff --git a/src/Data/Geometry/Ipe/PathParser.hs b/src/Data/Geometry/Ipe/PathParser.hs
--- a/src/Data/Geometry/Ipe/PathParser.hs
+++ b/src/Data/Geometry/Ipe/PathParser.hs
@@ -1,32 +1,21 @@
-{-# Language FlexibleInstances #-}
 {-# Language OverloadedStrings #-}
 module Data.Geometry.Ipe.PathParser where
 
-import           Numeric
-
-import           Data.Ext(ext)
-import           Control.Applicative
-import           Control.Monad
-
 import           Data.Bifunctor
-import           Data.Monoid(mconcat)
-import           Data.Semigroup
--- import           Data.Validation(AccValidation(..))
-
-import           Data.Char(isSpace)
-import           Data.Ratio
-
-import           Text.Parsec.Error(messageString, errorMessages)
-import           Data.Geometry.Point
+import           Data.Char (isSpace)
+import           Data.Ext (ext)
 import           Data.Geometry.Box
-import           Data.Geometry.Vector
-import           Data.Geometry.Transformation
 import           Data.Geometry.Ipe.ParserPrimitives
-import           Data.Geometry.Ipe.Types(Operation(..))
-import           Data.Text(Text)
+import           Data.Geometry.Ipe.Types (Operation(..))
+import           Data.Geometry.Point
+import           Data.Geometry.Transformation
+import           Data.Geometry.Vector
+import           Data.Ratio
+import           Data.Semigroup
+import           Data.Text (Text)
 import qualified Data.Text as T
-
--- type Matrix d m r = ()
+import           Numeric
+import           Text.Parsec.Error (messageString, errorMessages)
 
 
 -----------------------------------------------------------------------
@@ -144,9 +133,9 @@
                 pDecimal = pMaybe (pChar '.' *> pInteger)
 
 pRectangle :: Coordinate r => Parser (Rectangle () r)
-pRectangle = (\p q -> fromCornerPoints (ext p) (ext q)) <$> pPoint
-                                                        <*  pWhiteSpace
-                                                        <*> pPoint
+pRectangle = (\p q -> box (ext p) (ext q)) <$> pPoint
+                                           <*  pWhiteSpace
+                                           <*> pPoint
 
 pMatrix :: Coordinate r => Parser (Matrix 3 3 r)
 pMatrix = (\a b -> mkMatrix (a:b)) <$> pCoordinate
diff --git a/src/Data/Geometry/Ipe/Reader.hs b/src/Data/Geometry/Ipe/Reader.hs
--- a/src/Data/Geometry/Ipe/Reader.hs
+++ b/src/Data/Geometry/Ipe/Reader.hs
@@ -1,8 +1,6 @@
 {-# LANGUAGE UndecidableInstances #-}
 {-# LANGUAGE OverloadedStrings #-}
-{-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE ScopedTypeVariables #-}
-{-# LANGUAGE PolyKinds #-}
 module Data.Geometry.Ipe.Reader( -- * Reading ipe Files
                                  readRawIpeFile
                                , readIpeFile
@@ -26,45 +24,34 @@
                                , ipeReadRec
                                ) where
 
-import           Data.Proxy
-import           Data.Either(rights)
-import           Control.Applicative hiding (Const)
 import           Control.Lens hiding (Const, rmap)
-
+import qualified Data.ByteString as B
+import           Data.Either (rights)
 import           Data.Ext
-import qualified Data.Foldable as F
-import qualified Data.Traversable as Tr
-import           Data.Maybe(fromMaybe, isJust, mapMaybe)
-import qualified Data.List as L
-import           Data.Vinyl
-import           Data.Vinyl.Functor
-import           Data.Vinyl.TypeLevel
-
-import qualified Data.List.NonEmpty as NE
--- import           Data.Validation
-import qualified Data.Seq2     as S2
-
-import           Data.Geometry.Point
 import           Data.Geometry.Box
-import qualified Data.Geometry.Polygon as Polygon
-import           Data.Geometry.PolyLine
-import qualified Data.Geometry.Transformation as Trans
-import           Data.Geometry.Ipe.Types
 import           Data.Geometry.Ipe.Attributes
-import qualified Data.Geometry.Ipe.Attributes as IA
+import           Data.Geometry.Ipe.ParserPrimitives (pInteger)
 import           Data.Geometry.Ipe.PathParser
-import           Data.Geometry.Ipe.ParserPrimitives(pInteger)
-
-import qualified Data.ByteString as B
+import           Data.Geometry.Ipe.Types
+import           Data.Geometry.Point
+import           Data.Geometry.PolyLine
+import qualified Data.Geometry.Polygon as Polygon
+import qualified Data.Geometry.Transformation as Trans
+import qualified Data.List as L
+import qualified Data.List.NonEmpty as NE
+import           Data.Maybe (fromMaybe, mapMaybe)
 import           Data.Monoid
+import           Data.Proxy
+import qualified Data.Seq2 as S2
 import           Data.Singletons
-import           Data.Text(Text)
-
-
+import qualified Data.Text as T
+import           Data.Text (Text)
+import qualified Data.Traversable as Tr
+import           Data.Vinyl
+import           Data.Vinyl.Functor
+import           Data.Vinyl.TypeLevel
 import           Text.XML.Expat.Tree
 
-import qualified Data.Text as T
--- import qualified Data.Map as M
 
 --------------------------------------------------------------------------------
 
@@ -92,8 +79,7 @@
 
 -- | Given a Bytestring, try to parse the bytestring into anything that is
 -- IpeReadable, i.e. any of the Ipe elements.
-fromIpeXML   :: (Coordinate r, IpeRead (t r))
-             => B.ByteString -> Either ConversionError (t r)
+fromIpeXML   :: IpeRead (t r) => B.ByteString -> Either ConversionError (t r)
 fromIpeXML b = readXML b >>= ipeRead
 
 -- | Reads the data from a Bytestring into a proper Node
@@ -436,16 +422,16 @@
 
 validateAll' :: err -> Prism' (Operation r) (Point 2 r) -> [Operation r]
                -> Either [err] [Point 2 r]
-validateAll' err field = toEither . foldr (\op res -> f op <> res) (Right' [])
+validateAll' err field = toEither . foldr (\op' res -> f op' <> res) (Right' [])
   where
-    f op = maybe (Left' [err]) (\p -> Right' [p]) $ op ^? field
+    f op' = maybe (Left' [err]) (\p -> Right' [p]) $ op' ^? field
     toEither = either' Left Right
 
 -- This is a bit of a hack
 instance Coordinate r => IpeRead (PolyLine 2 () r) where
-  ipeRead (Element "path" ats ts) = ipeReadText . T.unlines . map unText $ ts
+  ipeRead (Element "path" _ ts) = ipeReadText . T.unlines . map unText $ ts
                                     -- apparently hexpat already splits the text into lines
-  ipeRead _                       = Left "iperead: no polyline."
+  ipeRead _                     = Left "iperead: no polyline."
 
 unText          :: Node t t1 -> t1
 unText (Text t) = t
@@ -457,30 +443,5 @@
 testP :: B.ByteString
 testP = "<path stroke=\"black\">\n128 656 m\n224 768 l\n304 624 l\n432 752 l\n</path>"
 
-
--- testPoly :: Either Text (Path Double)
--- testPoly = fromIpeXML testP
-
--- ipeRead' :: [Element Text Text]
--- ipeRead' = map ipeRead
-
--- instance IpeRead (IpePage gs) where
---   ipeRead (Element "page" ats chs) = Right . IpePage [] [] . fromList' . rights $ map ipeRead chs
---     where
---       fromList' = Group' . foldr (\x r -> (IpeObject x :& RNil) :& r) RNil
---   ipeRead _                        = Left "ipeRead: Not a page"
-
-readPolyLines :: Coordinate r => Node Text Text -> [PolyLine 2 () r]
-readPolyLines (Element "ipe" _ chs) = concatMap readPolyLines' chs
-
-
-readPolyLines' :: Coordinate r => Node Text Text -> [PolyLine 2 () r]
-readPolyLines' (Element "page" _ chs) = rights $ map ipeRead chs
-readPolyLines' _                      = []
-
-polylinesFromIpeFile :: (Coordinate r) => FilePath -> IO [PolyLine 2 () r]
-polylinesFromIpeFile = fmap readPolies . B.readFile
-  where
-    readPolies = either (const []) readPolyLines . parse' defaultParseOptions
 
 --------------------------------------------------------------------------------
diff --git a/src/Data/Geometry/Ipe/Types.hs b/src/Data/Geometry/Ipe/Types.hs
--- a/src/Data/Geometry/Ipe/Types.hs
+++ b/src/Data/Geometry/Ipe/Types.hs
@@ -5,9 +5,8 @@
 {-# LANGUAGE OverloadedStrings #-}
 module Data.Geometry.Ipe.Types where
 
-import           Control.Applicative
+
 import           Control.Lens
-import           Data.Monoid
 import           Data.Proxy
 import           Data.Vinyl
 
diff --git a/src/Data/Geometry/Ipe/Writer.hs b/src/Data/Geometry/Ipe/Writer.hs
--- a/src/Data/Geometry/Ipe/Writer.hs
+++ b/src/Data/Geometry/Ipe/Writer.hs
@@ -1,51 +1,42 @@
 {-# LANGUAGE OverloadedStrings #-}
-{-# LANGUAGE TupleSections #-}
 {-# LANGUAGE ScopedTypeVariables #-}
-{-# LANGUAGE TupleSections #-}
-{-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE UndecidableInstances #-}
 module Data.Geometry.Ipe.Writer where
 
-import           Control.Applicative hiding (Const(..))
-import           Control.Lens((^.),(^..),(.~),(&), to)
+import           Control.Lens ((^.),(^..),(.~),(&), to)
+import qualified Data.ByteString as B
+import qualified Data.ByteString.Char8 as C
 import           Data.Ext
 import           Data.Fixed
 import qualified Data.Foldable as F
+import           Data.Geometry.Box
+import           Data.Geometry.Ipe.Attributes
+import qualified Data.Geometry.Ipe.Attributes as IA
 import           Data.Geometry.Ipe.Types
 import qualified Data.Geometry.Ipe.Types as IT
 import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
 import           Data.Geometry.PolyLine
-import           Data.Geometry.Polygon(SimplePolygon, outerBoundary)
+import           Data.Geometry.Polygon (SimplePolygon, outerBoundary)
 import qualified Data.Geometry.Transformation as GT
-import           Data.Geometry.Point
-import           Data.Geometry.Box
 import           Data.Geometry.Vector
-import           Data.Maybe(catMaybes, mapMaybe, fromMaybe)
+import           Data.Maybe (catMaybes, mapMaybe, fromMaybe)
+import           Data.Proxy
 import           Data.Ratio
 import           Data.Semigroup
-import           Data.Proxy
+import qualified Data.Seq2 as S2
+import           Data.Singletons
+import qualified Data.Text as T
+import qualified Data.Text as Text
+import           Data.Text (Text)
 import qualified Data.Traversable as Tr
 import           Data.Vinyl
 import           Data.Vinyl.Functor
 import           Data.Vinyl.TypeLevel
-import           Data.Singletons
-import qualified Data.Geometry.Ipe.Attributes as IA
-import           Data.Geometry.Ipe.Attributes
-
-import qualified Data.ByteString as B
-import qualified Data.ByteString.Char8 as C
-import           Data.List(nub)
-import qualified Data.Seq2     as S2
-import           Data.Text(Text)
-import qualified Data.Text as Text
-
+import           System.IO (hPutStrLn,stderr)
+import           Text.XML.Expat.Format (format')
 import           Text.XML.Expat.Tree
-import           Text.XML.Expat.Format(format')
 
-import           System.IO(hPutStrLn,stderr)
-
-import qualified Data.Text as T
-
 --------------------------------------------------------------------------------
 
 -- | Given a prism to convert something of type g into an ipe file, a file path,
@@ -140,6 +131,9 @@
 instance IpeWriteText Int where
   ipeWriteText = writeByShow
 
+instance IpeWriteText Integer where
+  ipeWriteText = writeByShow
+
 instance HasResolution p => IpeWriteText (Fixed p) where
   ipeWriteText = writeByShow
 
@@ -293,8 +287,8 @@
                                    Element "image" [("rect", p <> " " <> q)] [Text dt]
                                  )
                                <$> ipeWriteText d
-                               <*> ipeWriteText (a^.core.(to getMin))
-                               <*> ipeWriteText (b^.core.(to getMax))
+                               <*> ipeWriteText (a^.core.cwMin)
+                               <*> ipeWriteText (b^.core.cwMax)
 
 -- TODO: Replace this one with s.t. that writes the actual image payload
 instance IpeWriteText () where
diff --git a/src/Data/Geometry/KDTree.hs b/src/Data/Geometry/KDTree.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Geometry/KDTree.hs
@@ -0,0 +1,186 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+module Data.Geometry.KDTree where
+
+import           Control.Lens hiding (imap, element, Empty, (:<))
+import           Data.BinaryTree
+import           Data.Coerce
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry.Box
+import           Data.Geometry.Point
+import           Data.Geometry.Properties
+import           Data.Geometry.Vector
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Maybe (fromJust)
+import           Data.Proxy
+import           Data.Seq (LSeq(..), ViewL(..))
+import qualified Data.Seq as Seq
+import           Data.Util
+import qualified Data.Vector.Fixed as FV
+import           GHC.TypeLits
+import           Prelude hiding (replicate)
+
+--------------------------------------------------------------------------------
+
+newtype Coord (d :: Nat) = Coord { unCoord ::  Int}
+
+instance KnownNat d => Eq (Coord d) where
+  (Coord i) == (Coord j) = (i `mod` d) == (j `mod` d)
+    where
+      d = fromInteger . natVal $ (Proxy :: Proxy d)
+
+instance KnownNat d => Show (Coord d) where
+  show (Coord i) = show $ 1 + (i `mod` d)
+    where
+      d = fromInteger . natVal $ (Proxy :: Proxy d)
+
+instance KnownNat d => Enum (Coord d) where
+  toEnum i = Coord $ 1 + (i `mod` d)
+    where
+      d = fromInteger . natVal $ (Proxy :: Proxy d)
+  fromEnum = subtract 1 . unCoord
+
+
+data Split d r = Split !(Coord d) !r !(Box d () r)
+
+deriving instance (Show r, Arity d, KnownNat d) => Show (Split d r)
+deriving instance (Eq r, Arity d, KnownNat d)   => Eq (Split d r)
+
+
+type Split' d r = SP (Coord d) r
+
+newtype KDTree' d p r = KDT { unKDT :: BinLeafTree (Split d r) (Point d r :+ p) }
+
+deriving instance (Show p, Show r, Arity d, KnownNat d) => Show (KDTree' d p r)
+deriving instance (Eq p, Eq r, Arity d, KnownNat d)     => Eq   (KDTree' d p r)
+
+
+data KDTree d p r = Empty
+                  | Tree (KDTree' d p r)
+
+deriving instance (Show p, Show r, Arity d, KnownNat d) => Show (KDTree d p r)
+deriving instance (Eq p, Eq r, Arity d, KnownNat d)     => Eq   (KDTree d p r)
+
+
+toMaybe          :: KDTree d p r -> Maybe (KDTree' d p r)
+toMaybe Empty    = Nothing
+toMaybe (Tree t) = Just t
+
+
+-- | Expects the input to be a set, i.e. no duplicates
+--
+-- running time: \(O(n \log n)\)
+buildKDTree :: (Arity d, KnownNat d, Index' 0 d, Ord r)
+            => [Point d r :+ p] -> KDTree d p r
+buildKDTree = maybe Empty (Tree . buildKDTree') . NonEmpty.nonEmpty
+
+buildKDTree' :: (Arity d, KnownNat d, Index' 0 d, Ord r)
+             => NonEmpty.NonEmpty (Point d r :+ p) -> KDTree' d p r
+buildKDTree' = KDT . addBoxes . build (Coord 1) . toPointSet . Seq.fromNonEmpty
+  where     -- compute one tree with bounding boxes, then merge them together
+    addBoxes t = let bbt = foldUpData (\l _ r -> boundingBoxList' [l,r])
+                                      (boundingBox . (^.core)) t
+                 in zipExactWith (\(SP c m) b -> Split c m b) const t bbt
+
+
+-- | Nub by sorting first
+ordNub :: Ord a => NonEmpty.NonEmpty a -> NonEmpty.NonEmpty a
+ordNub = fmap NonEmpty.head . NonEmpty.group1 . NonEmpty.sort
+
+
+
+toPointSet :: (Arity d, Ord r)
+           => LSeq n (Point d r :+ p) -> PointSet (LSeq n) d p r
+toPointSet = FV.imap sort . FV.replicate
+  where
+    sort i = Seq.unstableSortBy (compareOn $ 1 + i)
+
+
+compareOn       :: (Ord r, Arity d)
+                => Int -> Point d r :+ e -> Point d r :+ e -> Ordering
+compareOn i p q = let f = (^.core.unsafeCoord i)
+                  in (f p, p^.core) `compare` (f q, q^.core)
+
+
+build      :: (Index' 0 d, Arity d, KnownNat d, Ord r)
+           => Coord d
+           -> PointSet (LSeq 1) d p r
+           -> BinLeafTree (Split' d r) (Point d r :+ p)
+build i ps = case asSingleton ps of
+    Left p    -> Leaf p
+    Right ps' -> let (l,m,r) = splitOn i ps'
+                     j       = succ i
+                   -- the pattern match proves tha tthe seq has >= 2 elements
+                 in Node (build j l) m (build j r)
+
+
+--------------------------------------------------------------------------------
+
+reportSubTree :: KDTree' d p r -> NonEmpty.NonEmpty (Point d r :+ p)
+reportSubTree = NonEmpty.fromList . F.toList . unKDT
+
+-- | Searches in a KDTree
+--
+-- running time: \(O(n^{(d-1)/d} + k)\)
+searchKDTree    :: (Arity d, Ord r)
+                => Box d q r -> KDTree d p r -> [Point d r :+ p]
+searchKDTree qr = maybe [] (searchKDTree' qr) . toMaybe
+
+searchKDTree'                  :: (Arity d, Ord r)
+                              => Box d q r -> KDTree' d p r -> [Point d r :+ p]
+searchKDTree' qr = search . unKDT
+  where
+    search (Leaf p)
+      | (p^.core) `intersects` qr = [p]
+      | otherwise                 = []
+    search t@(Node l (Split _ _ b) r)
+      | b `containedIn` qr        = F.toList t
+      | otherwise                 = l' ++ r'
+      where
+        l' = if qr `intersects` boxOf l then search l else []
+        r' = if qr `intersects` boxOf r then search r else []
+
+
+boxOf :: (Arity d, Ord r) => BinLeafTree (Split d r) (Point d r :+ p) -> Box d () r
+boxOf (Leaf p)                 = boundingBox (p^.core)
+boxOf (Node _ (Split _ _ b) _) = b
+
+containedIn :: (Arity d, Ord r) => Box d q r -> Box d p r -> Bool
+(Box (CWMin p :+ _) (CWMax q :+ _)) `containedIn` b = all (`intersects` b) [p,q]
+
+--------------------------------------------------------------------------------
+
+
+type PointSet seq d p r = Vector d (seq (Point d r :+ p))
+
+-- | running time: \(O(n)\)
+splitOn                 :: (Arity d, KnownNat d, Ord r)
+                        => Coord d
+                        -> PointSet (LSeq 2) d p r
+                        -> ( PointSet (LSeq 1) d p r
+                           , Split' d r
+                           , PointSet (LSeq 1) d p r)
+splitOn c@(Coord i) pts = (l, SP c (m^.core.unsafeCoord i), r)
+  where
+    -- i = traceShow (c,j) j
+
+    m = let xs = fromJust $ pts^?element' (i-1)
+        in xs `Seq.index` (F.length xs `div` 2)
+
+    -- Since the input seq has >= 2 elems, F.length xs / 2 >= 1. It follows
+    -- that the both sets thus have at least one elemnt.
+    -- f :: LSeq 2 _ -> (LSeq 1 _, LSeq 1 _)
+    f = bimap Seq.promise Seq.promise
+      . Seq.partition (\p -> compareOn i p m == LT)
+
+    (l,r) = unzip' . fmap f $ pts
+
+    -- unzip' :: Vector d (a,b) -> (Vector d a, Vector d b)
+    unzip' = bimap vectorFromListUnsafe vectorFromListUnsafe . unzip . F.toList
+
+
+asSingleton   :: (Index' 0 d, Arity d) => PointSet (LSeq 1) d p r
+              -> Either (Point d r :+ p) (PointSet (LSeq 2) d p r)
+asSingleton v = case Seq.viewl $ v^.element (C :: C 0) of
+                  _ :< _ Seq.:<< _ -> Right $ coerce v
+                  p :< _           -> Left p -- only one element
diff --git a/src/Data/Geometry/Line/Internal.hs b/src/Data/Geometry/Line/Internal.hs
--- a/src/Data/Geometry/Line/Internal.hs
+++ b/src/Data/Geometry/Line/Internal.hs
@@ -1,34 +1,39 @@
 {-# LANGUAGE TemplateHaskell  #-}
+{-# LANGUAGE DeriveAnyClass  #-}
 {-# LANGUAGE ScopedTypeVariables  #-}
 {-# LANGUAGE UndecidableInstances #-}
 module Data.Geometry.Line.Internal where
 
+import           Control.DeepSeq
 import           Control.Lens
 import qualified Data.Foldable as F
 import           Data.Geometry.Point
 import           Data.Geometry.Properties
 import           Data.Geometry.Vector
-import           Data.Ord(comparing)
+import           Data.Ord (comparing)
 import qualified Data.Traversable as T
 import           Data.Vinyl
 import           Frames.CoRec
+import           GHC.Generics (Generic)
 
+
 --------------------------------------------------------------------------------
 -- * d-dimensional Lines
 
