manifolds-0.2.2.0: Data/Manifold/TreeCover.hs
-- |
-- Module : Data.Manifold.TreeCover
-- Copyright : (c) Justus Sagemüller 2015
-- License : GPL v3
--
-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
-- Stability : experimental
-- Portability : portable
--
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE DeriveFoldable #-}
{-# LANGUAGE DeriveTraversable #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE ParallelListComp #-}
{-# LANGUAGE MonadComprehensions #-}
{-# LANGUAGE UnicodeSyntax #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE ViewPatterns #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE LiberalTypeSynonyms #-}
{-# LANGUAGE RecordWildCards #-}
{-# LANGUAGE DataKinds #-}
module Data.Manifold.TreeCover (
-- * Shades
Shade(..), pattern(:±), Shade'(..), (|±|), IsShade
-- ** Lenses
, shadeCtr, shadeExpanse, shadeNarrowness
-- ** Construction
, fullShade, fullShade', pointsShades, pointsCovers
-- ** Evaluation
, occlusion
-- ** Misc
, factoriseShade, intersectShade's, Refinable, refineShade', convolveShade', coerceShade
-- * Shade trees
, ShadeTree(..), fromLeafPoints, onlyLeaves, indexShadeTree
-- * View helpers
, onlyNodes
-- ** Auxiliary types
, SimpleTree, Trees, NonEmptyTree, GenericTree(..)
-- * Misc
, sShSaw, chainsaw, HasFlatView(..), shadesMerge, smoothInterpolate
, twigsWithEnvirons, completeTopShading, flexTwigsShading
, WithAny(..), Shaded, stiAsIntervalMapping, spanShading
, constShaded, stripShadedUntopological
, DifferentialEqn, filterDEqnSolution_loc
-- ** Triangulation-builders
, TriangBuild, doTriangBuild, singleFullSimplex, autoglueTriangulation
, AutoTriang, elementaryTriang, breakdownAutoTriang
) where
import Data.List hiding (filter, all, elem, sum, foldr1)
import Data.Maybe
import qualified Data.Map as Map
import qualified Data.Vector as Arr
import Data.List.NonEmpty (NonEmpty(..))
import Data.List.FastNub
import qualified Data.List.NonEmpty as NE
import Data.Semigroup
import Data.Ord (comparing)
import Control.DeepSeq
import Data.VectorSpace
import Data.AffineSpace
import Data.LinearMap
import Data.LinearMap.HerMetric
import Data.LinearMap.Category
import Data.AffineSpace
import Data.Basis
import Data.Complex hiding (magnitude)
import Data.Void
import Data.Tagged
import Data.Proxy
import Data.SimplicialComplex
import Data.Manifold.Types
import Data.Manifold.Types.Primitive ((^), empty)
import Data.Manifold.PseudoAffine
import Data.Function.Differentiable
import Data.Function.Differentiable.Data
import Data.SetLike.Intersection
import Data.Embedding
import Data.CoNat
import qualified Prelude as Hask hiding(foldl, sum, sequence)
import qualified Control.Applicative as Hask
import qualified Control.Monad as Hask hiding(forM_, sequence)
import Data.Functor.Identity
import Control.Monad.Trans.State
import Control.Monad.Trans.Writer
import Control.Monad.Trans.Maybe
import Control.Monad.Trans.Class
import qualified Data.Foldable as Hask
import Data.Foldable (all, elem, toList, sum, foldr1)
import qualified Data.Traversable as Hask
import Data.Traversable (forM)
import qualified Numeric.LinearAlgebra.HMatrix as HMat
import Control.Category.Constrained.Prelude hiding
((^), all, elem, sum, forM, Foldable(..), foldr1, Traversable, traverse)
import Control.Arrow.Constrained
import Control.Monad.Constrained hiding (forM)
import Data.Foldable.Constrained
import Data.Traversable.Constrained (traverse)
import GHC.Generics (Generic)
-- | Possibly / Partially / asymPtotically singular metric.
data PSM x = PSM {
psmExpanse :: !(Metric' x)
, relevantEigenspan :: ![Needle' x]
}
-- | A 'Shade' is a very crude description of a region within a manifold. It
-- can be interpreted as either an ellipsoid shape, or as the Gaussian peak
-- of a normal distribution (use <http://hackage.haskell.org/package/manifold-random>
-- for actually sampling from that distribution).