 -- | A line is given by an anchor point and a vector indicating the
 -- direction.
-data Line d r = Line { _anchorPoint :: Point  d r
-                     , _direction   :: Vector d r
-                     }
+data Line d r = Line { _anchorPoint :: !(Point  d r)
+                     , _direction   :: !(Vector d r)
+                     } deriving Generic
 makeLenses ''Line
 
 instance (Show r, Arity d) => Show (Line d r) where
   show (Line p v) = concat [ "Line (", show p, ") (", show v, ")" ]
-deriving instance (Eq r,   Arity d) => Eq            (Line d r)
-deriving instance Arity d           => Functor       (Line d)
-deriving instance Arity d           => F.Foldable    (Line d)
-deriving instance Arity d           => T.Traversable (Line d)
+deriving instance (Eq r,   Arity d)   => Eq            (Line d r)
+deriving instance (NFData r, Arity d) => NFData        (Line d r)
+deriving instance Arity d             => Functor       (Line d)
+deriving instance Arity d             => F.Foldable    (Line d)
+deriving instance Arity d             => T.Traversable (Line d)
 
 
 type instance Dimension (Line d r) = d
@@ -82,7 +87,6 @@
 -- False
 onLine                :: (Eq r, Fractional r, Arity d) => Point d r -> Line d r -> Bool
 p `onLine` (Line q v) = p == q || (p .-. q) `isScalarMultipleOf` v
-  -- TODO: Maybe use a specialize pragma for 2D with an implementation using ccw
 
 
 -- | The intersection of two lines is either: NoIntersection, a point or a line.
diff --git a/src/Data/Geometry/LineSegment.hs b/src/Data/Geometry/LineSegment.hs
--- a/src/Data/Geometry/LineSegment.hs
+++ b/src/Data/Geometry/LineSegment.hs
@@ -6,6 +6,7 @@
                                 , pattern ClosedLineSegment
 
                                 , _SubLine
+                                , module Data.Geometry.Interval
 
 
                                 , toLineSegment
@@ -18,7 +19,6 @@
 
 import           Data.Ord(comparing)
 import           Control.Arrow((&&&))
-import           Control.Applicative
 import           Control.Lens
 import           Data.Bifunctor
 import           Data.Semigroup
@@ -31,7 +31,6 @@
 import           Data.Geometry.SubLine
 import           Data.Geometry.Transformation
 import           Data.Geometry.Vector
-import           Data.Range
 import           Data.Vinyl
 import           Data.UnBounded
 import           Frames.CoRec
@@ -83,6 +82,7 @@
   type EndCore  (LineSegment d p r) = Point d r
   type EndExtra (LineSegment d p r) = p
   end = unLineSeg.end
+
 
 _SubLine :: (Fractional r, Eq r, Arity d) => Iso' (LineSegment d p r) (SubLine d p r)
 _SubLine = iso segment2SubLine subLineToSegment
diff --git a/src/Data/Geometry/PlanarSubdivision.hs b/src/Data/Geometry/PlanarSubdivision.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Geometry/PlanarSubdivision.hs
@@ -0,0 +1,132 @@
+{-# LANGUAGE TemplateHaskell #-}
+module Data.Geometry.PlanarSubdivision where
+
+import           Control.Lens
+import qualified Data.BalBST as SS
+import           Data.Bifunctor.Apply
+import qualified Data.CircularSeq as C
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry.Interval
+import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
+import           Data.Geometry.Polygon
+import           Data.Geometry.Properties
+import qualified Data.Map as M
+import           Data.PlanarGraph
+import           Data.PlaneGraph
+import           Data.Semigroup
+import           Data.Util
+import qualified Data.Vector as V
+
+
+-- | Note that the functor instance is in v
+data VertexData r v = VertexData { _location :: !(Point 2 r)
+                                 , _vData    :: !v
+                                 } deriving (Show,Eq,Ord,Functor,Foldable,Traversable)
+makeLenses ''VertexData
+
+instance Bifunctor VertexData where
+  bimap f g (VertexData p v) = VertexData (fmap f p) (g v)
+
+
+data EdgeType = Visible | Invisible deriving (Show,Read,Eq,Ord)
+
+data EdgeData e = EdgeData { _edgeType :: !EdgeType
+                           , _eData    :: !e
+                           } deriving (Show,Eq,Ord,Functor,Foldable,Traversable)
+makeLenses ''EdgeData
+
+
+-- | The Face data consists of the data itself and a list of holes
+data FaceData h f = FaceData { _holes :: [h]
+                             , _fData :: !f
+                             } deriving (Show,Eq,Ord,Functor,Foldable,Traversable)
+makeLenses ''FaceData
+
+
+newtype PlanarSubdivision s v e f r = PlanarSubdivision { _graph ::
+    PlanarGraph s Primal_ (VertexData r v) (EdgeData e) (FaceData (Dart s) f) }
+      deriving (Show,Eq)
+makeLenses ''PlanarSubdivision
+
+instance Functor (PlanarSubdivision s v e f) where
+  fmap f s = s&graph.vertexData.traverse.location %~ fmap f
+
+
+--------------------------------------------------------------------------------
+
+-- | Construct a planar subdivision from a polygon
+--
+-- running time: \(O(n)\).
+fromPolygon                            :: proxy s
+                                       -> SimplePolygon p r
+                                       -> f -- ^ data inside
+                                       -> f -- ^ data outside the polygon
+                                       -> PlanarSubdivision s p () f r
+fromPolygon p (SimplePolygon vs) iD oD = PlanarSubdivision g'
+  where
+    g      = fromVertices p vs
+    fData' = V.fromList [FaceData [] iD, FaceData [] oD]
+
+    g'     = g & faceData .~ fData'
+               & dartData.traverse._2 .~ EdgeData Visible ()
+-- The following does not really work anymore
+-- frompolygon p (MultiPolygon vs hs) iD oD = PlanarSubdivision g'
+--   where
+--     g      = fromVertices p vs
+--     hs'    = map (\h -> fromPolygon p h oD iD) hs
+--            -- note that oD and iD are exchanged
+--     fData' = V.fromList [FaceData iD hs', FaceData oD []]
+
+
+fromVertices      :: proxy s
+                  -> C.CSeq (Point 2 r :+ p)
+                  -> PlanarGraph s Primal_ (VertexData r p) () ()
+fromVertices _ vs = g&vertexData .~ vData'
+  where
+    n = length vs
+    g = planarGraph [ [ (Dart (Arc i)               Positive, ())
+                      , (Dart (Arc $ (i+1) `mod` n) Negative, ())
+                      ]
+                    | i <- [0..(n-1)]]
+    vData' = V.fromList . map (\(p :+ e) -> VertexData p e) . F.toList $ vs
+
+
+-- | Constructs a connected planar subdivision.
+--
+-- pre: the segments form a single connected component
+-- running time: \(O(n\log n)\)
+fromConnectedSegments       :: (Foldable f, Ord r, Num r)
+                            => proxy s
+                            -> f (LineSegment 2 p r :+ EdgeData e)
+                            -> PlanarSubdivision s [p] e () r
+fromConnectedSegments px ss = PlanarSubdivision $
+    fromConnectedSegments' px ss & faceData.traverse %~ FaceData []
+
+-- | Constructs a planar graph
+--
+-- pre: The segments form a single connected component
+--
+-- running time: \(O(n\log n)\)
+fromConnectedSegments'      :: (Foldable f, Ord r, Num r)
+                            => proxy s
+                            -> f (LineSegment 2 p r :+ e)
+                            -> PlanarGraph s Primal_ (VertexData r [p]) e ()
+fromConnectedSegments' _ ss = planarGraph dts & vertexData .~ vxData
+  where
+    pts         = M.fromListWith (<>) . concatMap f . zipWith g [0..] . F.toList $ ss
+    f (s :+ e)  = [ ( s^.start.core
+                    , SP [s^.start.extra] [(s^.end.core)   :+ h Positive e])
+                  , ( s^.end.core
+                    , SP [s^.end.extra]   [(s^.start.core) :+ h Negative e])
+                  ]
+    g i (s :+ e) = s :+ (Arc i :+ e)
+    h d (a :+ e) = (Dart a d, e)
+
+    vts    = map (\(p,sp) -> (p,map (^.extra) . sortArround (ext p) <$> sp))
+           . M.assocs $ pts
+    -- vertex Data
+    vxData = V.fromList . map (\(p,sp) -> VertexData p (sp^._1)) $ vts
+    -- The darts
+    dts    = map (^._2._2) vts
diff --git a/src/Data/Geometry/Point.hs b/src/Data/Geometry/Point.hs
--- a/src/Data/Geometry/Point.hs
+++ b/src/Data/Geometry/Point.hs
@@ -2,25 +2,21 @@
 {-# LANGUAGE UndecidableInstances #-}
 module Data.Geometry.Point where
 
-import           Control.Applicative
-import           Data.Monoid(mconcat)
+import           Control.DeepSeq
 import           Control.Lens
-import           Data.Proxy
-
-import qualified Data.Traversable as T
-import qualified Data.Foldable as F
-import qualified Data.Vector.Fixed as FV
-import qualified Data.List as L
-
+import qualified Data.CircularList as C
+import qualified Data.CircularList.Util as CU
 import           Data.Ext
+import qualified Data.Foldable as F
 import           Data.Geometry.Properties
 import           Data.Geometry.Vector
-import           GHC.TypeLits
-
 import qualified Data.Geometry.Vector as Vec
-import qualified Data.CircularList as C
-import qualified Data.CircularList.Util as CU
-
+import qualified Data.List as L
+import           Data.Proxy
+import qualified Data.Traversable as T
+import qualified Data.Vector.Fixed as FV
+import           GHC.Generics (Generic)
+import           GHC.TypeLits
 --------------------------------------------------------------------------------
 
 --------------------------------------------------------------------------------
@@ -36,7 +32,7 @@
 -- * A d-dimensional Point
 
 -- | A d-dimensional point.
-newtype Point d r = Point { toVec :: Vector d r }
+newtype Point d r = Point { toVec :: Vector d r } deriving (Generic)
 
 instance (Show r, Arity d) => Show (Point d r) where
   show (Point (Vector v)) = mconcat [ "Point", show $ FV.length v , " "
@@ -45,11 +41,12 @@
 
 
 
-deriving instance (Eq r, Arity d)   => Eq (Point d r)
-deriving instance (Ord r, Arity d)  => Ord (Point d r)
-deriving instance Arity d           => Functor (Point d)
-deriving instance Arity d           => F.Foldable (Point d)
-deriving instance Arity d           => T.Traversable (Point d)
+deriving instance (Eq r, Arity d)     => Eq (Point d r)
+deriving instance (Ord r, Arity d)    => Ord (Point d r)
+deriving instance Arity d             => Functor (Point d)
+deriving instance Arity d             => F.Foldable (Point d)
+deriving instance Arity d             => T.Traversable (Point d)
+deriving instance (Arity d, NFData r) => NFData (Point d r)
 
 
 type instance NumType (Point d r) = r
@@ -103,7 +100,12 @@
 coord   :: forall proxy i d r. (Index' (i-1) d, Arity d) => proxy i -> Lens' (Point d r) r
 coord _ = vector . Vec.element (Proxy :: Proxy (i-1))
 
-
+-- | Constructs a point from a list of coordinates
+--
+-- >>> pointFromList [1,2,3] :: Maybe (Point 3 Int)
+-- Just Point3 [1,2,3]
+pointFromList :: Arity d => [r] -> Maybe (Point d r)
+pointFromList = fmap Point . Vec.vectorFromList
 
 
 --------------------------------------------------------------------------------
@@ -124,6 +126,8 @@
 -- if we want.
 pattern Point2       :: r -> r -> Point 2 r
 pattern Point2 x y   <- (_point2 -> (x,y))
+  where
+    Point2 x y = point2 x y
 
 -- | Similarly, we can write:
 --
@@ -136,6 +140,8 @@
 -- 3
 pattern Point3       :: r -> r -> r -> Point 3 r
 pattern Point3 x y z <- (_point3 -> (x,y,z))
+  where
+    Point3 x y z = point3 x y z
 
 -- | Construct a 2 dimensional point
 --
@@ -234,21 +240,10 @@
 -- and r are colinear with p, the closest one to p is reported first.
 -- running time: O(n log n)
 sortArround   :: (Ord r, Num r)
-               => Point 2 r :+ p -> [Point 2 r :+ p] -> [Point 2 r :+ p]
+               => Point 2 r :+ q -> [Point 2 r :+ p] -> [Point 2 r :+ p]
 sortArround c = L.sortBy (ccwCmpAround c)
 
--- sortArround       :: (Ord r, Num r)
---                   => Point 2 r :+ p -> [Point 2 r :+ p] -> [Point 2 r :+ p]
--- sortArround p pts = concatMap sortArround' [ topR, topL, bottomL, bottomR ]
---   where
---     (topL, topR, bottomL, bottomR) = partitionIntoQuadrants p pts
---     sortArround' = L.sortBy cmp
 
---     cmp q r = case ccw (p^.core) (q^.core) (r^.core) of
---                 CCW      -> LT
---                 CW       -> GT
---                 CoLinear -> qdA (p^.core) (q^.core) `compare` qdA (p^.core) (r^.core)
-
 -- | Quadrants of two dimensional points. in CCW order
 data Quadrant = TopRight | TopLeft | BottomLeft | BottomRight
               deriving (Show,Read,Eq,Ord,Enum,Bounded)
@@ -256,7 +251,7 @@
 -- | Quadrants around point c; quadrants are closed on their "previous"
 -- boundary (i..e the boundary with the previous quadrant in the CCW order),
 -- open on next boundary. The origin itself is assigned the topRight quadrant
-quadrantWith                   :: (Arity d, Ord r, 1 <=. d, 2 <=. d)
+quadrantWith                   :: (Ord r, 1 <=. d, 2 <=. d)
                                => Point d r :+ q -> Point d r :+ p -> Quadrant
 quadrantWith (c :+ _) (p :+ _) = case ( (c^.xCoord) `compare` (p^.xCoord)
                                       , (c^.yCoord) `compare` (p^.yCoord) ) of
@@ -271,7 +266,7 @@
                                    (LT, GT) -> BottomRight
 
 -- | Quadrants with respect to the origin
-quadrant :: (Arity d, Ord r, Num r, 1 <=. d, 2 <=. d) => Point d r :+ p -> Quadrant
+quadrant :: (Ord r, Num r, 1 <=. d, 2 <=. d) => Point d r :+ p -> Quadrant
 quadrant = quadrantWith (ext origin)
 
 -- | Given a center point c, and a set of points, partition the points into
@@ -279,7 +274,7 @@
 -- reported in the order topLeft, topRight, bottomLeft, bottomRight. The points
 -- are in the same order as they were in the original input lists.
 -- Points with the same x-or y coordinate as p, are "rounded" to above.
-partitionIntoQuadrants       :: (Ord r, Arity d, 1 <=. d, 2 <=. d)
+partitionIntoQuadrants       :: (Ord r, 1 <=. d, 2 <=. d)
                              => Point d r :+ q
                              -> [Point d r :+ p]
                              -> ( [Point d r :+ p], [Point d r :+ p]
diff --git a/src/Data/Geometry/PolyLine.hs b/src/Data/Geometry/PolyLine.hs
--- a/src/Data/Geometry/PolyLine.hs
+++ b/src/Data/Geometry/PolyLine.hs
@@ -3,21 +3,19 @@
 {-# LANGUAGE UndecidableInstances  #-}
 module Data.Geometry.PolyLine where
 
-import           Control.Applicative
 import           Control.Lens
 import           Data.Bifunctor
 import           Data.Ext
-import qualified Data.Foldable as F
 import           Data.Geometry.Box
 import           Data.Geometry.LineSegment
 import           Data.Geometry.Point
 import           Data.Geometry.Properties
 import           Data.Geometry.Transformation
 import           Data.Geometry.Vector
+import qualified Data.List.NonEmpty as NE
+import           Data.Semigroup
 import qualified Data.Seq2 as S2
 import qualified Data.Sequence as Seq
-import           Data.Semigroup
-import qualified Data.List.NonEmpty as NE
 
 --------------------------------------------------------------------------------
 -- * d-dimensional Polygonal Lines (PolyLines)
diff --git a/src/Data/Geometry/Polygon.hs b/src/Data/Geometry/Polygon.hs
--- a/src/Data/Geometry/Polygon.hs
+++ b/src/Data/Geometry/Polygon.hs
@@ -1,29 +1,27 @@
 {-# LANGUAGE ScopedTypeVariables  #-}
 module Data.Geometry.Polygon where
 
-
-import           Control.Applicative
 import           Control.Lens hiding (Simple)
 import           Data.Bifunctor
+import qualified Data.CircularSeq as C
 import           Data.Ext
-import           Data.Semigroup
+import qualified Data.List.NonEmpty as NonEmpty
 import qualified Data.Foldable as F
-import           Data.Geometry.Box
-import           Data.Geometry.Vector
 import           Data.Geometry.Boundary
+import           Data.Geometry.Box
+import           Data.Geometry.Line
 import           Data.Geometry.LineSegment
 import           Data.Geometry.Point
-import           Data.Geometry.Line
 import           Data.Geometry.Properties
 import           Data.Geometry.Transformation
-import           Data.Maybe(mapMaybe)
+import           Data.Geometry.Vector
+import           Data.Maybe (mapMaybe)
 import           Data.Proxy
-import           Data.Range
-import           Frames.CoRec(asA)
-import qualified Data.CircularSeq as C
--- import qualified Data.CircularList as C
-import           Linear.Vector(Additive(..), (^*), (^/))
+import           Data.Semigroup
+import           Data.Util
+import           Frames.CoRec (asA)
 
+
 --------------------------------------------------------------------------------
 -- * Polygons
 
@@ -61,7 +59,6 @@
 instance (Eq p, Eq r) => Eq (Polygon t p r) where
   (SimplePolygon vs)   == (SimplePolygon vs')    = vs == vs'
   (MultiPolygon vs hs) == (MultiPolygon vs' hs') = vs == vs' && hs == hs'
-  _                    == _                      = False
 
 instance PointFunctor (Polygon t p) where
   pmap f (SimplePolygon vs)   = SimplePolygon (fmap (first f) vs)
@@ -70,6 +67,8 @@
 instance Num r => IsTransformable (Polygon t p r) where
   transformBy = transformPointFunctor
 
+instance IsBoxable (Polygon t p r) where
+  boundingBox = boundingBoxList' . toListOf (outerBoundary.traverse.core)
 
 -- * Functions on Polygons
 
@@ -99,7 +98,7 @@
 outerVertex   :: Int -> Lens' (Polygon t p r) (Point 2 r :+ p)
 outerVertex i = outerBoundary.C.item i
 
--- running time: $O(\log i)$
+-- running time: \(O(\log i)\)
 outerBoundaryEdge     :: Int -> Polygon t p r -> LineSegment 2 p r
 outerBoundaryEdge i p = let u = p^.outerVertex i
                             v = p^.outerVertex (i+1)
@@ -114,9 +113,11 @@
 
 -- | The vertices in the polygon. No guarantees are given on the order in which
 -- they appear!
-vertices :: Polygon t p r -> [Point 2 r :+ p]
-vertices (SimplePolygon vs)   = F.toList vs
-vertices (MultiPolygon vs hs) = F.toList vs ++ concatMap vertices hs
+polygonVertices                      :: Polygon t p r
+                                     -> NonEmpty.NonEmpty (Point 2 r :+ p)
+polygonVertices (SimplePolygon vs)   = C.toNonEmpty vs
+polygonVertices (MultiPolygon vs hs) =
+  sconcat $ C.toNonEmpty vs NonEmpty.:| map polygonVertices hs
 
 
 
@@ -127,16 +128,18 @@
 outerBoundaryEdges :: Polygon t p r -> C.CSeq (LineSegment 2 p r)
 outerBoundaryEdges = toEdges . (^.outerBoundary)
 
--- | Gets the i^th edge on the outer boundary of the polygon, that is the edge
--- with vertices i and i+1 with respect to the current focus. All indices
--- modulo n.
---
 
+-- -- | Gets the i^th edge on the outer boundary of the polygon, that is the edge
+---- with vertices i and i+1 with respect to the current focus. All indices
+-- -- modulo n.
+-- --
+
 -- | Given the vertices of the polygon. Produce a list of edges. The edges are
 -- half-open.
 toEdges    :: C.CSeq (Point 2 r :+ p) -> C.CSeq (LineSegment 2 p r)
-toEdges vs = let vs' = F.toList vs in
-  C.fromList $ zipWith (\p q -> LineSegment (Closed p) (Open q)) vs' (tail vs' ++ vs')
+toEdges vs = C.zipLWith (\p q -> LineSegment (Closed p) (Open q)) vs (C.rotateR vs)
+  -- let vs' = F.toList vs in
+  -- C.fromList $ zipWith (\p q -> LineSegment (Closed p) (Open q)) vs' (tail vs' ++ vs')
 
 
 -- | Test if q lies on the boundary of the polygon. Running time: O(n)
@@ -231,11 +234,6 @@
                              GT -> SP lts       (gts + 1)) (SP 0 0)
 
 
-
-
-data SP a b = SP !a !b
-
-
 -- | Test if a point lies strictly inside the polgyon.
 insidePolygon        :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool
 q `insidePolygon` pg = q `inPolygon` pg == Inside
@@ -286,7 +284,7 @@
 -- | Test if the outer boundary of the polygon is in clockwise or counter
 -- clockwise order.
 --
--- running time: $O(1)$
+-- running time: \(O(1)\)
 --
 isCounterClockwise :: (Eq r, Fractional r) => Polygon t p r -> Bool
 isCounterClockwise = (\x -> x == abs x) . signedArea
@@ -294,11 +292,17 @@
 
 
 -- | Orient the outer boundary to clockwise order
-toClockwiseOrder   :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r
+toClockwiseOrder         :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r
 toClockwiseOrder p
   | isCounterClockwise p = p&outerBoundary %~ C.reverseDirection
   | otherwise            = p
 