--
-- For a /precise/ description of an arbitrarily-shaped connected subset of a manifold,
-- there is 'Region', whose implementation is vastly more complex.
data Shade x = Shade { _shadeCtr :: !(Interior x)
, _shadeExpanse :: !(Metric' x) }
deriving instance (Show x, Show (Needle x), WithField ℝ Manifold x) => Show (Shade x)
-- | A “co-shade” can describe ellipsoid regions as well, but unlike
-- 'Shade' it can be unlimited / infinitely wide in some directions.
-- It does OTOH need to have nonzero thickness, which 'Shade' needs not.
data Shade' x = Shade' { _shade'Ctr :: !(Interior x)
, _shade'Narrowness :: !(Metric x) }
deriving instance (Show x, Show (DualSpace (Needle x)), WithField ℝ Manifold x)
=> Show (Shade' x)
class IsShade shade where
-- type (*) shade :: *->*
-- | Access the center of a 'Shade' or a 'Shade''.
shadeCtr :: Functor f (->) (->) => (Interior x->f (Interior x)) -> shade x -> f (shade x)
-- -- | Convert between 'Shade' and 'Shade' (which must be neither singular nor infinite).
-- unsafeDualShade :: WithField ℝ Manifold x => shade x -> shade* x
-- | Check the statistical likelihood-density of a point being within a shade.
-- This is taken as a normal distribution.
occlusion :: ( Manifold x, s ~ (Scalar (Needle x)), RealDimension s )
=> shade x -> x -> s
factoriseShade :: ( Manifold x, RealDimension (Scalar (Needle x))
, Manifold y, RealDimension (Scalar (Needle y)) )
=> shade (x,y) -> (shade x, shade y)
coerceShade :: (Manifold x, Manifold y, LocallyCoercible x y) => shade x -> shade y
instance IsShade Shade where
shadeCtr f (Shade c e) = fmap (`Shade`e) $ f c
occlusion (Shade p₀ δ) = occ
where occ p = case p .-~. p₀ of
Option(Just vd) | mSq <- metricSq δinv vd
, mSq == mSq -- avoid NaN
-> exp (negate mSq)
_ -> zeroV
δinv = recipMetric δ
factoriseShade (Shade (x₀,y₀) δxy) = (Shade x₀ δx, Shade y₀ δy)
where (δx,δy) = factoriseMetric' δxy
coerceShade (Shade x (HerMetric' δxym))
= Shade (locallyTrivialDiffeomorphism x) (HerMetric' $ unsafeCoerceLinear<$>δxym)
shadeExpanse :: Functor f (->) (->) => (Metric' x -> f (Metric' x)) -> Shade x -> f (Shade x)
shadeExpanse f (Shade c e) = fmap (Shade c) $ f e
instance IsShade Shade' where
shadeCtr f (Shade' c e) = fmap (`Shade'`e) $ f c
occlusion (Shade' p₀ δinv) = occ
where occ p = case p .-~. p₀ of
Option(Just vd) | mSq <- metricSq δinv vd
, mSq == mSq -- avoid NaN
-> exp (negate mSq)
_ -> zeroV
factoriseShade (Shade' (x₀,y₀) δxy) = (Shade' x₀ δx, Shade' y₀ δy)
where (δx,δy) = factoriseMetric δxy
coerceShade (Shade' x (HerMetric δxym))
= Shade' (locallyTrivialDiffeomorphism x) (HerMetric $ unsafeCoerceLinear<$>δxym)
shadeNarrowness :: Functor f (->) (->) => (Metric x -> f (Metric x)) -> Shade' x -> f (Shade' x)
shadeNarrowness f (Shade' c e) = fmap (Shade' c) $ f e
instance (AffineManifold x) => Semimanifold (Shade x) where
type Needle (Shade x) = Diff x
fromInterior = id
toInterior = pure
translateP = Tagged (.+~^)
Shade c e .+~^ v = Shade (c.+^v) e
Shade c e .-~^ v = Shade (c.-^v) e
fullShade :: WithField ℝ Manifold x => x -> Metric' x -> Shade x
fullShade ctr expa = Shade ctr expa
fullShade' :: WithField ℝ Manifold x => x -> Metric x -> Shade' x
fullShade' ctr expa = Shade' ctr expa
-- | Span a 'Shade' from a center point and multiple deviation-vectors.