+-- | Orient the outer boundary to counter clockwise order
+toCounterClockWiseOrder    :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r
+toCounterClockWiseOrder p
+  | not $ isCounterClockwise p = p&outerBoundary %~ C.reverseDirection
+  | otherwise                  = p
+
 -- | Convert a Polygon to a simple polygon by forgetting about any holes.
 asSimplePolygon                        :: Polygon t p r -> SimplePolygon p r
 asSimplePolygon poly@(SimplePolygon _) = poly
@@ -314,7 +318,7 @@
 
 -- | Finds the extreme points, minimum and maximum, in a given direction
 --
--- running time: $O(n)$
+-- running time: \(O(n)\)
 extremesLinear     :: (Ord r, Num r) => Vector 2 r -> Polygon t p r
                    -> (Point 2 r :+ p, Point 2 r :+ p)
 extremesLinear u p = let vs = p^.outerBoundary
diff --git a/src/Data/Geometry/Polygon/Convex.hs b/src/Data/Geometry/Polygon/Convex.hs
--- a/src/Data/Geometry/Polygon/Convex.hs
+++ b/src/Data/Geometry/Polygon/Convex.hs
@@ -1,61 +1,90 @@
 {-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TemplateHaskell #-}
 {-|
 Module    : Data.Geometry.Polygon.Convex
 Description: Convex Polygons
 Copyright : (c) Frank Staals
 License : See LICENCE file
 -}
-module Data.Geometry.Polygon.Convex( ConvexPolygon
+module Data.Geometry.Polygon.Convex( ConvexPolygon(..), simplePolygon
                                    , merge
                                    , lowerTangent, upperTangent
                                    , isLeftOf, isRightOf
 
                                    , extremes
                                    , maxInDirection
+
+                                   , leftTangent, rightTangent
+
+                                   , minkowskiSum
+                                   , bottomMost
                                    ) where
 
 import           Control.Lens hiding ((:<), (:>))
-import qualified Data.CircularSeq as C
 import           Data.CircularSeq (focus,CSeq)
+import qualified Data.CircularSeq as C
 import           Data.Ext
 import qualified Data.Foldable as F
 import           Data.Function (on, )
 import           Data.Geometry
+import           Data.Geometry.Box (IsBoxable(..))
 import           Data.Geometry.Polygon (fromPoints, cmpExtreme)
+import           Data.Geometry.Properties
 import           Data.Maybe (fromJust)
 import           Data.Ord (comparing)
-import qualified Data.Sequence as S
 import           Data.Sequence (viewl,viewr, ViewL(..), ViewR(..))
+import qualified Data.Sequence as S
 
-import           Data.Geometry.Ipe
-import Debug.Trace
+-- import           Data.Geometry.Ipe
+import           Debug.Trace
 --------------------------------------------------------------------------------
 
-type ConvexPolygon = SimplePolygon
+newtype ConvexPolygon p r = ConvexPolygon {_simplePolygon :: SimplePolygon p r }
+                          deriving (Show,Eq,PointFunctor)
+makeLenses ''ConvexPolygon
 
+-- | Polygons are per definition 2 dimensional
+type instance Dimension (ConvexPolygon p r) = 2
+type instance NumType   (ConvexPolygon p r) = r
 
-mainWith inFile outFile = do
-    ePage <- readSinglePageFile inFile
-    case ePage of
-      Left err                         -> error "" -- err
-      Right (page :: IpePage Rational) -> case page^..content.traverse._withAttrs _IpePath _asSimplePolygon.core of
-        []         -> error "No points found"
-        polies@(_:_) -> do
-           -- let out  = [asIpe drawTriangulation dt, asIpe drawTree' emst]
-           -- print $ length $ edges' dt
-           -- print $ toPlaneGraph (Proxy :: Proxy DT) dt
-           -- writeIpeFile outFile . singlePageFromContent $ out
-           -- mapM_ (print . extremesNaive (v2 1 0)) polies
-           pure $ map (maxInDirection (v2 (-1) 0)) polies
 
+instance Num r => IsTransformable (ConvexPolygon p r) where
+  transformBy = transformPointFunctor
 
+instance IsBoxable (ConvexPolygon p r) where
+  boundingBox = boundingBox . _simplePolygon
 
 
+-- convexPolygon   :: SimplePolygon p r -> Maybe (ConvexPolygon p r)
+-- convexPolygon p = if isConvex p then Just p else Nothing
+
+-- isConvex   :: SimplePolygon p r -> Bool
+-- isConvex p = let ch = convexHull $ p^.vertices
+--              in p^.vertices.size == ch^.simplePolygon.vertices.size
+
+
+-- mainWith inFile outFile = do
+--     ePage <- readSinglePageFile inFile
+--     case ePage of
+--       Left err                         -> error "" -- err
+--       Right (page :: IpePage Rational) -> case page^..content.traverse._withAttrs _IpePath _asSimplePolygon.core of
+--         []         -> error "No points found"
+--         polies@(_:_) -> do
+--            -- let out  = [asIpe drawTriangulation dt, asIpe drawTree' emst]
+--            -- print $ length $ edges' dt
+--            -- print $ toPlaneGraph (Proxy :: Proxy DT) dt
+--            -- writeIpeFile outFile . singlePageFromContent $ out
+--            -- mapM_ (print . extremesNaive (v2 1 0)) polies
+--            pure $ map (flip rightTangent (point2 80 528)) polies
+
+
+
+
 -- | Finds the extreme points, minimum and maximum, in a given direction
 --
 -- pre: The input polygon is strictly convex.
 --
--- running time: $O(\log n)$
+-- running time: \(O(\log n)\)
 --
 --
 extremes     :: (Num r, Ord r) => Vector 2 r -> ConvexPolygon p r
@@ -68,10 +97,16 @@
 --
 -- pre: The input polygon is strictly convex.
 --
--- running time: $O(\log^2 n)$
-maxInDirection     :: (Num r, Ord r) => Vector 2 r -> ConvexPolygon p r -> Point 2 r :+ p
-maxInDirection u p = findMaxStart . C.rightElements $ p^.outerBoundary
+-- running time: \(O(\log^2 n)\)
+maxInDirection   :: (Num r, Ord r) => Vector 2 r -> ConvexPolygon p r -> Point 2 r :+ p
+maxInDirection u = findMaxWith (cmpExtreme u)
+
+findMaxWith       :: (Point 2 r :+ p -> Point 2 r :+ p -> Ordering)
+                  -> ConvexPolygon p r -> Point 2 r :+ p
+findMaxWith cmp p = findMaxStart . C.rightElements . getVertices $ p
   where
+    p' >=. q = (p' `cmp` q) /= LT
+
     findMaxStart s@(viewl -> (a:<r))
       | isLocalMax r a r = a
       | otherwise        = findMax s
@@ -91,7 +126,6 @@
         (True,False,_)      -> findMax (ac |> c)
         (True,True,True)    -> findMax (ac |> c)
         (True,True,False)   -> findMax (c <| cb)
-
         (False,True,_)      -> findMax (c <| cb)
         (False,False,False) -> findMax (ac |> c)
         (False,False,True)  -> findMax (c <| cb)
@@ -103,22 +137,33 @@
     isLocalMax _                 _ _                 = True
 
     -- the Edge from a to b is upwards w.r.t b if a is not larger than b
-    isUpwards a (viewl -> b :< _) = cmpExtreme u a b /= GT
+    isUpwards a (viewl -> b :< _) = (a `cmp` b) /= GT
     isUpwards _ _                 = error "isUpwards: no edge endpoint"
 
-    p' >=. q = cmpExtreme u p' q /= LT
 
+tangentCmp       :: (Num r, Ord r)
+                 => Point 2 r -> Point 2 r :+ p -> Point 2 r :+ q -> Ordering
+tangentCmp o p q = case ccw o (p^.core) (q^.core) of
+                     CCW      -> LT -- q is left of the line from o to p
+                     CoLinear -> EQ -- q is *on* the line from o to p
+                     CW       -> GT -- q is right of the line from o to p
 
+
 --  | Given a convex polygon poly, and a point outside the polygon, find the
---  right tangent of q and the polygon, i.e. the vertex v of the convex polygon
+--  left tangent of q and the polygon, i.e. the vertex v of the convex polygon
 --  s.t. the polygon lies completely to the right of the line from q to v.
 --
--- running time: $O(\log^2 n)$.
--- rightTangent        :: ConvexPolygon p r -> Point 2 r -> Point 2 r :+ p
--- rightTangent poly q = undefined
--- TODO: same as maxInDirection, but with a slightly different ordering
-
+-- running time: \(O(\log^2 n)\).
+leftTangent        :: (Ord r, Num r) => ConvexPolygon p r -> Point 2 r -> Point 2 r :+ p
+leftTangent poly q = findMaxWith (tangentCmp q) poly
 
+--  | Given a convex polygon poly, and a point outside the polygon, find the
+--  right tangent of q and the polygon, i.e. the vertex v of the convex polygon
+--  s.t. the polygon lies completely to the left of the line from q to v.
+--
+-- running time: \(O(\log^2 n)\).
+rightTangent        :: (Ord r, Num r) => ConvexPolygon p r -> Point 2 r -> Point 2 r :+ p
+rightTangent poly q = findMaxWith (flip $ tangentCmp q) poly
 
 
 
@@ -146,17 +191,17 @@
 -- Running time: O(n+m), where n and m are the sizes of the two polygons respectively
 merge       :: (Num r, Ord r) => ConvexPolygon p r  -> ConvexPolygon p r
             -> (ConvexPolygon p r, LineSegment 2 p r, LineSegment 2 p r)
-merge lp rp = (fromPoints $ r' ++ l', lt, ut)
+merge lp rp = (ConvexPolygon . fromPoints $ r' ++ l', lt, ut)
   where
     lt@(ClosedLineSegment a b) = lowerTangent lp rp
     ut@(ClosedLineSegment c d) = upperTangent lp rp
 
-
     takeUntil p xs = let (xs',x:_) = break p xs in xs' ++ [x]
     rightElems  = F.toList . C.rightElements
+    takeAndRotate x y = takeUntil (coreEq x) . rightElems . rotateTo' y . getVertices
 
-    r' = takeUntil (coreEq b) . rightElems . rotateTo' d $ rp^.outerBoundary
-    l' = takeUntil (coreEq c) . rightElems . rotateTo' a $ lp^.outerBoundary
+    r' = takeAndRotate b d rp
+    l' = takeAndRotate c a lp
 
 
 rotateTo'   :: Eq a => (a :+ b) -> CSeq (a :+ b) -> CSeq (a :+ b)
@@ -166,6 +211,8 @@
 coreEq :: Eq a => (a :+ b) -> (a :+ b) -> Bool
 coreEq = (==) `on` (^.core)
 
+
+
 -- | Compute the lower tangent of the two polgyons
 --
 --   pre: - polygons lp and rp have at least 1 vertex
@@ -174,11 +221,11 @@
 --        - The vertices of the polygons are given in clockwise order
 --
 -- Running time: O(n+m), where n and m are the sizes of the two polygons respectively
-lowerTangent                                     :: (Num r, Ord r)
-                                                 => ConvexPolygon p r
-                                                 -> ConvexPolygon p r
-                                                 -> LineSegment 2 p r
-lowerTangent (SimplePolygon l) (SimplePolygon r) = rotate xx yy zz zz''
+lowerTangent                                       :: (Num r, Ord r)
+                                                   => ConvexPolygon p r
+                                                   -> ConvexPolygon p r
+                                                   -> LineSegment 2 p r
+lowerTangent (getVertices -> l) (getVertices -> r) = rotate xx yy zz zz''
   where
     xx = rightMost l
     yy = leftMost r
@@ -186,6 +233,7 @@
     zz   = pred' yy
     zz'' = succ' xx
 
+
     rotate x y z z''
       | focus z   `isRightOf` (focus x, focus y) = rotate x   z (pred' z) z''
                                                       -- rotate the right polygon CCW
@@ -208,11 +256,11 @@
 --        - The vertices of the polygons are given in clockwise order
 --
 -- Running time: O(n+m), where n and m are the sizes of the two polygons respectively
-upperTangent                                     :: (Num r, Ord r)
-                                                 => ConvexPolygon p r
-                                                 -> ConvexPolygon p r
-                                                 -> LineSegment 2 p r
-upperTangent (SimplePolygon l) (SimplePolygon r) = rotate xx yy zz zz'
+upperTangent                                       :: (Num r, Ord r)
+                                                   => ConvexPolygon p r
+                                                   -> ConvexPolygon p r
+                                                   -> LineSegment 2 p r
+upperTangent (getVertices -> l) (getVertices -> r) = rotate xx yy zz zz'
   where
     xx = rightMost l
     yy = leftMost r
@@ -236,18 +284,60 @@
                     => Point 2 r :+ p -> (Point 2 r :+ p', Point 2 r :+ p'') -> Bool
 a `isLeftOf` (b,c) = ccw (b^.core) (c^.core) (a^.core) == CCW
 
+--------------------------------------------------------------------------------
 
+-- | Computes the Minkowski sum of the two input polygons with $n$ and $m$
+-- vertices respectively.
+--
+-- pre: input polygons are in CCW order.
+--
+-- running time: \(O(n+m)\).
+minkowskiSum     :: (Ord r, Num r)
+                 => ConvexPolygon p r -> ConvexPolygon q r -> ConvexPolygon (p,q) r
+minkowskiSum p q = ConvexPolygon . fromPoints $ merge' (f p) (f q)
+  where
+    f p' = let xs@(S.viewl -> (v :< _)) = C.asSeq . bottomMost . getVertices $ p'
+           in F.toList $ xs |> v
+    (v :+ ve) .+. (w :+ we) = v .+^ (toVec w) :+ (ve,we)
+
+    cmpAngle v v' w w' =
+      ccwCmpAround (ext $ origin) (ext . Point $ v' .-. v) (ext . Point $ w' .-. w)
+
+    merge' [_]       [_]       = []
+    merge' vs@[v]    (w:ws)    = v .+. w : merge' vs ws
+    merge' (v:vs)    ws@[w]    = v .+. w : merge' vs ws
+    merge' (v:v':vs) (w:w':ws) = v .+. w :
+      case cmpAngle (v^.core) (v'^.core) (w^.core) (w'^.core) of
+        LT -> merge' (v':vs)   (w:w':ws)
+        GT -> merge' (v:v':vs) (w':ws)
+        EQ -> merge' (v':vs)   (w':ws)
+    merge' _         _         = error $ "minkowskiSum: Should not happen"
+
+
+
+
 --------------------------------------------------------------------------------
 
--- | Rotate to the rightmost point
+-- | Rotate to the rightmost point (rightmost and topmost in case of ties)
 rightMost    :: Ord r => CSeq (Point 2 r :+ p) -> CSeq (Point 2 r :+ p)
-rightMost xs = let m = F.maximumBy (comparing (^.core.xCoord)) xs in rotateTo' m xs
+rightMost xs = let m = F.maximumBy (comparing (^.core)) xs in rotateTo' m xs
 
--- | Rotate to the leftmost point
+-- | Rotate to the leftmost point (and bottommost in case of ties)
 leftMost    :: Ord r => CSeq (Point 2 r :+ p) -> CSeq (Point 2 r :+ p)
-leftMost xs = let m = F.minimumBy (comparing (^.core.xCoord)) xs in rotateTo' m xs
+leftMost xs = let m = F.minimumBy (comparing (^.core)) xs in rotateTo' m xs
 
+-- | Rotate to the bottommost point (and leftmost in case of ties)
+bottomMost    :: Ord r => CSeq (Point 2 r :+ p) -> CSeq (Point 2 r :+ p)
+bottomMost xs = let f p = (p^.core.yCoord,p^.core.xCoord)
+                    m   = F.minimumBy (comparing f) xs
+                in rotateTo' m xs
 
+
+
+-- | Helper to get the vertices of a convex polygon
+getVertices :: ConvexPolygon p r -> CSeq (Point 2 r :+ p)
+getVertices = view (simplePolygon.outerBoundary)
+
 -- -- | rotate right while p 'current' 'rightNeibhour' is true
 -- rotateRWhile      :: (a -> a -> Bool) -> C.CList a -> C.CList a
 -- rotateRWhile p lst
@@ -258,3 +348,15 @@
 --                   xs' = C.rotR xs
 --                   nxt = focus' xs'
 --               in if p cur nxt then go xs' else xs
+
+test1 :: Num r => ConvexPolygon () r
+test1 = ConvexPolygon . fromPoints . map ext . reverse $ [origin, point2 1 4, point2 5 6, point2 10 3]
+
+test2 :: Num r => ConvexPolygon () r
+test2 = ConvexPolygon . fromPoints . map ext . reverse $ [point2 11 6, point2 10 10, point2 15 18, point2 12 5]
+
+testA :: Num r => ConvexPolygon () r
+testA = ConvexPolygon . fromPoints . map ext $ [origin, point2 5 1, point2 2 2]
+
+testB :: Num r => ConvexPolygon () r
+testB = ConvexPolygon . fromPoints . map ext $ [origin, point2 5 3, point2 (-2) 2, point2 (-2) 1]
diff --git a/src/Data/Geometry/Properties.hs b/src/Data/Geometry/Properties.hs
--- a/src/Data/Geometry/Properties.hs
+++ b/src/Data/Geometry/Properties.hs
@@ -4,8 +4,7 @@
 {-# LANGUAGE DefaultSignatures #-}
 module Data.Geometry.Properties where
 
-import           Control.Applicative
-import           Data.Maybe(isJust)
+import           Data.Maybe(isNothing)
 import           Data.Proxy
 import           Data.Vinyl.Core
 import           Data.Vinyl.Functor
@@ -66,13 +65,13 @@
 type AlwaysTrueIntersection g h = RecApplicative (IntersectionOf g h)
 
 
--- | Returns True iff the resultt is a NoIntersection
+-- | Returns True iff the result is *not* a NoIntersection
 defaultNonEmptyIntersection :: forall g h proxy.
                             ( NoIntersection ∈ IntersectionOf g h
                             , RecApplicative (IntersectionOf g h)
                             )
                             => proxy g -> proxy h -> Intersection g h -> Bool
-defaultNonEmptyIntersection _ _ = isJust . asA (Proxy :: Proxy NoIntersection)
+defaultNonEmptyIntersection _ _ = isNothing . asA (Proxy :: Proxy NoIntersection)
 
 
 -- type IsAlwaysTrueFromEither a b = (VTL.RIndex b [a,b] ~ ((VTL.S VTL.Z)))
diff --git a/src/Data/Geometry/SegmentTree.hs b/src/Data/Geometry/SegmentTree.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Geometry/SegmentTree.hs
@@ -0,0 +1,18 @@
+{-# LANGUAGE TemplateHaskell #-}
+module Data.Geometry.SegmentTree( module Data.Geometry.SegmentTree.Generic
+                                ) where
+
+
+import           Data.Ext
+import           Data.Semigroup
+import           Control.Lens
+import           Data.List.NonEmpty (NonEmpty)
+import qualified Data.List.NonEmpty as NonEmpty
+import qualified Data.List as List
+import           Data.BinaryTree
+import           Data.Range
+import           Data.Geometry.Interval
+import           Data.Geometry.SegmentTree.Generic
+import           Data.Geometry.Interval.Util
+import           Data.Geometry.Properties
+import           Data.Geometry.IntervalTree (IntervalLike(..))
diff --git a/src/Data/Geometry/SegmentTree/Generic.hs b/src/Data/Geometry/SegmentTree/Generic.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Geometry/SegmentTree/Generic.hs
@@ -0,0 +1,307 @@
+{-# LANGUAGE TemplateHaskell #-}
+module Data.Geometry.SegmentTree.Generic( NodeData(..), splitPoint, range, assoc
+                                        , LeafData(..), atomicRange, leafAssoc
+
+                                        , SegmentTree(..), unSegmentTree
+                                        , Assoc(..)
+
+                                        , createTree, fromIntervals
+                                        , insert, delete
+
+                                        , search, stab
+
+                                        , I(..), fromIntervals'
+
+                                        , Count(..)
+                                        ) where
+
+
+import           Control.DeepSeq
+import           Control.Lens
+import           Data.BinaryTree
+import           Data.Ext
+import           Data.Geometry.Interval
+import           Data.Geometry.Interval.Util
+import           Data.Geometry.IntervalTree (IntervalLike(..))
+import           Data.Geometry.Properties
+import qualified Data.List as List
+import           Data.List.NonEmpty (NonEmpty)
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Range
+import           Data.Semigroup
+import           GHC.Generics (Generic)
+
+--------------------------------------------------------------------------------
+
+-- | Internal nodes store a split point, the range, and an associated data structure
+data NodeData v r = NodeData { _splitPoint :: !(EndPoint r)
+                             , _range      :: !(Range r)
+                             , _assoc      :: !v
+                             } deriving (Show,Eq,Functor,Generic)
+makeLenses ''NodeData
+instance (NFData v, NFData r) => NFData (NodeData v r)
+
+-- | We store atomic ranges a bit more efficiently.
+data AtomicRange r = Singleton !r | AtomicRange deriving (Show,Eq,Functor,Generic)
+instance NFData r => NFData (AtomicRange r)
+
+-- | Leaf nodes store an atomic range, and an associated data structure.
+data LeafData v r = LeafData  { _atomicRange :: !(AtomicRange r)
+                              , _leafAssoc   :: !v
+                              } deriving (Show,Eq,Functor,Generic)
+makeLenses ''LeafData
+instance (NFData v, NFData r) => NFData (LeafData v r)
+
+--------------------------------------------------------------------------------
+
+-- | Segment tree on a Fixed set of endpoints
+newtype SegmentTree v r =
+  SegmentTree { _unSegmentTree :: BinLeafTree (NodeData v r) (LeafData v r) }
+    deriving (Show,Eq,Generic,NFData)
+makeLenses ''SegmentTree
+
+
+-- rangeOf :: BinLeafTree (NodeData v r) (LeafData v r) -> Range (UnBounded r)
+-- rangeOf (Node _ x _) = Val <$> x^.range
+-- rangeOf (Leaf x)     = case x^.atomicRange of
+--                         Singleton r -> ClsoedRange (Val r)     (Val r)
+--                         AtomicRange -> OpenRange   MinInfinity MaxInfinity
+
+
+data BuildLeaf a = LeafSingleton a | LeafRange a a deriving (Show,Eq)
+
+-- | Given a sorted list of endpoints, without duplicates, construct a segment tree
+--
+--
+-- \(O(n)\) time
+createTree                      :: NonEmpty r -> v -> SegmentTree v r
+-- createTree (r NonEmpty.:| []) v = SegmentTree . Leaf $ LeafData (Singleton r) v
+createTree pts                v = SegmentTree . fmap h . foldUpData f g . fmap _unElem
+                                . asBalancedBinLeafTree $ ranges
+  where
+    h (LeafSingleton r) = LeafData (Singleton r) v
+    h (LeafRange   _ _) = LeafData AtomicRange   v
+
+    f l _ r = let m  = l^.range.upper
+                  ll = l^.range.lower
+                  rr = r^.range.upper
+              in NodeData m (Range ll rr) v
+    -- | Singletons map to closed singleton ranges, Ranges map to open ranges
+    g (LeafSingleton r) = NodeData (Closed r) (ClosedRange r r) v
+    g (LeafRange   s r) = NodeData (Open r)   (OpenRange   s r) v
+
+
+    ranges  = interleave (fmap LeafSingleton pts) ranges'
+    ranges' = zipWith LeafRange (NonEmpty.toList pts) (NonEmpty.tail pts)
+
+
+
+
+-- | Interleaves the two lists
+--
+-- >>> interleave (NonEmpty.fromList ["0","1","2"]) ["01","12"]
+-- "0" :| ["01","1","12","2"]
+interleave                       :: NonEmpty a -> [a] -> NonEmpty a
+interleave (x NonEmpty.:| xs) ys = x NonEmpty.:| concat (zipWith (\a b -> [a,b]) ys xs)
+
+
+-- | Build a SegmentTree
+--
+-- \(O(n \log n)\)
+fromIntervals      :: (Ord r, Eq p, Assoc v i, IntervalLike i, Monoid v, NumType i ~ r)
+                   => (Interval p r -> i)
+                   -> NonEmpty (Interval p r) -> SegmentTree v r
+fromIntervals f is = foldr (insert . f) (createTree pts mempty) is
+  where
+    endPoints (toRange -> Range' a b) = [a,b]
+    pts = nub' . NonEmpty.sort . NonEmpty.fromList . concatMap endPoints $ is
+    nub' = fmap NonEmpty.head . NonEmpty.group1
+
+-- | lists all intervals
+toList :: SegmentTree v r -> [i]
+toList = undefined
+
+--------------------------------------------------------------------------------
+-- * Searching
+
+-- | Search for all intervals intersecting x
+--
+-- \(O(\log n + k)\) where \(k\) is the output size
+search   :: (Ord r, Monoid v) => r -> SegmentTree v r -> v
+search x = mconcat . stab x
+
+
+
+inAtomicRange                   :: Eq r => r -> AtomicRange r -> Bool
+x `inAtomicRange` (Singleton r) = x == r
+_ `inAtomicRange` AtomicRange   = True
+
+
+-- | Returns the associated values of the nodes on the search path to x
+--
+-- \(O(\log n)\)
+stab                   :: Ord r => r -> SegmentTree v r -> [v]
+stab x (SegmentTree t) = stabRoot t
+  where
+    stabRoot (Leaf (LeafData rr v))
+      | x `inAtomicRange` rr = [v]
+      | otherwise            = []
+    stabRoot (Node l (NodeData m rr v) r) = case (x `inRange` rr, Closed x <= m) of
+      (False,_)   -> []
+      (True,True) -> v : stab' l
+      _           -> v : stab' r
+
+    stab' (Leaf (LeafData rr v))      | x `inAtomicRange` rr = [v]
+                                      | otherwise            = []
+    stab' (Node l (NodeData m _ v) r) | Closed x <= m        = v : stab' l
+                                      | otherwise            = v : stab' r
+
+
+--------------------------------------------------------------------------------
+-- * Inserting intervals
+
+-- | Class for associcated data structures
+class Measured v i => Assoc v i where
+  insertAssoc :: i -> v -> v
+  deleteAssoc :: i -> v -> v
+
+-- | Gets the range associated with this node
+getRange  :: BinLeafTree (NodeData v r) (LeafData t r) -> Maybe (Range r)
+getRange (Leaf (LeafData (Singleton r) _)) = Just $ Range (Closed r) (Closed r)
+getRange (Leaf _)                          = Nothing
+getRange (Node _ nd _)                     = Just $ nd^.range
+
+coversAtomic                      :: Ord r
+                                  => Range r -> Range r -> AtomicRange r -> Bool
+coversAtomic ri _   (Singleton r) = r `inRange` ri
+coversAtomic ri inR AtomicRange   = ri `covers` inR
+
+-- | Pre: the interval should have one of the endpoints on which the tree is built.
+insert                   :: (Assoc v i, NumType i ~ r, Ord r, IntervalLike i)
+                         => i -> SegmentTree v r -> SegmentTree v r
+insert i (SegmentTree t) = SegmentTree $ insertRoot t
+  where
+    ri@(Range a b) = toRange i
+    insertRoot t' = maybe t' (flip insert' t') $ getRange t'
+
+    insert' inR         lf@(Leaf nd@(LeafData rr _))
+      | coversAtomic ri inR rr = Leaf $ nd&leafAssoc %~ insertAssoc i
+      | otherwise              = lf
+    insert' (Range c d) (Node l nd@(NodeData m rr _) r)
+      | ri `covers` rr       = Node l (nd&assoc %~ insertAssoc i) r
+      | otherwise            = Node l' nd r'
+      where
+           -- check if the range intersects the range of the left subtree
+        l'  = if a <= m then insert' (Range c          m) l else l
+        r'  = if m < b  then insert' (Range (toOpen m) d) r else r
+
+    toOpen = Open . view unEndPoint
+
+-- | Delete an interval from the tree
+--
+-- pre: The segment is in the tree!
+delete :: (Assoc v i, NumType i ~ r, Ord r, IntervalLike i)
+          => i -> SegmentTree v r -> SegmentTree v r
+delete i (SegmentTree t) = SegmentTree $ delete' t
+  where
+    (Range a b) = toRange i
+
+    delete' (Leaf ld) = Leaf $ ld&leafAssoc %~ deleteAssoc i
+    delete' (Node l nd@(_splitPoint -> m) r)
+      | b <= m    = Node (delete' l) (nd&assoc %~ deleteAssoc i) r
+      | otherwise = Node l           (nd&assoc %~ deleteAssoc i) (delete' r)
+
+    -- delete'' (Leaf ld)     = Leaf $ ld&leafAssoc %~ deleteAssoc i
+    -- delete'' (Node l nd r) = Node l (nd&assoc %~ deleteAssoc i) r
+
+    -- deleteL (Leaf ld)     = Leaf $ ld&leafAssoc %~ deleteAssoc i
+    -- deleteL (Node l nd@(_splitPoint -> m) r)
+    --   | a <= m    = Node (deleteL l) (nd&assoc %~ deleteAssoc i) (delete'' r)
+    --   | otherwise = Node l nd (deleteL r)
+
+    -- deleteR (Leaf ld)     = Leaf $ ld&leafAssoc %~ deleteAssoc i
+    -- deleteR (Node l nd@(_splitPoint -> m) r)
+    --   | m <= b    = Node (delete'' l) (nd&assoc %~ deleteAssoc i) (deleteR r)
+    --   | otherwise = Node (deleteR l) nd r
+
+
+--------------------------------------------------------------------------------
+-- * Listing the intervals stabbed
+
+-- | Interval
+newtype I a = I { _unI :: a} deriving (Show,Read,Eq,Ord,Generic,NFData)
+
+type instance NumType (I a) = NumType a
+
+instance Measured [I a] (I a) where
+  measure = (:[])
+
+instance Eq a => Assoc [I a] (I a) where
+  insertAssoc = (:)
+  deleteAssoc = List.delete
+
+-- instance Measured [Interval p r] (Interval p r) where
+--   measure = (:[])
+
+-- instance (Eq p, Eq r) => Assoc [Interval p r] (Interval p r) where
+--   insertAssoc = (:)
+--   deleteAssoc = List.delete
+
+
+instance IntervalLike a => IntervalLike (I a) where
+  toRange = toRange . _unI
+
+
+fromIntervals' :: (Eq p, Ord r)
+               => NonEmpty (Interval p r) -> SegmentTree [I (Interval p r)] r
+fromIntervals' = fromIntervals I
+
+
+--------------------------------------------------------------------------------
+-- * Counting the number of segments intersected
+
+newtype Count = Count { getCount :: Int}
+              deriving (Show,Eq,Ord,Num,Integral,Enum,Real,Generic,NFData)
+
+newtype C a = C { _unC :: a} deriving (Show,Read,Eq,Ord,Generic,NFData)
+
+instance Semigroup Count where
+  a <> b = Count $ getCount a + getCount b
+
+instance Measured Count (C i) where
+  measure _ = 1
+
+instance Assoc Count (C i) where
+  insertAssoc _ v = v + 1
+  deleteAssoc _ v = v - 1
+
+
+--------------------------------------------------------------------------------
+-- * Testing stuff
+
+test'' = fromIntervals' . NonEmpty.fromList $ test
+test = [Interval (Closed (238 :+ ())) (Open (309 :+ ())), Interval (Closed (175 :+ ())) (Closed (269 :+ ())),Interval (Closed (255 :+ ())) (Open (867 :+ ())),Interval (Open (236 :+ ())) (Closed (863 :+ ())),Interval (Open (150 :+ ())) (Closed (161 :+ ())),Interval (Closed (35 :+ ())) (Closed (77 :+ ()))]
+
+
+-- q =        [78]
+
+-- test = fromIntervals' . NonEmpty.fromList $ [ closedInterval 0 10
+--                                             , closedInterval 5 15
+--                                             , closedInterval 1 4
+--                                             , closedInterval 3 9
+--                                             ]
+tst = fromIntervals' . NonEmpty.fromList $ [ closedInterval 1 6
+                                           , closedInterval 2 6
+                                           -- , Interval (Closed $ ext 0) (Open $ ext 1)
+                                           ]
+
+
+
+closedInterval a b = ClosedInterval (ext a) (ext b)
+
+showT :: (Show r, Show v) => SegmentTree v r -> String
+showT = drawTree . _unSegmentTree
+
+
+test' :: (Show r, Num r, Ord r, Enum r) => SegmentTree [I (Interval () r)] r
+test' = insert (I $ closedInterval 6 14) $ createTree (NonEmpty.fromList [2,4..20]) []
diff --git a/src/Data/Geometry/Slab.hs b/src/Data/Geometry/Slab.hs
--- a/src/Data/Geometry/Slab.hs
+++ b/src/Data/Geometry/Slab.hs
@@ -2,25 +2,20 @@
 {-# Language TemplateHaskell #-}
 module Data.Geometry.Slab where
 
-
-import           Control.Applicative
-import           Control.Lens(makeLenses, (^.),(%~),(.~),(&), Lens', both)
-import           Data.Bitraversable
+import           Control.Lens (makeLenses, (^.),(%~),(.~),(&), both)
+import           Data.Bifunctor
 import           Data.Ext
 import qualified Data.Foldable as F
-import           Data.Geometry.Properties
-import           Data.Geometry.Interval
-import           Data.Geometry.Point
 import           Data.Geometry.Box.Internal
-import           Data.Geometry.LineSegment
+import           Data.Geometry.Interval
 import           Data.Geometry.Line
+import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
+import           Data.Geometry.Properties
 import           Data.Geometry.SubLine
-import           Data.Range
-import           Data.Semigroup
 import qualified Data.Traversable as T
 import           Data.Vinyl
 import           Frames.CoRec
-import           Data.Bifunctor
 
 --------------------------------------------------------------------------------
 
@@ -72,7 +67,7 @@
 instance (Slab Horizontal a r) `IsIntersectableWith` (Slab Vertical a r) where
   nonEmptyIntersection _ _ _ = True
 
-  (Slab h) `intersect` (Slab v) = coRec $ fromCornerPoints low high
+  (Slab h) `intersect` (Slab v) = coRec $ box low high
     where
       low  = point2 (v^.start.core) (h^.start.core) :+ (v^.start.extra, h^.start.extra)
       high = point2 (v^.end.core)   (h^.end.core)   :+ (v^.end.extra,   h^.end.extra)
@@ -127,11 +122,26 @@
 
 
 type instance IntersectionOf (SubLine 2 p r) (Slab o a r) =
-  [NoIntersection, SubLine 2 a r, LineSegment 2 a r]
+  [NoIntersection, SubLine 2 () r]
 
 instance (Fractional r, Ord r, HasBoundingLines o) =>
          SubLine 2 a r `IsIntersectableWith` (Slab o a r) where
 
   nonEmptyIntersection = defaultNonEmptyIntersection
 
-  (SubLine l r) `intersect` (Slab i) = undefined
+  sl@(SubLine l _) `intersect` s = match (l `intersect` s) $
+       (H $ \NoIntersection -> coRec NoIntersection)
+    :& (H $ \(Line _ _)     -> coRec $ dropExtra sl)
+    :& (H $ \seg            -> match (sl `intersect` (seg^._SubLine)) $
+                                    (H $ \NoIntersection -> coRec NoIntersection)
+                                 :& (H $ \p@(Point2 _ _) -> coRec $ singleton p)
+                                 :& (H $ \ss             -> coRec $ dropExtra ss)
+                                 :& RNil)
+    :& RNil
+    where
+      dropExtra sub = sub&subRange %~ bimap (const ()) id
+      singleton p = let x = ext $ toOffset p l in SubLine l (ClosedInterval x x)
+
+
+test :: SubLine 2 () Double
+test = (ClosedLineSegment (ext origin) (ext origin))^._SubLine
diff --git a/src/Data/Geometry/SubLine.hs b/src/Data/Geometry/SubLine.hs
--- a/src/Data/Geometry/SubLine.hs
+++ b/src/Data/Geometry/SubLine.hs
@@ -1,20 +1,18 @@
 {-# LANGUAGE TemplateHaskell  #-}
 module Data.Geometry.SubLine where
 
-import           Control.Applicative
+import           Control.Lens
+import           Data.Ext
 import qualified Data.Foldable as F
+import           Data.Geometry.Interval
+import           Data.Geometry.Line.Internal
+import           Data.Geometry.Point
+import           Data.Geometry.Properties
+import           Data.Geometry.Vector
 import qualified Data.Traversable as T
-import Control.Lens
-import Data.Ext
-import Data.Geometry.Interval
-import Data.Geometry.Line.Internal
-import Data.Geometry.Point
-import Data.Geometry.Properties
-import Data.Geometry.Vector
-import Data.Range
-import Data.UnBounded
+import           Data.UnBounded
+import           Data.Vinyl
 import           Frames.CoRec
-import Data.Vinyl
 
 --------------------------------------------------------------------------------
 
diff --git a/src/Data/Geometry/Transformation.hs b/src/Data/Geometry/Transformation.hs
--- a/src/Data/Geometry/Transformation.hs
+++ b/src/Data/Geometry/Transformation.hs
@@ -7,19 +7,15 @@
 {-# LANGUAGE DeriveFunctor  #-}
 module Data.Geometry.Transformation where
 
-
-import           Control.Applicative
-import           Control.Lens(lens,Lens',set)
+import           Control.Lens (lens,Lens',set)
 import           Data.Geometry.Point
 import           Data.Geometry.Properties
 import           Data.Geometry.Vector
+import qualified Data.Geometry.Vector as V
 import           Data.Proxy
-
-import           GHC.TypeLits
-import           Linear.Matrix((!*),(!*!))
-
 import qualified Data.Vector.Fixed as FV
-import qualified Data.Geometry.Vector as V
+import           GHC.TypeLits
+import           Linear.Matrix ((!*),(!*!))
 
 --------------------------------------------------------------------------------
 -- * Matrices
diff --git a/src/Data/Geometry/Vector.hs b/src/Data/Geometry/Vector.hs
--- a/src/Data/Geometry/Vector.hs
+++ b/src/Data/Geometry/Vector.hs
@@ -7,15 +7,16 @@
                            , scalarMultiple
                            ) where
 
-import           Data.Monoid
-import qualified Data.Foldable                    as F
+import qualified Data.Foldable as F
 import           Data.Geometry.Vector.VectorFixed
 import           Data.Geometry.Vector.VectorFixed as GV
 import           Data.Maybe
-import qualified Data.Vector.Fixed                as FV
-import           Linear.Affine(Affine(..), qdA, distanceA)
-import           Linear.Metric(dot,norm)
+import qualified Data.Vector.Fixed as FV
+import           Linear.Affine (Affine(..), qdA, distanceA)
+import           Linear.Metric (dot,norm)
 import           Linear.Vector as LV
+
+--------------------------------------------------------------------------------
 
 
 -- | Test if v is a scalar multiple of u.
diff --git a/src/Data/Geometry/Vector/VectorFixed.hs b/src/Data/Geometry/Vector/VectorFixed.hs
--- a/src/Data/Geometry/Vector/VectorFixed.hs
+++ b/src/Data/Geometry/Vector/VectorFixed.hs
@@ -1,27 +1,19 @@
-{-# LANGUAGE MultiParamTypeClasses  #-}
 {-# LANGUAGE ScopedTypeVariables  #-}
 {-# LANGUAGE UndecidableInstances #-}
 module Data.Geometry.Vector.VectorFixed where
 
-import           Data.Monoid
-import           Control.Applicative
+import           Control.DeepSeq
 import           Control.Lens
-import           Data.Foldable
-import           Data.Traversable
-
+import qualified Data.Foldable as F
+import qualified Data.Vector.Fixed as V
 import           Data.Vector.Fixed.Boxed
-import           Data.Vector.Fixed.Cont(Z, S, ToPeano)
-
+import           Data.Vector.Fixed.Cont (Z, S, ToPeano)
+import           GHC.Generics (Generic)
 import           GHC.TypeLits
-
-import           Linear.Affine
+import           Linear.Affine (Affine(..))
 import           Linear.Metric
-import           Linear.Vector
-
-import qualified Data.Vector.Fixed as V
-
 import qualified Linear.V3 as L3
-
+import           Linear.Vector
 
 --------------------------------------------------------------------------------
 
@@ -34,7 +26,7 @@
 -- | Datatype representing d dimensional vectors. Our implementation wraps the
 -- implementation provided by fixed-vector.
 newtype Vector (d :: Nat)  (r :: *) = Vector { _unV :: Vec (ToPeano d) r }
-
+                                    deriving (Generic)
 
 unV :: Lens' (Vector d r) (Vec (ToPeano d) r)
 unV = lens _unV (const Vector)
@@ -59,9 +51,16 @@
   | otherwise                                     = pure v
 
 
+vectorFromList :: Arity d => [a] -> Maybe (Vector d a)
+vectorFromList = fmap Vector . V.fromListM
+
+vectorFromListUnsafe :: Arity d => [a] -> Vector d a
+vectorFromListUnsafe = Vector . V.fromList
+
+
 instance (Show r, Arity d) => Show (Vector d r) where
   show (Vector v) = mconcat [ "Vector", show $ V.length v , " "
-                            , show $ toList v
+                            , show $ F.toList v
                             ]
 
 deriving instance (Eq r, Arity d)   => Eq (Vector d r)
@@ -74,7 +73,9 @@
 instance Arity d => Traversable (Vector d) where
   traverse f (Vector v) = Vector <$> traverse f v
 
+deriving instance (Arity d, NFData r) => NFData (Vector d r)
 
+
 instance Arity d => Additive (Vector d) where
   zero = pure 0
   (Vector u) ^+^ (Vector v) = Vector $ V.zipWith (+) u v
@@ -144,7 +145,7 @@
 instance Prefix Z d where
   prefix' _ = V.vector V.empty
 
-instance (V.Arity i, V.Arity d, V.Index i d, Prefix i d) => Prefix (S i) (S d) where
+instance (V.Arity i, V.Arity d, Prefix i d) => Prefix (S i) (S d) where
   prefix' v = V.vector $ V.head v `V.cons` (prefix' $ V.tail v)
 
 
diff --git a/src/Data/Permutation.hs b/src/Data/Permutation.hs
--- a/src/Data/Permutation.hs
+++ b/src/Data/Permutation.hs
@@ -54,6 +54,8 @@
 
 -- | Lookup the indices of an element, i.e. in which orbit the item is, and the
 -- index within the orbit.
+--
+-- runnign time: \(O(1)\)
 lookupIdx        :: Enum a => Permutation a -> a -> (Int,Int)
 lookupIdx perm x = perm^.indexes.ix' (fromEnum x)
 
diff --git a/src/Data/PlanarGraph.hs b/src/Data/PlanarGraph.hs
--- a/src/Data/PlanarGraph.hs
+++ b/src/Data/PlanarGraph.hs
@@ -1,4 +1,5 @@
 {-# LANGUAGE TemplateHaskell #-}
+{-# LANGUAGE ScopedTypeVariables #-}
 module Data.PlanarGraph( Arc(..)
                        , Direction(..), rev
 
@@ -15,7 +16,7 @@
                        , embedding, vertexData, dartData, faceData
                        , edgeData
 
-                       , planarGraph, planarGraph'
+                       , planarGraph, planarGraph', fromAdjacencyLists
 
                        , numVertices, numDarts, numEdges, numFaces
                        , darts', darts, edges', edges, vertices', vertices, faces', faces
@@ -29,16 +30,38 @@
                        , dual
 