pattern (:±) :: () => WithField ℝ Manifold x => x -> [Needle x] -> Shade x
pattern x :± shs <- Shade x (eigenSpan -> shs)
where x :± shs = fullShade x $ projector's shs
-- | Similar to ':±', but instead of expanding the shade, each vector /restricts/ it.
-- Iff these form a orthogonal basis (in whatever sense applicable), then both
-- methods will be equivalent.
--
-- Note that '|±|' is only possible, as such, in an inner-product space; in
-- general you need reciprocal vectors ('Needle'') to define a 'Shade''.
(|±|) :: WithField ℝ EuclidSpace x => x -> [Needle x] -> Shade' x
x |±| shs = Shade' x $ projectors [v^/(v<.>v) | v<-shs]
subshadeId' :: WithField ℝ Manifold x
=> x -> NonEmpty (Needle' x) -> x -> (Int, HourglassBulb)
subshadeId' c expvs x = case x .-~. c of
Option (Just v) -> let (iu,vl) = maximumBy (comparing $ abs . snd)
$ zip [0..] (map (v <.>^) $ NE.toList expvs)
in (iu, if vl>0 then UpperBulb else LowerBulb)
_ -> (-1, error "Trying to obtain the subshadeId of a point not actually included in the shade.")
subshadeId :: WithField ℝ Manifold x => Shade x -> x -> (Int, HourglassBulb)
subshadeId (Shade c expa) = subshadeId' c . NE.fromList $ eigenCoSpan expa
-- | Attempt to find a 'Shade' that describes the distribution of given points.
-- At least in an affine space (and thus locally in any manifold), this can be used to
-- estimate the parameters of a normal distribution from which some points were
-- sampled. Note that some points will be “outside” of the shade,
-- as happens for a normal distribution with some statistical likelyhood.
-- (Use 'pointsCovers' if you need to prevent that.)
--
-- For /nonconnected/ manifolds it will be necessary to yield separate shades
-- for each connected component. And for an empty input list, there is no shade!
-- Hence the result type is a list.
pointsShades :: WithField ℝ Manifold x => [x] -> [Shade x]
pointsShades = map snd . pointsShades' zeroV
-- | Like 'pointsShades', but ensure that all points are actually in
-- the shade, i.e. if @['Shade' x₀ ex]@ is the result then
-- @'metric' (recipMetric ex) (p-x₀) ≤ 1@ for all @p@ in the list.
pointsCovers :: ∀ x . WithField ℝ Manifold x => [x] -> [Shade x]
pointsCovers = map guaranteeIn . pointsShades' zeroV
where guaranteeIn (ps, Shade x₀ ex)
= case ps >>= \p -> let Option (Just v) = p.-~.x₀
in guard (metric ex' v > 1) >> [(p,projector' v)]
of [] -> Shade x₀ ex
outs -> guaranteeIn ( fst<$>outs
, Shade x₀
$ ex ^+^ sumV (snd<$>outs)
^/ fromIntegral (2 * length outs) )
where ex' = recipMetric ex
pointsShade's :: WithField ℝ Manifold x => [x] -> [Shade' x]
pointsShade's = map (\(Shade c e) -> Shade' c $ recipMetric e) . pointsShades
pointsCover's :: WithField ℝ Manifold x => [x] -> [Shade' x]
pointsCover's = map (\(Shade c e) -> Shade' c $ recipMetric e) . pointsCovers
pseudoECM :: WithField ℝ Manifold x => NonEmpty x -> (x, ([x],[x]))
pseudoECM (p₀ NE.:| psr) = foldl' ( \(acc, (rb,nr)) (i,p)
-> case p.-~.acc of
Option (Just δ) -> (acc .+~^ δ^/i, (p:rb, nr))
_ -> (acc, (rb, p:nr)) )
(p₀, mempty)
( zip [1..] $ p₀:psr )
pointsShades' :: WithField ℝ Manifold x => Metric' x -> [x] -> [([x], Shade x)]
pointsShades' _ [] = []
pointsShades' minExt ps = case expa of
Option (Just e) -> (ps, fullShade ctr e)
: pointsShades' minExt unreachable
_ -> pointsShades' minExt inc'd
++ pointsShades' minExt unreachable
where (ctr,(inc'd,unreachable)) = pseudoECM $ NE.fromList ps
expa = ( (^+^minExt) . (^/ fromIntegral(length ps)) . projector's )
<$> mapM (.-~.ctr) ps
-- | Attempt to reduce the number of shades to fewer (ideally, a single one).