                        , FaceId(..)
-                       , leftFace, rightFace, boundary
+                       , leftFace, rightFace, boundary, boundaryVertices
+
+
+
+                       , EdgeOracle
+                       , edgeOracle, buildEdgeOracle
+                       , findEdge
+                       , hasEdge
                        ) where
 
+import           Control.Applicative (Alternative(..))
 import           Control.Lens
 import           Control.Monad (forM_)
+import           Control.Monad.ST (ST)
+import           Control.Monad.State.Strict
+import           Data.Bifunctor
+import           Data.Bitraversable
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Maybe (catMaybes, isJust)
 import           Data.Permutation
+import           Data.Semigroup (Semigroup(..))
+import           Data.Traversable (fmapDefault,foldMapDefault)
+import           Data.Util
 import qualified Data.Vector as V
-import qualified Data.Foldable as F
+import qualified Data.Vector.Generic as GV
 import qualified Data.Vector.Mutable as MV
+import qualified Data.Vector.Unboxed as UV
+import qualified Data.Vector.Unboxed.Mutable as UMV
 
+
+import           Debug.Trace
 --------------------------------------------------------------------------------
 
 --------------------------------------------------------------------------------
@@ -47,23 +70,23 @@
 -- let dart i s = Dart (Arc i) (read s)
 --     (aA:aB:aC:aD:aE:aG:_) = take 6 [Arc 0..]
 --     myGraph :: PlanarGraph Test Primal_ () String ()
---     myGraph = planarGraph' [ [ (Dart aA Negative, "a-")
---                              , (Dart aC Positive, "c+")
---                              , (Dart aB Positive, "b+")
---                              , (Dart aA Positive, "a+")
---                              ]
---                            , [ (Dart aE Negative, "e-")
---                              , (Dart aB Negative, "b-")
---                              , (Dart aD Negative, "d-")
---                              , (Dart aG Positive, "g+")
---                              ]
---                            , [ (Dart aE Positive, "e+")
---                              , (Dart aD Positive, "d+")
---                              , (Dart aC Negative, "c-")
---                              ]
---                            , [ (Dart aG Negative, "g-")
---                              ]
---                            ]
+--     myGraph = planarGraph [ [ (Dart aA Negative, "a-")
+--                             , (Dart aC Positive, "c+")
+--                             , (Dart aB Positive, "b+")
+--                             , (Dart aA Positive, "a+")
+--                             ]
+--                           , [ (Dart aE Negative, "e-")
+--                             , (Dart aB Negative, "b-")
+--                             , (Dart aD Negative, "d-")
+--                             , (Dart aG Positive, "g+")
+--                             ]
+--                           , [ (Dart aE Positive, "e+")
+--                             , (Dart aD Positive, "d+")
+--                             , (Dart aC Negative, "c-")
+--                             ]
+--                           , [ (Dart aG Negative, "g-")
+--                             ]
+--                           ]
 -- :}
 
 -- TODO: Add a fig. of the Graph
@@ -145,6 +168,7 @@
 -- by the phantom type s, and to a particular world w.
 newtype VertexId s (w :: World) = VertexId { _unVertexId :: Int } deriving (Eq,Ord,Enum)
 -- VertexId's are in the range 0...|orbits|-1
+makeLenses ''VertexId
 
 instance Show (VertexId s w) where
   show (VertexId i) = "VertexId " ++ show i
@@ -191,8 +215,8 @@
 
 
 -- | Construct a planar graph
-planarGraph      :: Permutation (Dart s) -> PlanarGraph s Primal_ () () ()
-planarGraph perm = PlanarGraph perm vData eData fData
+planarGraph'      :: Permutation (Dart s) -> PlanarGraph s w () () ()
+planarGraph' perm = PlanarGraph perm vData eData fData
   where
     d = size perm
     e = d `div` 2
@@ -205,16 +229,127 @@
 
 
 
--- | Construct a planar graph.
+-- | Construct a planar graph, given the darts in cyclic order around each
+-- vertex.
 --
-planarGraph'    :: [[(Dart s,e)]] -> PlanarGraph s Primal_ () e ()
-planarGraph' ds = (planarGraph perm)&dartData .~ (V.fromList . concat $ ds)
+-- running time: \(O(n)\).
+planarGraph    :: [[(Dart s,e)]] -> PlanarGraph s Primal_ () e ()
+planarGraph ds = (planarGraph' perm)&dartData .~ (V.fromList . concat $ ds)
   where
     n     = sum . map length $ ds
     perm  = toCycleRep n $ map (map fst) ds
 
 
 
+-- | Construct a planar graph from a adjacency matrix. For every vertex, all
+-- vertices should be given in counter clockwise order.
+--
+-- running time: \(O(n)\).
+fromAdjacencyLists      :: forall s w f. (Foldable f, Functor f)
+                        => [(VertexId s w, f (VertexId s w))]
+                        -> PlanarGraph s w () () ()
+fromAdjacencyLists adjM = planarGraph' . toCycleRep n $ perm
+  where
+    n    = sum . fmap length $ perm
+    perm = map toOrbit  $ adjM'
+
+    adjM' = fmap (second F.toList) adjM
+
+    -- -- | Assign Arcs
+    -- adjM' = (^._1) . foldr assignArcs (SP [] 0) $ adjM
+
+    -- Build an edgeOracle, so that we can query the arcId assigned to
+    -- an edge in O(1) time.
+    oracle :: EdgeOracle s w Int
+    oracle = fmap (^.core) . assignArcs . buildEdgeOracle
+           . map (second $ map ext)  $ adjM'
+
+    toOrbit (u,adjU) = concatMap (toDart u) adjU
+
+    -- if u = v we have a self-loop, so we add both a positive and a negative dart
+    toDart u v = let Just a = findEdge u v oracle
+                 in case u `compare` v of
+                      LT -> [Dart (Arc a) Positive]
+                      EQ -> [Dart (Arc a) Positive, Dart (Arc a) Negative]
+                      GT -> [Dart (Arc a) Negative]
+
+
+assignArcs   :: EdgeOracle s w e -> EdgeOracle s w (Int :+ e)
+assignArcs o = evalState (traverse f o) 0
+  where
+    f   :: e -> State Int (Int :+ e)
+    f e = do i <- get ; put (i+1) ; pure (i :+ e)
+
+
+
+
+--     -- Go through all of the edges we find an edge (u,v), with u <= v,
+--     -- assign an ArcId to this edge (and increment the next available arcId).
+--     -- if u > v. Don't assign anything.
+-- assignArcs                     :: (Ord v, Foldable f)
+--                                    => (v, f v)
+--                                    -> SP [(v, [v :+ Maybe Int])] Int
+--                                    -> SP [(v, [v :+ Maybe Int])] Int
+-- assignArcs (u,adjU) (SP acc i) = first (\adjU' -> (u,adjU'):acc)
+--                                    . foldr (assignArcs' u) (SP [] i)
+--                                    . F.toList $ adjU
+
+--     -- Go through all edges that have u as one endpoint.
+-- assignArcs'   :: Ord v => v -> v -> SP [v :+ Maybe Int] Int
+--                                  -> SP [v :+ Maybe Int] Int
+-- assignArcs' u v (SP acc i)
+--       | u <= v    = SP ((v :+ Just i)  : acc) (i+1)
+--       | otherwise = SP ((v :+ Nothing) : acc) i
+
+
+
+
+
+
+
+-- -- - m: a Map, mapping edges, represented by a pair of vertexId's (u,v) with
+-- --            u < v, to arcId's.
+-- -- - a: the next available unused arcID
+-- -- - x: the data value we are interested in computing
+-- type STR' s b = STR (SM.Map (VertexId s Primal_,VertexId s Primal_) Int) Int b
+
+-- -- | Construct a planar graph from a adjacency matrix. For every vertex, all
+-- -- vertices should be given in counter clockwise order.
+-- --
+-- -- running time: \(O(n \log n)\).
+-- fromAdjacencyLists      :: forall s.
+--                            [(VertexId s Primal_, C.CList (VertexId s Primal_))]
+--                         -> PlanarGraph s Primal_ () () ()
+-- fromAdjacencyLists adjM = planarGraph' . toCycleRep n $ perm
+--   where
+--     n    = sum . fmap length $ adjM
+--     perm = trd' . foldr toOrbit (STR mempty 0 mempty) $ adjM
+
+
+--     -- | Given a vertex with its adjacent vertices (u,vs) (in CCW order) convert this
+--     -- vertex with its adjacent vertices into an Orbit
+--     toOrbit                     :: (VertexId s Primal_, C.CList (VertexId s Primal_))
+--                                 -> STR' s [[Dart s]]
+--                                 -> STR' s [[Dart s]]
+--     toOrbit (u,vs) (STR m a dss) =
+--       let (STR m' a' ds') = foldr (toDart . (u,)) (STR m a mempty) . C.toList $ vs
+--       in STR m' a' (ds':dss)
+
+
+--     -- | Given an edge (u,v) and a triplet (m,a,ds) we construct a new dart
+--     -- representing this edge.
+--     toDart                    :: (VertexId s Primal_,VertexId s Primal_)
+--                               -> STR' s [Dart s]
+--                               -> STR' s [Dart s]
+--     toDart (u,v) (STR m a ds) = let dir = if u < v then Positive else Negative
+--                                     t'  = (min u v, max u v)
+--                                in case SM.lookup t' m of
+--       Just a' -> STR m                  a     (Dart (Arc a') dir : ds)
+--       Nothing -> STR (SM.insert t' a m) (a+1) (Dart (Arc a)  dir : ds)
+
+
+--------------------------------------------------------------------------------
+
 -- | Get the number of vertices
 --
 -- >>> numVertices myGraph
@@ -305,19 +440,26 @@
 
 -- | The tail of a dart, i.e. the vertex this dart is leaving from
 --
+-- running time: \(O(1)\)
 tailOf     :: Dart s -> PlanarGraph s w v e f -> VertexId s w
 tailOf d g = VertexId . fst $ lookupIdx (g^.embedding) d
 
 -- | The vertex this dart is heading in to
+--
+-- running time: \(O(1)\)
 headOf   :: Dart s -> PlanarGraph s w v e f -> VertexId s w
 headOf d = tailOf (twin d)
 
 -- | endPoints d g = (tailOf d g, headOf d g)
+--
+-- running time: \(O(1)\)
 endPoints :: Dart s -> PlanarGraph s w v e f -> (VertexId s w, VertexId s w)
 endPoints d g = (tailOf d g, headOf d g)
 
 
 -- | All edges incident to vertex v, in counterclockwise order around v.
+--
+-- running time: \(O(k)\), where \(k\) is the output size
 incidentEdges                :: VertexId s w -> PlanarGraph s w v e f
                              -> V.Vector (Dart s)
 incidentEdges (VertexId v) g = g^.embedding.orbits.ix' v
@@ -333,6 +475,8 @@
 
 -- | Gets the neighbours of a particular vertex, in counterclockwise order
 -- around the vertex.
+--
+-- running time: \(O(k)\), where \(k\) is the output size
 neighboursOf     :: VertexId s w -> PlanarGraph s w v e f -> V.Vector (VertexId s w)
 neighboursOf v g = otherVtx <$> incidentEdges v g
   where
@@ -350,14 +494,20 @@
 
 -- | Get the vertex data associated with a node. Note that updating this data may be
 -- expensive!!
+--
+-- running time: \(O(1)\)
 vDataOf              :: VertexId s w -> Lens' (PlanarGraph s w v e f) v
 vDataOf (VertexId i) = vertexData.ix' i
 
 -- | Edge data of a given dart
+--
+-- running time: \(O(1)\)
 eDataOf   :: Dart s -> Lens' (PlanarGraph s w v e f) e
 eDataOf d = rawDartData.ix' (fromEnum d)
 
 -- | Data of a face of a given face
+--
+-- running time: \(O(1)\)
 fDataOf                       :: FaceId s w -> Lens' (PlanarGraph s w v e f) f
 fDataOf (FaceId (VertexId i)) = faceData.ix' i
 
@@ -368,6 +518,8 @@
 
 
 -- | Data corresponding to the endpoints of the dart
+--
+-- running time: \(O(1)\)
 endPointData     :: Dart s -> PlanarGraph s w v e f -> (v,v)
 endPointData d g = let (u,v) = endPoints d g in (g^.vDataOf u, g^.vDataOf v)
 
@@ -386,6 +538,8 @@
 --  in (dual myGraph)^.embedding.orbits == answer
 -- :}
 -- True
+--
+-- running time: \(O(n)\).
 dual   :: PlanarGraph s w v e f -> PlanarGraph s (Dual w) f e v
 dual g = let perm = g^.embedding
          in PlanarGraph (cycleRep (elems perm) (apply perm . twin))
@@ -417,6 +571,8 @@
 -- FaceId 2
 -- >>> leftFace (dart 0 "+1") myGraph
 -- FaceId 0
+--
+-- running time: \(O(1)\).
 leftFace     :: Dart s -> PlanarGraph s w v e f -> FaceId s w
 leftFace d g = FaceId . headOf d $ dual g
 
@@ -431,6 +587,8 @@
 -- FaceId 1
 -- >>> rightFace (dart 0 "+1") myGraph
 -- FaceId 1
+--
+-- running time: \(O(1)\).
 rightFace     :: Dart s -> PlanarGraph s w v e f -> FaceId s w
 rightFace d g = FaceId . tailOf d $ dual g
 
@@ -439,54 +597,203 @@
 -- the outer face in counter clockwise order.
 --
 --
-boundary     :: FaceId s w -> PlanarGraph s w v e f -> V.Vector (Dart s)
+-- running time: \(O(k)\), where \(k\) is the output size.
+boundary              :: FaceId s w -> PlanarGraph s w v e f -> V.Vector (Dart s)
 boundary (FaceId v) g = incidentEdges v $ dual g
 
 
-
+-- | The vertices bounding this face, for internal faces in clockwise order, for
+-- the outer face in counter clockwise order.
+--
+--
+-- running time: \(O(k)\), where \(k\) is the output size.
+boundaryVertices     :: FaceId s w -> PlanarGraph s w v e f -> V.Vector (VertexId s w)
+boundaryVertices f g = fmap (flip tailOf g) $ boundary f g
 
 --------------------------------------------------------------------------------
 -- Testing stuff
 
-testPerm :: Permutation (Dart s)
-testPerm = let (a:b:c:d:e:g:_) = take 6 [Arc 0..]
-           in toCycleRep 12 [ [ Dart a Negative
-                              , Dart c Positive
-                              , Dart b Positive
-                              , Dart a Positive
-                              ]
-                            , [ Dart e Negative
-                              , Dart b Negative
-                              , Dart d Negative
-                              , Dart g Positive
-                              ]
-                            , [ Dart e Positive
-                              , Dart d Positive
-                              , Dart c Negative
-                              ]
-                            , [ Dart g Negative
-                              ]
-                            ]
+-- testPerm :: Permutation (Dart s)
+-- testPerm = let (a:b:c:d:e:g:_) = take 6 [Arc 0..]
+--            in toCycleRep 12 [ [ Dart a Negative
+--                               , Dart c Positive
+--                               , Dart b Positive
+--                               , Dart a Positive
+--                               ]
+--                             , [ Dart e Negative
+--                               , Dart b Negative
+--                               , Dart d Negative
+--                               , Dart g Positive
+--                               ]
+--                             , [ Dart e Positive
+--                               , Dart d Positive
+--                               , Dart c Negative
+--                               ]
+--                             , [ Dart g Negative
+--                               ]
+--                             ]
 
 data Test
 
-testG :: PlanarGraph Test Primal_ () String ()
-testG = planarGraph' [ [ (Dart aA Negative, "a-")
-                       , (Dart aC Positive, "c+")
-                       , (Dart aB Positive, "b+")
-                       , (Dart aA Positive, "a+")
-                       ]
-                     , [ (Dart aE Negative, "e-")
-                       , (Dart aB Negative, "b-")
-                       , (Dart aD Negative, "d-")
-                       , (Dart aG Positive, "g+")
-                       ]
-                     , [ (Dart aE Positive, "e+")
-                       , (Dart aD Positive, "d+")
-                       , (Dart aC Negative, "c-")
-                       ]
-                     , [ (Dart aG Negative, "g-")
-                       ]
-                     ]
+-- testG :: PlanarGraph Test Primal_ () String ()
+-- testG = planarGraph' [ [ (Dart aA Negative, "a-")
+--                        , (Dart aC Positive, "c+")
+--                        , (Dart aB Positive, "b+")
+--                        , (Dart aA Positive, "a+")
+--                        ]
+--                      , [ (Dart aE Negative, "e-")
+--                        , (Dart aB Negative, "b-")
+--                        , (Dart aD Negative, "d-")
+--                        , (Dart aG Positive, "g+")
+--                        ]
+--                      , [ (Dart aE Positive, "e+")
+--                        , (Dart aD Positive, "d+")
+--                        , (Dart aC Negative, "c-")
+--                        ]
+--                      , [ (Dart aG Negative, "g-")
+--                        ]
+--                      ]
+--   where
+--     (aA:aB:aC:aD:aE:aG:_) = take 6 [Arc 0..]
+
+
+--------------------------------------------------------------------------------
+
+
+
+-- type ArcID = Int
+
+-- -- | ST' is a strict triple (m,a,x) containing:
+-- --
+-- -- - m: a Map, mapping edges, represented by a pair of vertexId's (u,v) with
+-- --            u < v, to arcId's.
+-- -- - a: the next available unused arcID
+-- -- - x: the data value we are interested in computing
+-- type ST' a = ST (SM.Map (VertexID,VertexID) ArcID) ArcID a
+
+
+--------------------------------------------------------------------------------
+-- * Testing Connectivity
+
+
+-- | Edge Oracle:
+--
+-- main idea: store adjacency lists in such a way that we store an edge (u,v)
+-- either in u's adjacency list or in v's. This can be done s.t. all adjacency
+-- lists have length at most 6.
+--
+-- note: Every edge is stored exactly once (i.e. either at u or at v, but not both)
+newtype EdgeOracle s w a =
+  EdgeOracle { _unEdgeOracle :: V.Vector (V.Vector (VertexId s w :+ a)) }
+                         deriving (Show,Eq)
+
+instance Functor (EdgeOracle s w) where
+  fmap = fmapDefault
+
+instance Foldable (EdgeOracle s w) where
+  foldMap = foldMapDefault
+
+instance Traversable (EdgeOracle s w) where
+  traverse f (EdgeOracle v) = EdgeOracle <$> traverse g v
+    where
+      -- g   :: V.Vector (VertexId :+ a) -> f (V.Vector (VertexId :+ b))
+      g = traverse (bitraverse pure f)
+
+
+edgeOracle   :: PlanarGraph s w v e f -> EdgeOracle s w ()
+edgeOracle g = buildEdgeOracle [ (v, ext <$> neighboursOf v g)
+                               | v <- F.toList $ vertices' g
+                               ]
+
+-- | Builds an edge oracle that can be used to efficiently test if two vertices
+-- are connected by an edge.
+--
+-- running time: \(O(n)\)
+buildEdgeOracle        :: forall f s w e. (Foldable f)
+                       => [(VertexId s w, f (VertexId s w :+ e))] -> EdgeOracle s w e
+buildEdgeOracle inAdj' = EdgeOracle $ V.create $ do
+                          counts <- UV.thaw initCounts
+                          marks  <- UMV.replicate (UMV.length counts) False
+                          outV   <- MV.new (UMV.length counts)
+                          build counts marks outV initQ
+                          pure outV
+    -- main idea: maintain a vector with counts; i.e. how many unprocessed
+    -- vertices are adjacent to u, and a bit vector with marks to keep track if
+    -- a vertex has been processed yet. When we process a vertex, we keep only
+    -- the adjacencies of unprocessed verticese.
   where
-    (aA:aB:aC:aD:aE:aG:_) = take 6 [Arc 0..]
+    -- Convert to a vector representation
+    inAdj = V.create $ do
+              mv <- MV.new (length inAdj')
+              forM_ inAdj' $ \(VertexId i,adjI) ->
+                MV.write mv i (V.fromList . F.toList $ adjI)
+              pure mv
+
+    initCounts = V.convert . fmap GV.length $ inAdj
+    -- initial vertices available for processing
+    initQ = GV.ifoldr (\i k q -> if k <= 6 then i : q else q) [] initCounts
+
+    -- | Construct the adjacencylist for vertex i. I.e. by retaining only adjacent
+    -- vertices that have not been processed yet.
+    extractAdj         :: UMV.MVector s' Bool -> Int
+                       -> ST s' (V.Vector (VertexId s w :+ e))
+    extractAdj marks i = let p = fmap not . UMV.read marks . (^.core.unVertexId)
+                         in GV.filterM  p $ inAdj V.! i
+
+    -- | Decreases the number of adjacencies that vertex j has
+    -- if it has <= 6 adjacencies left it has become available for processing
+    decrease                          :: UMV.MVector s' Int -> (VertexId s w :+ e')
+                                      -> ST s' (Maybe Int)
+    decrease counts (VertexId j :+ _) = do k <- UMV.read counts j
+                                           let k'  = k - 1
+                                           UMV.write counts j k'
+                                           pure $ if k' <= 6 then Just j else Nothing
+
+    -- The actual algorithm that builds the items
+    build :: UMV.MVector s' Int -> UMV.MVector s' Bool
+          -> MV.MVector s' (V.Vector (VertexId s w :+ e)) -> [Int] -> ST s' ()
+    build _      _     _    []    = pure ()
+    build counts marks outV (i:q) = do
+             b <- UMV.read marks i
+             nq <- if b then pure []
+                        else do
+                          adjI <- extractAdj marks i
+                          MV.write outV i adjI
+                          UMV.write marks i True
+                          V.toList <$> mapM (decrease counts) adjI
+             build counts marks outV (catMaybes nq <> q)
+
+
+
+-- | Test if u and v are connected by an edge.
+--
+-- running time: \(O(1)\)
+hasEdge     :: VertexId s w -> VertexId s w -> EdgeOracle s w a -> Bool
+hasEdge u v = isJust . findEdge u v
+
+
+-- | Find the edge data corresponding to edge (u,v) if such an edge exists
+--
+-- running time: \(O(1)\)
+findEdge :: VertexId s w -> VertexId s w -> EdgeOracle s w a -> Maybe a
+findEdge  (VertexId u) (VertexId v) (EdgeOracle os) = find' u v <|> find' v u
+  where
+    find' j i = fmap (^.extra) . F.find (\(VertexId k :+ _) -> j == k) $ os V.! i
+
+
+
+--------------------------------------------------------------------------------
+
+data TestG
+
+type Vertex = VertexId TestG Primal_
+
+testEdges :: [(Vertex,[Vertex])]
+testEdges = map (\(i,vs) -> (VertexId i, map VertexId vs))
+            [ (0, [1])
+            , (1, [0,1,2,4])
+            , (2, [1,3,4])
+            , (3, [2,5])
+            , (4, [1,2,5])
+            , (5, [3,4])
+            ]
diff --git a/src/Data/PlaneGraph.hs b/src/Data/PlaneGraph.hs
--- a/src/Data/PlaneGraph.hs
+++ b/src/Data/PlaneGraph.hs
@@ -2,19 +2,25 @@
                       , PlaneGraph
 