-- In the simplest cases these should guaranteed cover the same area;
-- for non-flat manifolds it only works in a heuristic sense.
shadesMerge :: WithField ℝ Manifold x
=> ℝ -- ^ How near (inverse normalised distance, relative to shade expanse)
-- two shades must be to be merged. If this is zero, any shades
-- in the same connected region of a manifold are merged.
-> [Shade x] -- ^ A list of /n/ shades.
-> [Shade x] -- ^ /m/ ≤ /n/ shades which cover at least the same area.
shadesMerge fuzz (sh₁@(Shade c₁ e₁) : shs) = case extractJust tryMerge shs of
(Just mg₁, shs') -> shadesMerge fuzz
$ shs'++[mg₁] -- Append to end to prevent undue weighting
-- of first shade and its mergers.
(_, shs') -> sh₁ : shadesMerge fuzz shs'
where tryMerge (Shade c₂ e₂)
| Option (Just v) <- c₁.-~.c₂
, Option (Just v') <- c₂.-~.c₁
, [e₁',e₂'] <- recipMetric<$>[e₁, e₂]
, b₁ <- metric e₂' v
, b₂ <- metric e₁' v
, fuzz*b₁*b₂ <= b₁ + b₂
= Just $ let cc = c₂ .+~^ v ^/ 2
Option (Just cv₁) = c₁.-~.cc
Option (Just cv₂) = c₂.-~.cc
in Shade cc $ e₁ ^+^ e₂ ^+^ projector's [cv₁, cv₂]
| otherwise = Nothing
shadesMerge _ shs = shs
-- | Evaluate the shade as a quadratic form; essentially
-- @
-- minusLogOcclusion sh x = x <.>^ (sh^.shadeExpanse $ x - sh^.shadeCtr)
-- @
-- where 'shadeExpanse' gives a metric (matrix) that characterises the
-- width of the shade.
minusLogOcclusion' :: ( Manifold x, s ~ (Scalar (Needle x)), RealDimension s )
=> Shade' x -> x -> s
minusLogOcclusion' (Shade' p₀ δinv) = occ
where occ p = case p .-~. p₀ of
Option(Just vd) | mSq <- metricSq δinv vd
, mSq == mSq -- avoid NaN
-> mSq
_ -> 1/0
minusLogOcclusion :: ( Manifold x, s ~ (Scalar (Needle x)), RealDimension s )
=> Shade x -> x -> s
minusLogOcclusion (Shade p₀ δ) = occ
where occ p = case p .-~. p₀ of
Option(Just vd) | mSq <- metricSq δinv vd
, mSq == mSq -- avoid NaN
-> mSq
_ -> 1/0
δinv = recipMetric δ
-- | Hourglass as the geometric shape (two opposing ~conical volumes, sharing
-- only a single point in the middle); has nothing to do with time.