                       , withEdgeDistances
+                      , faceToSimplePolygon
                       ) where
 
 import Data.Ext
 import Control.Lens
 import Data.PlanarGraph
 import Data.Geometry.Point
+import Data.Geometry.Polygon(fromPoints, SimplePolygon)
+import Data.Geometry.Properties
+import qualified Data.Vector as V
 
 
 --------------------------------------------------------------------------------
 
 type PlaneGraph s w v e f r = PlanarGraph s w (Point 2 r :+ v) e f
 
+type instance NumType (PlaneGraph s w v e f r) = r
 
+
 -- | Labels the edges of a plane graph with their distances, as specified by
 -- the distance function.
 withEdgeDistances     :: (Point 2 r ->  Point 2 r -> a)
@@ -22,3 +28,10 @@
 withEdgeDistances f g = g&dartData %~ fmap (\(d,x) -> (d,len d :+ x))
   where
     len d = uncurry f . over both (^.core) $ endPointData d g
+
+
+faceToSimplePolygon      :: FaceId s w -> PlaneGraph s w p e f r
+                         -> SimplePolygon p r :+ f
+faceToSimplePolygon i gr = poly (boundaryVertices i gr) :+ gr^.fDataOf i
+  where
+    poly = fromPoints . V.toList . fmap (\j -> gr^.vDataOf j)
diff --git a/src/Data/Range.hs b/src/Data/Range.hs
--- a/src/Data/Range.hs
+++ b/src/Data/Range.hs
@@ -1,4 +1,5 @@
 {-# LANGUAGE TemplateHaskell   #-}
+{-# LANGUAGE DeriveAnyClass  #-}
 module Data.Range( EndPoint(..)
                  , isOpen, isClosed
                  , unEndPoint
@@ -7,25 +8,32 @@
                  , lower, upper
                  , pattern OpenRange, pattern ClosedRange, pattern Range'
                  , inRange, width, clipLower, clipUpper, midPoint
-                 , isValid
+                 , isValid, covers
 
                  , shiftLeft, shiftRight
                  ) where
 
-import           Control.Applicative
-import           Control.Arrow((&&&))
-import           Control.Lens
-import qualified Data.Foldable as F
-import           Data.Geometry.Properties
-import qualified Data.Traversable as T
+import Control.Lens
+import Data.Geometry.Properties
+import Frames.CoRec
+import Text.Printf(printf)
+import GHC.Generics (Generic)
+import Control.DeepSeq
 
 --------------------------------------------------------------------------------
 
 
-data EndPoint a = Open   a
-                | Closed a
-                deriving (Show,Read,Eq,Functor,F.Foldable,T.Traversable)
+data EndPoint a = Open   !a
+                | Closed !a
+                deriving (Show,Read,Eq,Functor,Foldable,Traversable,Generic,NFData)
 
+instance Ord a => Ord (EndPoint a) where
+  -- | order on the actual value, and Open before Closed
+  a `compare` b = f a `compare` f b
+    where
+      f (Open x)   = (x,False)
+      f (Closed x) = (x,True)
+
 _unEndPoint            :: EndPoint a -> a
 _unEndPoint (Open a)   = a
 _unEndPoint (Closed a) = a
@@ -47,19 +55,26 @@
 
 --------------------------------------------------------------------------------
 
-data Range a = Range { _lower :: EndPoint a
-                     , _upper :: EndPoint a
+data Range a = Range { _lower :: !(EndPoint a)
+                     , _upper :: !(EndPoint a)
                      }
-               deriving (Show,Read,Eq,Functor,F.Foldable,T.Traversable)
-
+               deriving (Eq,Functor,Foldable,Traversable,Generic,NFData)
 makeLenses ''Range
 
+instance Show a => Show (Range a) where
+  show (Range l u) = printf "Range (%s) (%s)" (show l) (show u)
+
+type instance NumType (Range a) = a
+
+pattern OpenRange       :: a -> a -> Range a
 pattern OpenRange   l u = Range (Open l)   (Open u)
+
+pattern ClosedRange     :: a -> a -> Range a
 pattern ClosedRange l u = Range (Closed l) (Closed u)
 
 -- | A range from l to u, ignoring/forgetting the type of the enpoints
-pattern Range' l u <- (_lower &&& _upper -> (l,u))
-
+pattern Range'     :: a -> a -> Range a
+pattern Range' l u <- ((\r -> (r^.lower.unEndPoint,r^.upper.unEndPoint) -> (l,u)))
 
 
 prettyShow             :: Show a => Range a -> String
@@ -133,6 +148,10 @@
 -- either open, ), or closed, ],
 clipUpper     :: Ord a => EndPoint a -> Range a -> Maybe (Range a)
 clipUpper u r = let r' = clipUpper' u r in if isValid r' then Just r' else Nothing
+
+-- | Wether or not the first range completely covers the second one
+covers       :: (Ord a) => Range a -> Range a -> Bool
+a `covers` b = maybe False (== b) . asA (Identity a) $ a `intersect` b
 
 
 -- | Check if the range is valid and nonEmpty, i.e. if the lower endpoint is
diff --git a/src/Data/Seq.hs b/src/Data/Seq.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Seq.hs
@@ -0,0 +1,235 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+module Data.Seq( LSeq
+               , toSeq
+               , empty
+               , fromList
+               , fromNonEmpty
+               , fromSeq
+               , toNonEmpty
+
+               , (<|), (|>)
+               , (><)
+               , eval
+
+               , index
+               , adjust
+               , partition
+               , mapWithIndex
+               , take
+               , drop
+               , unstableSortBy
+
+               , ViewL(..)
+               , viewl
+               , pattern (:<|)
+
+               , pattern (:<<)
+               , pattern EmptyL
+
+               , ViewR(..)
+               , viewr
+               , pattern (:|>)
+
+
+               , promise
+               ) where
+
+import           Control.Lens ((%~), (&), (<&>), (^?), Lens', bimap)
+import           Control.Lens.At (Ixed(..), Index, IxValue)
+import qualified Data.Foldable as F
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Maybe (fromJust)
+import           Data.Proxy
+import           Data.Semigroup
+import qualified Data.Sequence as S
+import qualified Data.Traversable as Tr
+import           GHC.TypeLits
+import           Prelude hiding (drop,take)
+
+
+-- | LSeq n a certifies that the sequence has *at least* n items
+newtype LSeq (n :: Nat) a = LSeq { toSeq :: S.Seq a}
+                          deriving (Show,Read,Eq,Ord,Foldable,Functor,Traversable)
+
+instance Semigroup (LSeq n a) where
+  (LSeq s) <> (LSeq s') = LSeq (s <> s')
+
+instance Monoid (LSeq 0 a) where
+  mempty = empty
+  mappend = (<>)
+
+type instance Index   (LSeq n a) = Int
+type instance IxValue (LSeq n a) = a
+instance Ixed (LSeq n a) where
+  ix i f s@(LSeq xs)
+    | 0 <= i && i < S.length xs = f (S.index xs i) <&> \x -> LSeq $ S.update i x xs
+    | otherwise                 = pure s
+
+empty :: LSeq 0 a
+empty = LSeq S.empty
+
+(<|) :: a -> LSeq n a -> LSeq (1 + n) a
+x <| xs = LSeq (x S.<| toSeq xs)
+
+(|>)    :: LSeq n a -> a -> LSeq (1 + n) a
+xs |> x = LSeq (toSeq xs S.|> x)
+
+infixr 5 <|
+infixl 5 |>
+
+(><) :: LSeq n a -> LSeq m a -> LSeq (n + m) a
+xs >< ys = LSeq (toSeq xs <> toSeq ys)
+
+infix 5 ><
+
+
+eval :: forall proxy n m a. KnownNat n => proxy n -> LSeq m a -> Maybe (LSeq n a)
+eval n (LSeq xs)
+  | toInteger (S.length xs) >= natVal n = Just $ LSeq xs
+  | otherwise                           = Nothing
+
+
+
+
+
+-- | Promises that the length of this LSeq is actually n. This is not
+-- checked.
+--
+-- This function should be a noop
+promise :: LSeq m a -> LSeq n a
+promise = LSeq . toSeq
+
+toNonEmpty :: LSeq (1 + n) a -> NonEmpty.NonEmpty a
+toNonEmpty = NonEmpty.fromList . F.toList
+
+
+--------------------------------------------------------------------------------
+
+-- | get the element with index i, counting from the left and starting at 0.
+-- O(log(min(i,n-i)))
+index     :: LSeq n a -> Int -> a
+index s i = fromJust $ s^?ix i
+
+adjust       :: (a -> a) -> Int -> LSeq n a -> LSeq n a
+adjust f i s = s&ix i %~ f
+
+
+partition   :: (a -> Bool) -> LSeq n a -> (LSeq 0 a, LSeq 0 a)
+partition p = bimap LSeq LSeq . S.partition p . toSeq
+
+mapWithIndex   :: (Int -> a -> b) -> LSeq n a -> LSeq n b
+mapWithIndex f = wrapUnsafe (S.mapWithIndex f)
+
+take   :: Int -> LSeq n a -> LSeq 0 a
+take i = wrapUnsafe (S.take i)
+
+drop   :: Int -> LSeq n a -> LSeq 0 a
+drop i = wrapUnsafe (S.drop i)
+
+
+unstableSortBy   :: (a -> a -> Ordering) -> LSeq n a -> LSeq n a
+unstableSortBy f = wrapUnsafe (S.unstableSortBy f)
+
+
+wrapUnsafe :: (S.Seq a -> S.Seq b) -> LSeq n a -> LSeq m b
+wrapUnsafe f = LSeq . f . toSeq
+
+--------------------------------------------------------------------------------
+
+fromSeq :: S.Seq a -> LSeq 0 a
+fromSeq = LSeq
+
+fromList :: Foldable f => f a -> LSeq 0 a
+fromList = LSeq . S.fromList . F.toList
+
+fromNonEmpty :: NonEmpty.NonEmpty a -> LSeq 1 a
+fromNonEmpty = LSeq . S.fromList . F.toList
+
+
+--------------------------------------------------------------------------------
+
+data ViewL n a where
+  (:<) :: a -> LSeq n a -> ViewL (1 + n) a
+
+infixr 5 :<
+
+instance Semigroup (ViewL n a) where
+  (x :< xs) <> (y :< ys) = x :< LSeq (toSeq xs <> (y S.<| toSeq ys))
+
+deriving instance Show a => Show (ViewL n a)
+instance Functor (ViewL n) where
+  fmap = Tr.fmapDefault
+instance Foldable (ViewL n) where
+  foldMap = Tr.foldMapDefault
+instance Traversable (ViewL n) where
+  traverse f (x :< xs) = (:<) <$> f x <*> traverse f xs
+instance Eq a => Eq (ViewL n a) where
+  s == s' = F.toList s == F.toList s'
+instance Ord a => Ord (ViewL n a) where
+  s `compare` s' = F.toList s `compare` F.toList s'
+
+
+viewl :: LSeq (1 + n) a -> ViewL (1 + n) a
+viewl xs = let ~(x S.:< ys) = S.viewl $ toSeq xs in x :< LSeq ys
+
+infixr 5 :<|
+
+-- pattern (:<|) :: a -> LSeq n a -> LSeq (1 + n) a
+pattern x :<| xs <- (viewl -> x :< xs)
+  where
+    x :<| xs = x <| xs
+
+infixr 5 :<<
+
+pattern x :<< xs <- (viewLSeq -> Just (x,xs))
+
+pattern EmptyL   <- (viewLSeq -> Nothing)
+
+viewLSeq          :: LSeq n a -> Maybe (a,LSeq 0 a)
+viewLSeq (LSeq s) = case S.viewl s of
+                      S.EmptyL    -> Nothing
+                      (x S.:< xs) -> Just (x,LSeq xs)
+
+
+--------------------------------------------------------------------------------
+
+data ViewR n a where
+  (:>) :: LSeq n a -> a -> ViewR (1 + n) a
+
+infixl 5 :>
+
+instance Semigroup (ViewR n a) where
+  (xs :> x) <> (ys :> y) = LSeq ((toSeq xs S.|> x) <> toSeq ys) :> y
+
+deriving instance Show a => Show (ViewR n a)
+instance Functor (ViewR n) where
+  fmap = Tr.fmapDefault
+instance Foldable (ViewR n) where
+  foldMap = Tr.foldMapDefault
+instance Traversable (ViewR n) where
+  traverse f (xs :> x) = (:>) <$> traverse f xs <*> f x
+instance Eq a => Eq (ViewR n a) where
+  s == s' = F.toList s == F.toList s'
+instance Ord a => Ord (ViewR n a) where
+  s `compare` s' = F.toList s `compare` F.toList s'
+
+viewr :: LSeq (1 + n) a -> ViewR (1 + n) a
+viewr xs = let ~(ys S.:> x) = S.viewr $ toSeq xs in LSeq ys :> x
+
+
+infixl 5 :|>
+
+-- pattern (:|>) :: LSeq n a -> a -> LSeq (1 + n) a
+pattern xs :|> x <- (viewr -> xs :> x)
+  where
+    xs :|> x = xs |> x
+
+--------------------------------------------------------------------------------
+
+testL = (eval (Proxy :: Proxy 2) $ fromList [1..5])
+
+testL' :: LSeq 2 Integer
+testL' = fromJust testL
+
+test            :: Show a => LSeq (1 + n) a -> String
+test (x :<| xs) = show x ++ show xs
diff --git a/src/Data/Seq2.hs b/src/Data/Seq2.hs
--- a/src/Data/Seq2.hs
+++ b/src/Data/Seq2.hs
@@ -1,24 +1,21 @@
 module Data.Seq2 where
 
-import           Control.Applicative
 import           Control.Lens ((%~), (&), (<&>), (^?), Lens', lens)
 import           Control.Lens.At (Ixed(..), Index, IxValue)
+import qualified Data.Foldable as F
 import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Maybe (fromJust)
 import           Data.Semigroup
-import           Prelude hiding (foldr,foldl,head,tail,last,length)
-
-
-import qualified Data.Traversable as T
-import qualified Data.Foldable as F
 import qualified Data.Sequence as S
+import qualified Data.Traversable as T
+import           Prelude hiding (foldr,foldl,head,tail,last,length)
 
 --------------------------------------------------------------------------------
 
 -- | Basically Data.Sequence but with the guarantee that the list contains at
 -- least two elements.
 data Seq2 a = Seq2 a (S.Seq a) a
-                deriving (Eq,Ord,Show,Read)
+                deriving (Eq,Ord,Show)
 
 
 instance T.Traversable Seq2 where
@@ -57,7 +54,12 @@
 adjust       :: (a -> a) -> Int -> Seq2 a -> Seq2 a
 adjust f i s = s&ix i %~ f
 
-
+partition                :: (a -> Bool) -> Seq2 a -> (S.Seq a, S.Seq a)
+partition p (Seq2 x s y) = let (l,r) = S.partition p s in case (p x, p y) of
+    (False,False) -> ((x S.<| l) S.|> y, r)
+    (False,_)     -> (x S.<| l,          r S.|> y)
+    (True, False) -> (l S.|> y,          x S.<| r)
+    _             -> (l,                 (x S.<| r) S.|> y)
 
 
 (<|) :: a -> Seq2 a -> Seq2 a
@@ -111,7 +113,7 @@
 --------------------------------------------------------------------------------
 -- | Left views
 
-data ViewL2 a = a :<< ViewR1 a deriving (Show,Read,Eq,Ord)
+data ViewL2 a = a :<< ViewR1 a deriving (Show,Eq,Ord)
 
 -- | At least two elements
 instance T.Traversable ViewL2 where
@@ -129,8 +131,11 @@
 
 
 -- | At least one element
-data ViewL1 a = a :< S.Seq a deriving (Show,Read,Eq,Ord)
+data ViewL1 a = a :< S.Seq a deriving (Eq,Ord)
 
+instance Show a => Show (ViewL1 a) where
+  show (x :< xs) = concat [ show x, " :< ", show $ F.toList xs]
+
 instance T.Traversable ViewL1 where
   traverse f ~(a :< s) = (:<) <$> f a <*> T.traverse f s
 
@@ -157,7 +162,13 @@
 viewL1FromNonEmpty                     :: NonEmpty.NonEmpty a -> ViewL1 a
 viewL1FromNonEmpty ~(x NonEmpty.:| xs) = x :< S.fromList xs
 
+viewL1FromSeq   :: S.Seq a -> ViewL1 a
+viewL1FromSeq s = case S.viewl s of
+  S.EmptyL    -> error "viewL1FromSeq: Empty seq"
+  (x S.:< xs) -> x :< xs
 
+
+
 -- | O(1) get a left view
 viewl                 :: Seq2 a -> ViewL2 a
 viewl ~(Seq2 l s r) = l :<< (s :> r)
@@ -175,7 +186,7 @@
 
 -- | A view of the right end of the seq, with the guarantee that it
 -- has at least two elements
-data ViewR2 a = ViewL1 a :>> a deriving (Show,Read,Eq,Ord)
+data ViewR2 a = ViewL1 a :>> a deriving (Show,Eq,Ord)
 
 instance T.Traversable ViewR2 where
   traverse f ~(s :>> a) = (:>>) <$> T.traverse f s <*> f a
diff --git a/src/Data/Sequence/Util.hs b/src/Data/Sequence/Util.hs
--- a/src/Data/Sequence/Util.hs
+++ b/src/Data/Sequence/Util.hs
@@ -10,7 +10,7 @@
 --
 -- returns Nothing if no element satisfies p
 --
--- running time: $O(\log^2 n + T*\log n)$, where $T$ is the time to execute the
+-- running time: \(O(\log^2 n + T*\log n)\), where \(T\) is the time to execute the
 -- predicate.
 binarySearchSeq     :: (a -> Bool) -> Seq a -> Maybe Int
 binarySearchSeq p s = case S.viewr s of
@@ -31,7 +31,7 @@
 --
 -- all elements in s occur in either xs or ys.
 --
--- running time: $O(\log^2 n + T*\log n)$, where $T$ is the time to execute the
+-- running time: \(O(\log^2 n + T*\log n)\), where \(T\) is the time to execute the
 -- predicate.
 splitMonotone     :: (a -> Bool) -> Seq a -> (Seq a, Seq a)
 splitMonotone p s = case binarySearchSeq p s of
@@ -47,7 +47,7 @@
 -- Get the index h such that everything strictly smaller than h has: p i =
 -- False, and all i >= h, we have p h = True
 --
--- running time: $O(\log(u - l))$
+-- running time: \(O(\log(u - l))\)
 {-# SPECIALIZE binarySearch :: (Int -> Bool) -> Int -> Int -> Int #-}
 binarySearch       :: Integral a => (a -> Bool) -> a -> a -> a
 binarySearch p l u = let d = u - l
diff --git a/src/Data/UnBounded.hs b/src/Data/UnBounded.hs
--- a/src/Data/UnBounded.hs
+++ b/src/Data/UnBounded.hs
@@ -10,7 +10,6 @@
                      , unBoundedToMaybe
                      ) where
 
-import           Control.Applicative
 import           Control.Lens
 import qualified Data.Foldable as F
 import qualified Data.Traversable as T
@@ -26,7 +25,10 @@
 newtype Top a = GTop { topToMaybe :: Maybe a }
                 deriving (Eq,Functor,F.Foldable,T.Traversable,Applicative,Monad)
 
+pattern ValT  :: a -> Top a
 pattern ValT x = GTop (Just x)
+
+pattern Top    :: Top a
 pattern Top    = GTop Nothing
 
 instance Ord a => Ord (Top a) where
@@ -45,11 +47,14 @@
 -- i.e. an element that is smaller than any other element. We can think of
 -- `Bottom a` being defined as:
 --
--- >>> data Bottom a = ValB
+-- >>> data Bottom a = Bottom | ValB a
 newtype Bottom a = GBottom { bottomToMaybe :: Maybe a }
                  deriving (Eq,Ord,Functor,F.Foldable,T.Traversable,Applicative,Monad)
 