data Hourglass s = Hourglass { upperBulb, lowerBulb :: !s }
deriving (Generic, Hask.Functor, Hask.Foldable)
instance (NFData s) => NFData (Hourglass s)
instance (Semigroup s) => Semigroup (Hourglass s) where
Hourglass u l <> Hourglass u' l' = Hourglass (u<>u') (l<>l')
sconcat hgs = let (us,ls) = NE.unzip $ (upperBulb&&&lowerBulb) <$> hgs
in Hourglass (sconcat us) (sconcat ls)
instance (Monoid s, Semigroup s) => Monoid (Hourglass s) where
mempty = Hourglass mempty mempty; mappend = (<>)
mconcat hgs = let (us,ls) = unzip $ (upperBulb&&&lowerBulb) <$> hgs
in Hourglass (mconcat us) (mconcat ls)
instance Hask.Applicative Hourglass where
pure x = Hourglass x x
Hourglass f g <*> Hourglass x y = Hourglass (f x) (g y)
instance Foldable Hourglass (->) (->) where
ffoldl f (x, Hourglass a b) = f (f(x,a), b)
foldMap f (Hourglass a b) = f a `mappend` f b
flipHour :: Hourglass s -> Hourglass s
flipHour (Hourglass u l) = Hourglass l u
data HourglassBulb = UpperBulb | LowerBulb
oneBulb :: HourglassBulb -> (a->a) -> Hourglass a->Hourglass a
oneBulb UpperBulb f (Hourglass u l) = Hourglass (f u) l
oneBulb LowerBulb f (Hourglass u l) = Hourglass u (f l)
data ShadeTree x = PlainLeaves [x]
| DisjointBranches !Int (NonEmpty (ShadeTree x))
| OverlappingBranches !Int !(Shade x) (NonEmpty (DBranch x))
deriving (Generic)
data DBranch' x c = DBranch { boughDirection :: !(Needle' x)
, boughContents :: !(Hourglass c) }
deriving (Generic, Hask.Functor, Hask.Foldable)
type DBranch x = DBranch' x (ShadeTree x)
newtype DBranches' x c = DBranches (NonEmpty (DBranch' x c))
deriving (Generic, Hask.Functor, Hask.Foldable)
-- ^ /Unsafe/: this assumes the direction information of both containers to be equivalent.
instance (Semigroup c) => Semigroup (DBranches' x c) where
DBranches b1 <> DBranches b2 = DBranches $ NE.zipWith (\(DBranch d1 c1) (DBranch _ c2)
-> DBranch d1 $ c1<>c2 ) b1 b2
directionChoices :: WithField ℝ Manifold x
=> [DBranch x]
-> [ ( (Needle' x, ShadeTree x)
,[(Needle' x, ShadeTree x)] ) ]
directionChoices [] = []
directionChoices (DBranch ѧ (Hourglass t b) : hs)
= ( (ѧ,t), (v,b) : map fst uds)
: ((v,b), (ѧ,t) : map fst uds)
: map (second $ ((ѧ,t):) . ((v,b):)) uds
where v = negateV ѧ
uds = directionChoices hs
traverseDirectionChoices :: (WithField ℝ Manifold x, Hask.Applicative f)
=> ( (Int, (Needle' x, ShadeTree x))
-> [(Int, (Needle' x, ShadeTree x))]
-> f (ShadeTree x) )
-> [DBranch x]
-> f [DBranch x]
traverseDirectionChoices f dbs
= td [] . scanLeafNums 0
$ dbs >>= \(DBranch ѧ (Hourglass τ β))
-> [(ѧ,τ), (negateV ѧ,β)]
where td pds (ѧt@(_,(ѧ,_)):vb:vds)
= liftA3 (\t' b' -> (DBranch ѧ (Hourglass t' b') :))
(f ѧt $ vb:uds)
(f vb $ ѧt:uds)
$ td (ѧt:vb:pds) vds
where uds = pds ++ vds
td _ _ = pure []
scanLeafNums _ [] = []
scanLeafNums i₀ ((v,t):vts) = (i₀, (v,t)) : scanLeafNums (i₀ + nLeaves t) vts
instance (NFData x, NFData (Needle' x)) => NFData (ShadeTree x) where
rnf (PlainLeaves xs) = rnf xs
rnf (DisjointBranches n bs) = n `seq` rnf (NE.toList bs)
rnf (OverlappingBranches n sh bs) = n `seq` sh `seq` rnf (NE.toList bs)
instance (NFData x, NFData (Needle' x)) => NFData (DBranch x)
-- | Experimental. There might be a more powerful instance possible.
instance (AffineManifold x) => Semimanifold (ShadeTree x) where
type Needle (ShadeTree x) = Diff x
fromInterior = id
toInterior = pure
translateP = Tagged (.+~^)
PlainLeaves xs .+~^ v = PlainLeaves $ (.+^v)<$>xs
OverlappingBranches n sh br .+~^ v
= OverlappingBranches n (sh.+~^v)
$ fmap (\(DBranch d c) -> DBranch d $ (.+~^v)<$>c) br
DisjointBranches n br .+~^ v = DisjointBranches n $ (.+~^v)<$>br
-- | WRT union.
instance WithField ℝ Manifold x => Semigroup (ShadeTree x) where
PlainLeaves [] <> t = t
t <> PlainLeaves [] = t
t <> s = fromLeafPoints $ onlyLeaves t ++ onlyLeaves s
-- Could probably be done more efficiently
sconcat = mconcat . NE.toList
instance WithField ℝ Manifold x => Monoid (ShadeTree x) where
mempty = PlainLeaves []
mappend = (<>)
mconcat l = case filter ne l of
[] -> mempty
[t] -> t
l' -> fromLeafPoints $ onlyLeaves =<< l'
where ne (PlainLeaves []) = False; ne _ = True
-- | Build a quite nicely balanced tree from a cloud of points, on any real manifold.
--
-- Example: https://nbviewer.jupyter.org/github/leftaroundabout/manifolds/blob/master/test/Trees-and-Webs.ipynb#pseudorandomCloudTree
--
-- <<images/examples/simple-2d-ShadeTree.png>>
fromLeafPoints :: ∀ x. WithField ℝ Manifold x => [x] -> ShadeTree x
fromLeafPoints = fromLeafPoints' sShIdPartition
indexShadeTree :: ∀ x . WithField ℝ Manifold x
=> ShadeTree x -> Int -> Either Int ([ShadeTree x], x)
indexShadeTree _ i
| i<0 = Left i
indexShadeTree sh@(PlainLeaves lvs) i = case length lvs of
n | i<n -> Right ([sh], lvs!!i)
| otherwise -> Left $ i-n
indexShadeTree (DisjointBranches n brs) i
| i<n = foldl (\case
Left i' -> (`indexShadeTree`i')
result -> return result
) (Left i) brs
| otherwise = Left $ i-n
indexShadeTree sh@(OverlappingBranches n _ brs) i
| i<n = first (sh:) <$> foldl (\case
Left i' -> (`indexShadeTree`i')
result -> return result
) (Left i) (toList brs>>=toList)
| otherwise = Left $ i-n
fromFnGraphPoints :: ∀ x y . (WithField ℝ Manifold x, WithField ℝ Manifold y)
=> [(x,y)] -> ShadeTree (x,y)
fromFnGraphPoints = fromLeafPoints' fg_sShIdPart
where fg_sShIdPart :: Shade (x,y) -> [(x,y)] -> NonEmpty (DBranch' (x,y) [(x,y)])
fg_sShIdPart (Shade c expa) xs
| b:bs <- [DBranch (v, zeroV) mempty
| v <- eigenCoSpan
(transformMetric' fst expa :: Metric' x) ]
= sShIdPartition' c xs $ b:|bs
fromLeafPoints' :: ∀ x. WithField ℝ Manifold x =>
(Shade x -> [x] -> NonEmpty (DBranch' x [x])) -> [x] -> ShadeTree x
fromLeafPoints' sShIdPart = go zeroV
where go :: Metric' x -> [x] -> ShadeTree x
go preShExpa = \xs -> case pointsShades' (preShExpa^/10) xs of
[] -> mempty
[(_,rShade)] -> let trials = sShIdPart rShade xs
in case reduce rShade trials of
Just redBrchs
-> OverlappingBranches
(length xs) rShade
(branchProc (_shadeExpanse rShade) redBrchs)
_ -> PlainLeaves xs
partitions -> DisjointBranches (length xs)
. NE.fromList
$ map (\(xs',pShade) -> go zeroV xs') partitions
where
branchProc redSh = fmap (fmap $ go redSh)
reduce :: Shade x -> NonEmpty (DBranch' x [x])
-> Maybe (NonEmpty (DBranch' x [x]))
reduce sh@(Shade c _) brCandidates
= case findIndex deficient cards of
Just i | (DBranch _ reBr, o:ok)
<- amputateId i (NE.toList brCandidates)
-> reduce sh