+pattern Bottom :: Bottom a
 pattern Bottom = GBottom Nothing
+
+pattern ValB   :: a -> Bottom a
 pattern ValB x = GBottom (Just x)
 
 instance Show a => Show (Bottom a) where
diff --git a/src/Data/Util.hs b/src/Data/Util.hs
new file mode 100644
--- /dev/null
+++ b/src/Data/Util.hs
@@ -0,0 +1,41 @@
+module Data.Util where
+
+import Control.Lens
+import Data.Bifunctor
+import Data.Semigroup
+
+-- |  strict triple
+data STR a b c = STR { fst' :: !a, snd' :: !b , trd' :: !c}
+               deriving (Show,Eq,Ord,Functor)
+
+
+instance Field1 (STR a b c) (STR d b c) a d where
+  _1 = lens fst' (\(STR _ b c) d -> STR d b c)
+
+instance Field2 (STR a b c) (STR a d c) b d where
+  _2 = lens snd' (\(STR a _ c) d -> STR a d c)
+
+instance Field3 (STR a b c) (STR a b d) c d where
+  _3 = lens trd' (\(STR a b _) d -> STR a b d)
+
+
+
+
+
+-- | Strict pair
+data SP a b = SP !a !b deriving (Show,Eq,Ord,Functor)
+
+instance (Semigroup a, Semigroup b) => Semigroup (SP a b) where
+  (SP a b) <> (SP c d) = SP (a <> c) (b <> d)
+
+instance Field1 (SP a b) (SP c b) a c where
+  _1 = lens (\(SP a _) -> a) (\(SP _ b) c -> SP c b)
+
+instance Field2 (SP a b) (SP a c) b c where
+  _2 = lens (\(SP _ b) -> b) (\(SP a _) c -> SP a c)
+
+instance Bifunctor SP where
+  bimap f g (SP a b) = SP (f a) (g b)
+
+-- | Strict pair with both items the same
+type Two a = SP a a
diff --git a/test/Algorithms/Geometry/DelaunayTriangulation/DTSpec.hs b/test/Algorithms/Geometry/DelaunayTriangulation/DTSpec.hs
--- a/test/Algorithms/Geometry/DelaunayTriangulation/DTSpec.hs
+++ b/test/Algorithms/Geometry/DelaunayTriangulation/DTSpec.hs
@@ -1,22 +1,20 @@
 {-# LANGUAGE LambdaCase #-}
 module Algorithms.Geometry.DelaunayTriangulation.DTSpec where
 
-import Util
-
-import Test.Hspec
-import Control.Lens
-import Data.Geometry
-import Data.Maybe(mapMaybe, fromJust)
-import Algorithms.Geometry.DelaunayTriangulation.Types
 import qualified Algorithms.Geometry.DelaunayTriangulation.DivideAndConqueror as DC
-import qualified Algorithms.Geometry.DelaunayTriangulation.Naive              as Naive
-import qualified Data.List.NonEmpty as NonEmpty
-import Data.Geometry.Ipe
-import Data.Ext
-import Data.Traversable(traverse)
+import qualified Algorithms.Geometry.DelaunayTriangulation.Naive as Naive
+import           Algorithms.Geometry.DelaunayTriangulation.Types
+import           Control.Lens
 import qualified Data.CircularList.Util as CU
+import           Data.Ext
+import           Data.Geometry
+import           Data.Geometry.Ipe
+import qualified Data.List.NonEmpty as NonEmpty
 import qualified Data.Map as M
+import           Data.Maybe (mapMaybe, fromJust)
 import qualified Data.Vector as V
+import           Test.Hspec
+import           Util
 
 
 dtEdges :: (Fractional r, Ord r)
diff --git a/test/Algorithms/Geometry/LineSegmentIntersection/BentleyOttmannSpec.hs b/test/Algorithms/Geometry/LineSegmentIntersection/BentleyOttmannSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Algorithms/Geometry/LineSegmentIntersection/BentleyOttmannSpec.hs
@@ -0,0 +1,65 @@
+module Algorithms.Geometry.LineSegmentIntersection.BentleyOttmannSpec where
+
+import qualified Algorithms.Geometry.LineSegmentIntersection.BentleyOttmann as Sweep
+import qualified Algorithms.Geometry.LineSegmentIntersection.Naive as Naive
+import           Algorithms.Geometry.LineSegmentIntersection.Types
+import           Control.Lens
+import           Data.Ext
+import           Data.Geometry.Interval
+import           Data.Geometry.Ipe
+import           Data.Geometry.LineSegment
+import           Data.Geometry.Point
+import qualified Data.List as L
+import qualified Data.List.NonEmpty as NonEmpty
+import qualified Data.Map as Map
+import qualified Data.Set as Set
+import           Test.Hspec
+import           Test.QuickCheck
+import           Util
+
+spec :: Spec
+spec = do
+  describe "Testing Bentley Ottmann LineSegment Intersection" $ do
+    -- toSpec (TestCase "myPoints" myPoints)
+    -- toSpec (TestCase "myPoints'" myPoints')
+    ipeSpec
+
+ipeSpec :: Spec
+ipeSpec = testCases "test/Algorithms/Geometry/LineSegmentIntersection/manual.ipe"
+
+testCases    :: FilePath -> Spec
+testCases fp = (runIO $ readInput fp) >>= \case
+    Left e    -> it "reading LineSegment Intersection file" $
+                   expectationFailure $ "Failed to read ipe file " ++ show e
+    Right tcs -> mapM_ toSpec tcs
+
+
+-- | Point sets per color, Crosses form the solution
+readInput    :: FilePath -> IO (Either ConversionError [TestCase Rational])
+readInput fp = fmap f <$> readSinglePageFile fp
+  where
+    f page = [TestCase segs]
+      where
+        segs = page^..content.traverse._IpePath.core._asLineSegment
+
+
+
+data TestCase r = TestCase { _segments :: [LineSegment 2 () r]
+                           } deriving (Show,Eq)
+
+
+toSpec                 :: (Fractional r, Ord r, Show r) => TestCase r -> Spec
+toSpec (TestCase segs) = describe ("testing segments ") $ do
+                            samePointsAsNaive segs
+                            sameAsNaive segs
+
+-- | Test if we have the same intersection points
+samePointsAsNaive segs = it "Same points as Naive" $ do
+  (Map.keys $ Sweep.intersections segs) `shouldBe` (Map.keys $ Naive.intersections segs)
+
+-- | Test if they every intersection point has the right segments
+sameAsNaive      :: (Fractional r, Ord r, Eq p
+                    , Show p, Show r
+                    ) => [LineSegment 2 p r] -> Spec
+sameAsNaive segs = it "Same as Naive " $ do
+    (Sweep.intersections segs) `shouldBe` (Naive.intersections segs)
diff --git a/test/Algorithms/Geometry/LineSegmentIntersection/manual.ipe b/test/Algorithms/Geometry/LineSegmentIntersection/manual.ipe
new file mode 100644
--- /dev/null
+++ b/test/Algorithms/Geometry/LineSegmentIntersection/manual.ipe
@@ -0,0 +1,324 @@
+<?xml version="1.0"?>
+<!DOCTYPE ipe SYSTEM "ipe.dtd">
+<ipe version="70107" creator="Ipe 7.2.2">
+<info created="D:20160903131616" modified="D:20161022125153"/>
+<ipestyle name="basic">
+<symbol name="arrow/arc(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/farc(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/ptarc(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/fptarc(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="mark/circle(sx)" transformations="translations">
+<path fill="sym-stroke">
+0.6 0 0 0.6 0 0 e
+0.4 0 0 0.4 0 0 e
+</path>
+</symbol>
+<symbol name="mark/disk(sx)" transformations="translations">
+<path fill="sym-stroke">
+0.6 0 0 0.6 0 0 e
+</path>
+</symbol>
+<symbol name="mark/fdisk(sfx)" transformations="translations">
+<group>
+<path fill="sym-fill">
+0.5 0 0 0.5 0 0 e
+</path>
+<path fill="sym-stroke" fillrule="eofill">
+0.6 0 0 0.6 0 0 e
+0.4 0 0 0.4 0 0 e
+</path>
+</group>
+</symbol>
+<symbol name="mark/box(sx)" transformations="translations">
+<path fill="sym-stroke" fillrule="eofill">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+-0.4 -0.4 m
+0.4 -0.4 l
+0.4 0.4 l
+-0.4 0.4 l
+h
+</path>
+</symbol>
+<symbol name="mark/square(sx)" transformations="translations">
+<path fill="sym-stroke">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+</path>
+</symbol>
+<symbol name="mark/fsquare(sfx)" transformations="translations">
+<group>
+<path fill="sym-fill">
+-0.5 -0.5 m
+0.5 -0.5 l
+0.5 0.5 l
+-0.5 0.5 l
+h
+</path>
+<path fill="sym-stroke" fillrule="eofill">
+-0.6 -0.6 m
+0.6 -0.6 l
+0.6 0.6 l
+-0.6 0.6 l
+h
+-0.4 -0.4 m
+0.4 -0.4 l
+0.4 0.4 l
+-0.4 0.4 l
+h
+</path>
+</group>
+</symbol>
+<symbol name="mark/cross(sx)" transformations="translations">
+<group>
+<path fill="sym-stroke">
+-0.43 -0.57 m
+0.57 0.43 l
+0.43 0.57 l
+-0.57 -0.43 l
+h
+</path>
+<path fill="sym-stroke">
+-0.43 0.57 m
+0.57 -0.43 l
+0.43 -0.57 l
+-0.57 0.43 l
+h
+</path>
+</group>
+</symbol>
+<symbol name="arrow/fnormal(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/pointed(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/fpointed(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-0.8 0 l
+-1 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/linear(spx)">
+<path stroke="sym-stroke" pen="sym-pen">
+-1 0.333 m
+0 0 l
+-1 -0.333 l
+</path>
+</symbol>
+<symbol name="arrow/fdouble(spx)">
+<path stroke="sym-stroke" fill="white" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+-1 0 m
+-2 0.333 l
+-2 -0.333 l
+h
+</path>
+</symbol>
+<symbol name="arrow/double(spx)">
+<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
+0 0 m
+-1 0.333 l
+-1 -0.333 l
+h
+-1 0 m
+-2 0.333 l
+-2 -0.333 l
+h
+</path>
+</symbol>
+<pen name="heavier" value="0.8"/>
+<pen name="fat" value="1.2"/>
+<pen name="ultrafat" value="2"/>
+<symbolsize name="large" value="5"/>
+<symbolsize name="small" value="2"/>
+<symbolsize name="tiny" value="1.1"/>
+<arrowsize name="large" value="10"/>
+<arrowsize name="small" value="5"/>
+<arrowsize name="tiny" value="3"/>
+<color name="red" value="1 0 0"/>
+<color name="green" value="0 1 0"/>
+<color name="blue" value="0 0 1"/>
+<color name="yellow" value="1 1 0"/>
+<color name="orange" value="1 0.647 0"/>
+<color name="gold" value="1 0.843 0"/>
+<color name="purple" value="0.627 0.125 0.941"/>
+<color name="gray" value="0.745"/>
+<color name="brown" value="0.647 0.165 0.165"/>
+<color name="navy" value="0 0 0.502"/>
+<color name="pink" value="1 0.753 0.796"/>
+<color name="seagreen" value="0.18 0.545 0.341"/>
+<color name="turquoise" value="0.251 0.878 0.816"/>
+<color name="violet" value="0.933 0.51 0.933"/>
+<color name="darkblue" value="0 0 0.545"/>
+<color name="darkcyan" value="0 0.545 0.545"/>
+<color name="darkgray" value="0.663"/>
+<color name="darkgreen" value="0 0.392 0"/>
+<color name="darkmagenta" value="0.545 0 0.545"/>
+<color name="darkorange" value="1 0.549 0"/>
+<color name="darkred" value="0.545 0 0"/>
+<color name="lightblue" value="0.678 0.847 0.902"/>
+<color name="lightcyan" value="0.878 1 1"/>
+<color name="lightgray" value="0.827"/>
+<color name="lightgreen" value="0.565 0.933 0.565"/>
+<color name="lightyellow" value="1 1 0.878"/>
+<dashstyle name="dashed" value="[4] 0"/>
+<dashstyle name="dotted" value="[1 3] 0"/>
+<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
+<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
+<textsize name="large" value="\large"/>
+<textsize name="Large" value="\Large"/>
+<textsize name="LARGE" value="\LARGE"/>
+<textsize name="huge" value="\huge"/>
+<textsize name="Huge" value="\Huge"/>
+<textsize name="small" value="\small"/>
+<textsize name="footnote" value="\footnotesize"/>
+<textsize name="tiny" value="\tiny"/>
+<textstyle name="center" begin="\begin{center}" end="\end{center}"/>
+<textstyle name="itemize" begin="\begin{itemize}" end="\end{itemize}"/>
+<textstyle name="item" begin="\begin{itemize}\item{}" end="\end{itemize}"/>
+<gridsize name="4 pts" value="4"/>
+<gridsize name="8 pts (~3 mm)" value="8"/>
+<gridsize name="16 pts (~6 mm)" value="16"/>
+<gridsize name="32 pts (~12 mm)" value="32"/>
+<gridsize name="10 pts (~3.5 mm)" value="10"/>
+<gridsize name="20 pts (~7 mm)" value="20"/>
+<gridsize name="14 pts (~5 mm)" value="14"/>
+<gridsize name="28 pts (~10 mm)" value="28"/>
+<gridsize name="56 pts (~20 mm)" value="56"/>
+<anglesize name="90 deg" value="90"/>
+<anglesize name="60 deg" value="60"/>
+<anglesize name="45 deg" value="45"/>
+<anglesize name="30 deg" value="30"/>
+<anglesize name="22.5 deg" value="22.5"/>
+<opacity name="10%" value="0.1"/>
+<opacity name="30%" value="0.3"/>
+<opacity name="50%" value="0.5"/>
+<opacity name="75%" value="0.75"/>
+<tiling name="falling" angle="-60" step="4" width="1"/>
+<tiling name="rising" angle="30" step="4" width="1"/>
+</ipestyle>
+<ipestyle name="frank">
+<arrowsize name="normal" value="5"/>
+<arrowsize name="large" value="8"/>
+<arrowsize name="huge" value="10"/>
+<arrowsize name="small" value="3"/>
+<arrowsize name="tiny" value="1"/>
+<dashstyle name="dashed" value="[2 2] 0"/>
+<dashstyle name="dotted" value="[0.5 1] 0"/>
+<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
+<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
+<gridsize name="1 pts" value="1"/>
+<gridsize name="2 pts" value="2"/>
+<opacity name="10%" value="0.1"/>
+<opacity name="30%" value="0.3"/>
+<opacity name="50%" value="0.5"/>
+<opacity name="20%" value="0.2"/>
+<opacity name="40%" value="0.4"/>
+<opacity name="60%" value="0.6"/>
+<opacity name="70%" value="0.7"/>
+<opacity name="80%" value="0.8"/>
+<opacity name="90%" value="0.9"/>
+</ipestyle>
+<page>
+<layer name="alpha"/>
+<view layers="alpha" active="alpha"/>
+<path layer="alpha" stroke="black">
+16 16 m
+144 80 l
+</path>
+<path stroke="black">
+128 32 m
+256 112 l
+</path>
+<path stroke="black">
+144 80 m
+225.708 52.1085 l
+</path>
+<path matrix="2.39165 0 0 2.39165 -206.711 -120.784" stroke="black">
+144 80 m
+148.537 86.7918 l
+</path>
+<path stroke="black">
+144 80 m
+140.284 85.5945 l
+</path>
+<path stroke="black">
+137.686 70.5482 m
+173.031 38.3458 l
+</path>
+<path stroke="black">
+89.5694 78.2992 m
+186.333 44.2802 l
+</path>
+<path stroke="black">
+75.255 80.7127 m
+75.255 24.0132 l
+</path>
+<path stroke="black">
+35.9103 40.8867 m
+180.812 40.8867 l
+</path>
+<path stroke="black">
+82.6725 35.3346 m
+57.1425 25.6907 l
+</path>
+<path stroke="black">
+40.6648 34.541 m
+144.411 12.4306 l
+</path>
+</page>
+</ipe>
diff --git a/test/Algorithms/Geometry/WellSeparatedPairDecomposition/WSPDSpec.hs b/test/Algorithms/Geometry/WellSeparatedPairDecomposition/WSPDSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Algorithms/Geometry/WellSeparatedPairDecomposition/WSPDSpec.hs
@@ -0,0 +1,92 @@
+module Algorithms.Geometry.WellSeparatedPairDecomposition.WSPDSpec where
+
+import           Algorithms.Geometry.Diameter
+import           Algorithms.Geometry.WellSeparatedPairDecomposition.Types
+import           Algorithms.Geometry.WellSeparatedPairDecomposition.WSPD
+import           Control.Lens
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry
+import qualified Data.List.NonEmpty as NonEmpty
+import qualified Data.Seq2 as S2
+import qualified Data.Set as Set
+import qualified Data.Vector as V
+import           Test.Hspec
+import           Util
+
+--------------------------------------------------------------------------------
+
+spec :: Spec
+spec = do
+  reIndexTest
+  distributePointsTest
+
+reIndexTest :: Spec
+reIndexTest = describe "ReIndex tests" $ do
+    it "simple input reordering " $ do
+      reIndexPoints input `shouldBe` output
+  where
+    input = v2 (ptSeq [ origin :+ 1, point2 1 1 :+ 100, point2 5 5 :+ 101 ])
+               (ptSeq [ point2 1 1 :+ 100, point2 5 5 :+ 101, origin :+ 1 ])
+    output = v2 (ptSeq [ origin :+ 0, point2 1 1 :+ 1, point2 5 5 :+ 2 ])
+                (ptSeq [ point2 1 1 :+ 1, point2 5 5 :+ 2, origin :+ 0 ])
+
+
+
+
+
+
+distributePointsTest :: Spec
+distributePointsTest = describe "DistributePoints tests" $ do
+    it "distributePoints' on a single list " $ do
+      distributePoints' 3 levels input `shouldBe` output
+    it "distributePoints on multiple lists" $ do
+      distributePoints 3 levels (v2 input input) `shouldBe` output'
+
+  where
+    levels = V.fromList [Just $ Level 0 (Just 2),Just $ Level 1 (Just 1), Nothing]
+    input  = ptSeq [ origin :+ 0, point2 1 1 :+ 1, point2 2 2 :+ 2]
+    output = V.fromList [ ptSeq [origin :+ 0]
+                        , ptSeq [point2 1 1 :+ 1]
+                        , ptSeq [point2 2 2 :+ 2]
+                        ]
+    output' = fmap (\pts -> v2 pts pts) output
+
+    --     input = v2 (f [ origin :+ 1, point2 1 1 :+ 100, point2 5 5 :+ 101 ])
+--                (f [ point2 1 1 :+ 100, point2 5 5 :+ 101, origin :+ 1 ])
+--     output = v2 (f [ origin :+ 0, point2 1 1 :+ 1, point2 5 5 :+ 2 ])
+--                 (f [ point2 1 1 :+ 1, point2 5 5 :+ 2, origin :+ 0 ])
+
+--     f = S2.viewL1FromNonEmpty . NonEmpty.fromList . map (&extra %~ ext)
+
+ptSeq = S2.viewL1FromNonEmpty . NonEmpty.fromList . map (&extra %~ ext)
+
+-- coversAll
+
+points1 :: NonEmpty.NonEmpty (Point 2 Double :+ ())
+points1 = ext <$> NonEmpty.fromList [point2 0 0, point2 1 1, point2 2 100, point2 3 101]
+
+
+-- | Computes all pairs of points that are uncovered by the WSPD with separation s
+uncovered         :: (Floating r, Ord r, AlwaysTrueWSPD d, Ord p)
+                  => [Point d r :+ p] -> r -> SplitTree d p r a -> [(Point d r :+ p, Point d r :+ p)]
+uncovered pts s t = Set.toList $ allPairs `Set.difference` covered
+  where
+    allPairs = Set.fromList [ (p,q) | p <- pts, q <- pts, p < q ]
+    covered  = Set.unions [ mkSet as bs | (as,bs) <- wellSeparatedPairs s t]
+
+    mkSet as bs = Set.fromList [ (min a b,max a b) | a <- F.toList as, b <- F.toList bs]
+
+-- | Naively check if a WSP pair is actually well separated with respect to
+-- separation s. i.e. computes the maximum diameter of as and bs, and then
+-- tests by brute force if all pairs (a,b) from different sets are at distance
+-- at least s times the maximum diameter.
+isWellSeparated           :: (Floating r, Ord r, Arity d) => r -> WSP d p r a -> Bool
+isWellSeparated s (as,bs) =
+    and [ euclideanDist (a^.core) (b^.core) >= s*r | a <- F.toList as, b <- F.toList bs ]
+  where
+    r = (/2) . maximum . map (diameterNaive . F.toList) $ [as,bs]
+
+
+
+allCoveredTest = describe
diff --git a/test/Data/EdgeOracleSpec.hs b/test/Data/EdgeOracleSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Data/EdgeOracleSpec.hs
@@ -0,0 +1,63 @@
+module Data.EdgeOracleSpec where
+
+import           Control.Arrow
+import           Data.Ext
+import           Data.PlanarGraph
+import           Data.Semigroup
+import qualified Data.Set as S
+import           Test.Hspec
+
+--------------------------------------------------------------------------------
+
+data TestG
+
+type Vertex = VertexId TestG Primal_
+
+
+testEdges :: [(Vertex,[Vertex])]
+testEdges = map (\(i,vs) -> (VertexId i, map VertexId vs))
+            [ (0, [1])
+            , (1, [0,1,2,4])
+            , (2, [1,3,4])
+            , (3, [2,5])
+            , (4, [1,2,5])
+            , (5, [3,4])
+            ]
+
+buildEdgeOracle'  :: [(Vertex,[Vertex])] -> EdgeOracle TestG Primal_ ()
+buildEdgeOracle' = buildEdgeOracle . map (second $ fmap ext)
+
+-- | Flattens an adjacencylist representation into a set of edges
+flattenEdges :: [(t, [a])] -> [(t, a)]
+flattenEdges = concatMap (\(i,vs) -> map (i,) vs)
+
+-- | Given a set of edges, generates all non-edges, i.e. all pairs of vertices
+-- that do not form an edge
+nonEdges    :: [(VertexId s w, [VertexId s w])] -> [(VertexId s w, VertexId s w)]
+nonEdges es = flattenEdges . map (second $ f . S.fromList) $ es
+  where
+    f vs  = filter (`S.notMember` vs) allVs
+    allVs = map fst es
+
+-- | Retains only the edges in the graph
+hasEdges         :: EdgeOracle s w a -> [(VertexId s w, VertexId s w)]
+                 -> [(VertexId s w, VertexId s w)]
+oracle `hasEdges` es = filter (\(u,v) -> hasEdge u v oracle) es
+
+
+-- | Tests all edges es
+edgeOracleSpec      :: String -> [(Vertex, [Vertex])]  -> Spec
+edgeOracleSpec s es = do
+    let oracle = buildEdgeOracle' es
+    describe ("EdgeOracle on " <> s) $ do
+      it "test postitive edges" $
+          (oracle `hasEdges` flattenEdges es) `shouldBe` flattenEdges es
+      it "test negative edges " $
+          (oracle `hasEdges` nonEdges es) `shouldBe` []
+
+      -- it "test maximum adjacency-list lengths" $
+      --     (filter (\v -> length v > 6) . _unEdgeOracle $ oracle) `shouldBe` []
+
+spec :: Spec
+spec = do
+         edgeOracleSpec "testEdges" testEdges
diff --git a/test/Data/Geometry/BoxSpec.hs b/test/Data/Geometry/BoxSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Data/Geometry/BoxSpec.hs
@@ -0,0 +1,16 @@
+module Data.Geometry.BoxSpec where
+
+import Data.Geometry
+import Data.Geometry.Box
+import Data.Geometry.Properties
+import Test.Hspec
+
+
+spec :: Spec
+spec = do
+  describe "Box" $ do
+    it "intersect tests" $
+      ((boundingBoxList' $ [point2 (-4) (-3), point2 (-4) (10 :: Int)])
+       `intersects`
+       (boundingBoxList' $ [point2 (-5) 1, point2 (-4) (0 :: Int)]))
+      `shouldBe` True
diff --git a/test/Data/Geometry/IntervalSpec.hs b/test/Data/Geometry/IntervalSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Data/Geometry/IntervalSpec.hs
@@ -0,0 +1,64 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+module Data.Geometry.IntervalSpec where
+
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry
+import           Data.Geometry.Box
+import qualified Data.Geometry.IntervalTree as IntTree
+import           Data.Geometry.IntervalTree (IntervalTree)
+import qualified Data.Geometry.SegmentTree as SegTree
+import           Data.Geometry.SegmentTree (SegmentTree, I(..))
+import qualified Data.List.NonEmpty as NonEmpty
+import qualified Data.Seq as Seq
+import qualified Data.Set as Set
+import           GHC.TypeLits
+import           QuickCheck.Instances ()
+import           Test.Hspec
+import           Test.QuickCheck
+import           Util
+
+
+naive   :: (Ord r, Foldable f) => r -> f (Interval p r) -> [Interval p r]
+naive q = filter (q `inInterval`) . F.toList
+
+sameAsNaive                 :: (Ord r, Ord p, Foldable f)
+                            => f (Interval p r)
+                            -> (r -> t -> [Interval p r], t)
+                            -> r
+                            -> Bool
+sameAsNaive is (search,t) q = search q t `sameElems` naive q is
+
+
+sameElems    :: Eq a => [a] -> [a] -> Bool
+sameElems xs = null . difference xs
+
+
+allSameAsNaive       :: (Ord r, Ord p)
+                     => NonEmpty.NonEmpty (Interval p r) -> [r] -> Bool
+allSameAsNaive is = all (sameAsNaive is (\q t -> _unI <$> SegTree.search q t
+                                        , SegTree.fromIntervals' is))
+
+
+allSameAsNaiveIT       :: (Ord r, Ord p)
+                     => NonEmpty.NonEmpty (Interval p r) -> [r] -> Bool
+allSameAsNaiveIT is = all (sameAsNaive is (\q t -> IntTree.search q t
+                                         , IntTree.fromIntervals $ F.toList is))
+
+spec :: Spec
+spec = do
+  describe "Same as Naive" $ do
+    it "quickcheck segmentTree" $
+      property $ \(is :: NonEmpty.NonEmpty (Interval () Word)) -> allSameAsNaive is
+    it "quickcheck IntervalTree" $
+      property $ \(Intervals is :: Intervals Word) -> allSameAsNaiveIT is
+
+
+newtype Intervals r = Intervals (NonEmpty.NonEmpty (Interval () r)) deriving (Show,Eq)
+
+-- don't generate double open intervals
+instance (Arbitrary r, Ord r) => Arbitrary (Intervals r) where
+  arbitrary = Intervals . NonEmpty.fromList <$> listOf1 (suchThat arbitrary p)
+    where
+      p (OpenInterval _ _) = False
+      p _                  = True
diff --git a/test/Data/Geometry/KDTreeSpec.hs b/test/Data/Geometry/KDTreeSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Data/Geometry/KDTreeSpec.hs
@@ -0,0 +1,65 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+module Data.Geometry.KDTreeSpec where
+
+import           Data.Ext
+import qualified Data.Foldable as F
+import           Data.Geometry
+import           Data.Geometry.Box
+import           Data.Geometry.KDTree
+import qualified Data.Seq as Seq
+import qualified Data.Set as Set
+import           GHC.TypeLits
+import           QuickCheck.Instances()
+import           Test.Hspec
+import           Test.QuickCheck
+
+--------------------------------------------------------------------------------
+
+naive    :: (Arity d, Ord r) => Box d q r -> [Point d r :+ p] -> [Point d r :+ p]
+naive qr = filter (\(p :+ _) ->  p `intersects` qr)
+
+sameAsNaive         :: (Ord r, Ord p, Arity d)
+                    => [Point d r :+ p] -> KDTree d p r -> Box d q r -> Bool
+sameAsNaive pts t q = Set.fromList (searchKDTree q t) == Set.fromList (naive q pts)
+
+allSameAsNaive     :: (Ord r, Ord p, Arity d, KnownNat d, Index' 0 d, Foldable f)
+                   => f (Point d r :+ p) -> [Box d () r] -> Bool
+allSameAsNaive pts = let pts' = F.toList pts
+                     in  all (sameAsNaive pts' $ buildKDTree pts')
+
+allSame :: (Arity d, Eq a) => Vector d a -> Bool
+allSame v = case F.toList v of
+              []     -> True
+              (x:xs) -> all (== x) xs
+
+-- newtype Pts n d r = Pts (PointSet (Seq.LSeq n) d () r)
+-- deriving instance (Arity d, Show r) => Show (Pts n d r)
+
+-- instance (KnownNat n, Arity d, KnownNat d, Arbitrary r, Ord r) => Arbitrary (Pts n d r) where
+--   arbitrary = Pts . toPointSet . Seq.toNonEmpty <$> arbitrary
+
+
+spec :: Spec
+spec = do
+  describe "splitOn" $ do
+    it "quickheck: left set same points" $
+      property $ \c (pts :: Seq.LSeq 2 (Point 2 Int :+ ())) ->
+                   let (l,_,_) = splitOn (toEnum c) (toPointSet pts)
+                   in allSame . fmap (Set.fromList . F.toList) $ l
+    it "quickheck: right set same points" $
+      property $ \c (pts :: Seq.LSeq 2 (Point 2 Int :+ ())) ->
+                   let (_,_,r) = splitOn (toEnum c) (toPointSet pts)
+                   in allSame . fmap (Set.fromList . F.toList) $ r
+  describe "Same as Naive" $ do
+    it "quickcheck 1d" $
+      property $ \(pts :: Set.Set (Point 1 Int :+ ())) -> allSameAsNaive pts
+    it "quickcheck 2d" $
+      property $ \(pts :: Set.Set (Point 2 Int :+ ())) -> allSameAsNaive pts
+    it "quickcheck 3d" $
+      property $ \(pts :: Set.Set (Point 3 Int :+ ())) -> allSameAsNaive pts
+    it "quickcheck 8d" $
+      property $ \(pts :: Set.Set (Point 8 Int :+ ())) -> allSameAsNaive pts
+
+
+-- pts = map ext [point2 (-2) 2, point2 5 (-1)]
+-- boxx = box (ext $ point2 3 (-4)) (ext $ point2 5 4)
diff --git a/test/Data/Geometry/Polygon/Convex/ConvexSpec.hs b/test/Data/Geometry/Polygon/Convex/ConvexSpec.hs
--- a/test/Data/Geometry/Polygon/Convex/ConvexSpec.hs
+++ b/test/Data/Geometry/Polygon/Convex/ConvexSpec.hs
@@ -1,5 +1,7 @@
+{-# LANGUAGE ScopedTypeVariables #-}
 module Data.Geometry.Polygon.Convex.ConvexSpec where
 