$ sShIdPartition' c (fold reBr) (o:|ok)
| otherwise -> Nothing
_ -> Just brCandidates
where (cards, maxCard) = (NE.toList &&& maximum')
$ fmap (fmap length . boughContents) brCandidates
deficient (Hourglass u l) = any (\c -> c^2 <= maxCard + 1) [u,l]
maximum' = maximum . NE.toList . fmap (\(Hourglass u l) -> max u l)
sShIdPartition' :: WithField ℝ Manifold x
=> x -> [x] -> NonEmpty (DBranch' x [x])->NonEmpty (DBranch' x [x])
sShIdPartition' c xs st
= foldr (\p -> let (i,h) = ssi p
in asList $ update_nth (\(DBranch d c)
-> DBranch d (oneBulb h (p:) c))
i )
st xs
where ssi = subshadeId' c (boughDirection<$>st)
sShIdPartition :: WithField ℝ Manifold x => Shade x -> [x] -> NonEmpty (DBranch' x [x])
sShIdPartition (Shade c expa) xs
| b:bs <- [DBranch v mempty | v <- eigenCoSpan expa]
= sShIdPartition' c xs $ b:|bs
asList :: ([a]->[b]) -> NonEmpty a->NonEmpty b
asList f = NE.fromList . f . NE.toList
update_nth :: (a->a) -> Int -> [a] -> [a]
update_nth _ n l | n<0 = l
update_nth f 0 (c:r) = f c : r
update_nth f n [] = []
update_nth f n (l:r) = l : update_nth f (n-1) r
amputateId :: Int -> [a] -> (a,[a])
amputateId i l = let ([a],bs) = amputateIds [i] l in (a, bs)
deleteIds :: [Int] -> [a] -> [a]
deleteIds kids = snd . amputateIds kids
amputateIds :: [Int] -- ^ Sorted list of non-negative indices to extract
-> [a] -- ^ Input list
-> ([a],[a]) -- ^ (Extracted elements, remaining elements)
amputateIds = go 0
where go _ _ [] = ([],[])
go _ [] l = ([],l)
go i (k:ks) (x:xs)
| i==k = first (x:) $ go (i+1) ks xs
| otherwise = second (x:) $ go (i+1) (k:ks) xs
sortByKey :: Ord a => [(a,b)] -> [b]
sortByKey = map snd . sortBy (comparing fst)
trunks :: ∀ x. WithField ℝ Manifold x => ShadeTree x -> [Shade x]
trunks (PlainLeaves lvs) = pointsCovers lvs
trunks (DisjointBranches _ brs) = Hask.foldMap trunks brs
trunks (OverlappingBranches _ sh _) = [sh]
nLeaves :: ShadeTree x -> Int
nLeaves (PlainLeaves lvs) = length lvs
nLeaves (DisjointBranches n _) = n
nLeaves (OverlappingBranches n _ _) = n
overlappingBranches :: Shade x -> NonEmpty (DBranch x) -> ShadeTree x
overlappingBranches shx brs = OverlappingBranches n shx brs
where n = sum $ fmap (sum . fmap nLeaves) brs
unsafeFmapLeaves :: (x -> x) -> ShadeTree x -> ShadeTree x
unsafeFmapLeaves f (PlainLeaves lvs) = PlainLeaves $ fmap f lvs
unsafeFmapLeaves f (DisjointBranches n brs)
= DisjointBranches n $ unsafeFmapLeaves f <$> brs
unsafeFmapLeaves f (OverlappingBranches n sh brs)
= OverlappingBranches n sh $ fmap (unsafeFmapLeaves f) <$> brs
unsafeFmapTree :: (NonEmpty x -> NonEmpty y)
-> (Needle' x -> Needle' y)
-> (Shade x -> Shade y)
-> ShadeTree x -> ShadeTree y
unsafeFmapTree _ _ _ (PlainLeaves []) = PlainLeaves []
unsafeFmapTree f _ _ (PlainLeaves lvs) = PlainLeaves . toList . f $ NE.fromList lvs
unsafeFmapTree f fn fs (DisjointBranches n brs)
= let brs' = unsafeFmapTree f fn fs <$> brs
in DisjointBranches (sum $ nLeaves<$>brs') brs'
unsafeFmapTree f fn fs (OverlappingBranches n sh brs)
= let brs' = fmap (\(DBranch dir br)
-> DBranch (fn dir) (unsafeFmapTree f fn fs<$>br)
) brs
in overlappingBranches (fs sh) brs'
-- | Class of manifolds which can use 'Shade'' as a basic set type.