+import           Algorithms.Geometry.ConvexHull.GrahamScan (convexHull)
 import           Control.Applicative
 import           Control.Arrow ((&&&))
 import           Control.Lens
@@ -9,8 +11,11 @@
 import           Data.Geometry.Ipe
 import           Data.Geometry.Polygon (extremesLinear)
 import           Data.Geometry.Polygon.Convex
+import qualified Data.List.NonEmpty as NonEmpty
 import           Data.Traversable (traverse)
+import           QuickCheck.Instances
 import           Test.Hspec
+import           Test.QuickCheck
 
 
 
@@ -22,8 +27,8 @@
     Left e    -> it "reading ConvexTests file" $
                    expectationFailure . unwords $
                      [ "Failed to read ipe file", show fp, ":", show e]
-    Right tcs -> mapM_ toSpec tcs
-
+    Right tcs -> do mapM_ toSpec tcs
+                    minkowskiTests $ map _polygon tcs
 
 data TestCase r = TestCase { _polygon    :: ConvexPolygon () r
                            }
@@ -33,7 +38,8 @@
                     => ConvexPolygon q r -> Vector 2 r -> SpecWith ()
 toSingleSpec poly u = it msg $
   -- test that the reported extremes are equally far in direction u
-    F.all allEq (unzip [extremes u poly, extremesLinear u poly]) `shouldBe` True
+    F.all allEq (unzip [extremes u poly, extremesLinear u (poly^.simplePolygon)])
+    `shouldBe` True
   where
     allEq (p:ps) = all (\q -> cmpExtreme u p q == EQ) ps
     msg = "Extremes test with direction " ++ show u
@@ -54,6 +60,55 @@
 readInputFromFile    :: FilePath -> IO (Either ConversionError [TestCase Rational])
 readInputFromFile fp = fmap f <$> readSinglePageFile fp
   where
-    f page = [ TestCase poly | (poly :+ _) <- polies ]
+    f page = [ TestCase (ConvexPolygon poly) | (poly :+ _) <- polies ]
       where
         polies = page^..content.traverse._withAttrs _IpePath _asSimplePolygon
+
+
+--------------------------------------------------------------------------------
+
+minkowskiTests     ::  (Fractional r, Ord r, Show r) => [ConvexPolygon () r] -> Spec
+minkowskiTests pgs = do
+      minkowskiTests' "polygons in ipe file" pgs
+      it "quickcheck minkowskisum same as naive" $
+        property $ \(CP p :: CP Double) (CP q) ->
+          minkowskiSum p q == naiveMinkowski p q
+
+
+
+minkowskiTests'                      ::  (Fractional r, Ord r, Show r)
+                                     => String -> [ConvexPolygon () r] -> Spec
+minkowskiTests' s (map toCCW -> pgs) = describe ("Minkowskisums on " ++ s) $
+    mapM_ (uncurry minkowskiTest) [ (p,q) | p <- pgs, q <- pgs ]
+
+
+minkowskiTest     ::  (Fractional r, Ord r, Eq p, Show r, Show p)
+                  => ConvexPolygon p r -> ConvexPolygon p r -> Spec
+minkowskiTest p q = it "minkowskisum" $
+  F (p,q) (minkowskiSum p q) `shouldBe` F (p,q) (naiveMinkowski p q)
+
+naiveMinkowski     :: (Fractional r, Ord r)
+                   => ConvexPolygon p r -> ConvexPolygon q r -> ConvexPolygon (p, q) r
+naiveMinkowski p q = over (simplePolygon.outerBoundary) bottomMost
+                   . toCCW . convexHull . NonEmpty.fromList
+                   $ [ v .+. w | v <- p^..simplePolygon.outerBoundary.traverse
+                               , w <- q^..simplePolygon.outerBoundary.traverse
+                     ]
+  where
+    (v :+ ve) .+. (w :+ we) = v .+^ (toVec w) :+ (ve,we)
+
+
+toCCW :: (Fractional r, Eq r) => ConvexPolygon p r -> ConvexPolygon p r
+toCCW = over simplePolygon toCounterClockWiseOrder
+
+data F a b = F a b deriving (Show)
+
+instance Eq b => Eq (F a b) where
+  (F _ b1) == (F _ b2) = b1 == b2
+
+
+newtype CP r = CP (ConvexPolygon () r) deriving (Eq,Show)
+
+instance (Arbitrary r, Fractional r, Ord r) => Arbitrary (CP r) where
+  arbitrary =  CP . toCCW <$> suchThat (convexHull <$> arbitrary)
+                              (\p -> p^.simplePolygon.outerBoundary.to length > 2)
diff --git a/test/Data/PlanarGraphSpec.hs b/test/Data/PlanarGraphSpec.hs
new file mode 100644
--- /dev/null
+++ b/test/Data/PlanarGraphSpec.hs
@@ -0,0 +1,97 @@
+module Data.PlanarGraphSpec where
+
+
+import           Data.Util
+import           Data.PlanarGraph
+import           Data.Permutation(toCycleRep)
+import           Test.Hspec
+import qualified Data.Foldable as F
+import qualified Data.Set as S
+import qualified Data.Vector as V
+import qualified Data.Map.Strict as SM
+import           Data.Semigroup
+
+
+
+data TestG
+
+type Vertex = VertexId TestG Primal_
+
+-- | Report all adjacnecies from g missing in h
+missingAdjacencies     :: PlanarGraph s w v e f -> PlanarGraph s w v e f
+                    -> [(VertexId s w, VertexId s w)]
+missingAdjacencies g h = concatMap f . vertices' $ g
+  where
+    f u = let adjUh = S.fromList . F.toList $ neighboursOf u h
+          in F.toList . fmap (u,) . V.filter (`S.notMember` adjUh) $ neighboursOf u g
+
+
+sameGraphs s g h = do
+    describe ("Same Adjacencies " <> s) $ do
+      it "Missing edges from g in h" $
+          (missingAdjacencies g h) `shouldBe` []
+      it "Missing edges from h in g" $
+          (missingAdjacencies h g) `shouldBe` []
+
+
+
+
+spec :: Spec
+spec = sameGraphs "testEdges" (fromAdjacencyLists testEdges) (fromAdjacencyListsOld testEdges)
+
+
+
+
+testEdges :: [(Vertex,[Vertex])]
+testEdges = map (\(i,vs) -> (VertexId i, map VertexId vs))
+            [ (0, [1])
+            , (1, [0,2,4])
+            , (2, [1,3,4])
+            , (3, [2,5])
+            , (4, [1,2,5])
+            , (5, [3,4])
+            ]
+
+--------------------------------------------------------------------------------
+
+
+-- - m: a Map, mapping edges, represented by a pair of vertexId's (u,v) with
+--            u < v, to arcId's.
+-- - a: the next available unused arcID
+-- - x: the data value we are interested in computing
+type STR' s b = STR (SM.Map (VertexId s Primal_,VertexId s Primal_) Int) Int b
+
+-- | Construct a planar graph from a adjacency matrix. For every vertex, all
+-- vertices should be given in counter clockwise order.
+--
+-- running time: $O(n \log n)$.
+fromAdjacencyListsOld      :: forall s f.(Foldable f, Functor f)
+                        => [(VertexId s Primal_, f (VertexId s Primal_))]
+                        -> PlanarGraph s Primal_ () () ()
+fromAdjacencyListsOld adjM = planarGraph' . toCycleRep n $ perm
+  where
+    n    = sum . fmap length $ perm
+    perm = trd' . foldr toOrbit (STR mempty 0 mempty) $ adjM
+
+
+    -- | Given a vertex with its adjacent vertices (u,vs) (in CCW order) convert this
+    -- vertex with its adjacent vertices into an Orbit
+    toOrbit                     :: Foldable f
+                                => (VertexId s Primal_, f (VertexId s Primal_))
+                                -> STR' s [[Dart s]]
+                                -> STR' s [[Dart s]]
+    toOrbit (u,vs) (STR m a dss) =
+      let (STR m' a' ds') = foldr (toDart . (u,)) (STR m a mempty) . F.toList $ vs
+      in STR m' a' (ds':dss)
+
+
+    -- | Given an edge (u,v) and a triplet (m,a,ds) we construct a new dart
+    -- representing this edge.
+    toDart                    :: (VertexId s Primal_,VertexId s Primal_)
+                              -> STR' s [Dart s]
+                              -> STR' s [Dart s]
+    toDart (u,v) (STR m a ds) = let dir = if u < v then Positive else Negative
+                                    t'  = (min u v, max u v)
+                               in case SM.lookup t' m of
+      Just a' -> STR m                  a     (Dart (Arc a') dir : ds)
+      Nothing -> STR (SM.insert t' a m) (a+1) (Dart (Arc a)  dir : ds)
diff --git a/test/QuickCheck/Instances.hs b/test/QuickCheck/Instances.hs
new file mode 100644
--- /dev/null
+++ b/test/QuickCheck/Instances.hs
@@ -0,0 +1,91 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# OPTIONS_GHC -fno-warn-orphans #-}
+module QuickCheck.Instances where
+
+import           Control.Lens
+import           Data.BinaryTree
+import           Data.Ext
+import           Data.Geometry hiding (vector)
+import           Data.Geometry.Box
+import           Data.Geometry.Interval
+import           Data.Geometry.SubLine
+import           Data.Geometry.Vector
+import qualified Data.List.NonEmpty as NonEmpty
+import           Data.Proxy
+import           Data.Range
+import           Data.Semigroup
+import qualified Data.Seq as Seq
+import qualified Data.Seq2 as S2
+import           GHC.TypeLits
+import           Test.QuickCheck
+
+--------------------------------------------------------------------------------
+
+-- instance Arbitrary a => Arbitrary (NonEmpty.NonEmpty a) where
+--   arbitrary = NonEmpty.fromList <$> listOf1 arbitrary
+
+instance Arbitrary a => Arbitrary (S2.Seq2 a) where
+  arbitrary = S2.Seq2 <$> arbitrary <*> arbitrary <*> arbitrary
+
+instance Arbitrary a => Arbitrary (BinaryTree a) where
+  arbitrary = sized f
+    where f n | n <= 0    = pure Nil
+              | otherwise = do
+                              l <- choose (0,n-1)
+                              Internal <$> f l <*> arbitrary <*> f (n-l-1)
+
+instance (Arbitrary a, Arbitrary v) => Arbitrary (BinLeafTree v a) where
+  arbitrary = sized f
+    where f n | n <= 0    = Leaf <$> arbitrary
+              | otherwise = do
+                              l <- choose (0,n-1)
+                              Node <$> f l <*> arbitrary <*> f (n-l-1)
+
+
+instance (KnownNat n, Arbitrary a) => Arbitrary (Seq.LSeq n a) where
+  arbitrary = (\s s' -> Seq.promise . Seq.fromList $ s <> s')
+            <$> vector (fromInteger . natVal $ (Proxy :: Proxy n))
+            <*> arbitrary
+
+instance (Arbitrary r, Arity d) => Arbitrary (Vector d r) where
+  arbitrary = vectorFromListUnsafe <$> infiniteList
+
+instance (Arbitrary r, Arity d) => Arbitrary (Point d r) where
+  arbitrary = Point <$> arbitrary
+
+instance (Arbitrary r, Arity d, Num r) => Arbitrary (Line d r) where
+  arbitrary = lineThrough <$> arbitrary <*> arbitrary
+
+instance (Arbitrary r, Arity d, Ord r) => Arbitrary (Box d () r) where
+  arbitrary = (\p (q :: Point d r) -> boundingBoxList' [p,q]) <$> arbitrary <*> arbitrary
+
+
+instance Arbitrary r => Arbitrary (EndPoint r) where
+  arbitrary = frequency [ (1, Open   <$> arbitrary)
+                        , (9, Closed <$> arbitrary)
+                        ]
+
+instance (Arbitrary r, Ord r) => Arbitrary (Range r) where
+  arbitrary = do
+                l <- arbitrary
+                r <- suchThat arbitrary (p l)
+                return $ Range l r
+   where
+     p (Open l)   r = l <  r^.unEndPoint
+     p (Closed l) r = l <= r^.unEndPoint
+
+
+instance (Arbitrary c, Arbitrary e) => Arbitrary (c :+ e) where
+  arbitrary = (:+) <$> arbitrary <*> arbitrary
+
+instance (Arbitrary r, Arbitrary p, Ord r, Ord p) => Arbitrary (Interval p r) where
+  arbitrary = GInterval <$> arbitrary
+
+
+instance (Arbitrary r, Arbitrary p, Arity d, Ord r, Ord p, Num r)
+         => Arbitrary (SubLine d p r) where
+  arbitrary = SubLine <$> arbitrary <*> arbitrary
+
+
+instance (Arbitrary r, Arbitrary p, Arity d) => Arbitrary (LineSegment d p r) where
+  arbitrary = LineSegment <$> arbitrary <*> arbitrary
diff --git a/test/Util.hs b/test/Util.hs
--- a/test/Util.hs
+++ b/test/Util.hs
@@ -12,3 +12,13 @@
                . map (\x -> (x,lookup' x))
   where
     lookup' (_ :+ ats) = lookupAttr (Proxy :: Proxy Stroke) ats
+
+-- | Computes all elements on which the two lists differ
+difference :: Eq a => [a] -> [a] -> [a]
+difference xs ys = (xs L.\\ ys) ++ (ys L.\\ xs)
+
+-- differenceBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]
+
+
+diffBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]
+diffBy p xs ys = foldr (L.deleteBy p) ys xs