-- This is easily possible for vector spaces with the default implementations.
class (WithField ℝ Manifold y) => Refinable y where
-- | @a `subShade'` b ≡ True@ means @a@ is fully contained in @b@, i.e. from
-- @'minusLogOcclusion'' a p < 1@ follows also @minusLogOcclusion' b p < 1@.
subShade' :: Shade' y -> Shade' y -> Bool
subShade' (Shade' ac ae) tsh = all ((<1) . minusLogOcclusion' tsh)
[ ac.+~^σ*^v | σ<-[0,1], v<-eigenCoSpan' ae ]
-- | Specialised intersection operation. If @p@ is in @a@ and @b@, then it is
-- also in @refineShade' a b@. (The converse may not hold.)
refineShade' :: Shade' y -> Shade' y -> Option (Shade' y)
refineShade' (Shade' c e) (Shade' ζ η)
| μe < 1 && μη < 1 = return $ Shade' iCtr iExpa
| otherwise = empty
where [c', ζ'] = [ ctr.+~^linearCombo
[ (v, 1 / (1 + metricSq oExpa w))
| v <- (*^) <$> [-1,1] <*> span
, let p = ctr .+~^ v :: y
Option (Just w) = p.-~.oCtr
]
| ctr <- [c, ζ ]
| span <- eigenCoSpan'<$> [e, η ]
| (oCtr,oExpa) <- [(ζ,η), (c,e)]
]
Option (Just c'2ζ') = ζ'.-~.c'
Option (Just c2ζ') = ζ'.-~.c
Option (Just ζ2c') = c'.-~.ζ
μc = metricSq e c2ζ'
μζ = metricSq η ζ2c'
iCtr = c' .+~^ c'2ζ' ^* (μζ/(μc + μζ)) -- weighted mean between c' and ζ'.
Option (Just rc) = c.-~.iCtr
Option (Just rζ) = ζ.-~.iCtr
rcⰰ = toDualWith e rc
rζⰰ = toDualWith η rζ
μe = rcⰰ<.>^rc
μη = rζⰰ<.>^rζ
iExpa = (e^+^η)^/2 ^+^ projector rcⰰ^/(1-μe) ^+^ projector rζⰰ^/(1-μη)
-- | If @p@ is in @a@ (red) and @δ@ is in @b@ (green),
-- then @p.+~^δ@ is in @convolveShade' a b@ (blue).
--
-- Example: https://nbviewer.jupyter.org/github/leftaroundabout/manifolds/blob/master/test/ShadeCombinations.ipynb#shadeConvolutions
--
-- <<images/examples/ShadeCombinations/2Dconvolution-skewed.png>>
convolveShade' :: Shade' y -> Shade' (Needle y) -> Shade' y
convolveShade' (Shade' y₀ ey) (Shade' δ₀ eδ)
= Shade' (y₀.+~^δ₀)
( projectors [ f ^* ζ crl
| (f,_) <- eδsp
| crl <- corelap ] )
where (_,eδsp) = eigenSystem (ey,eδ)
corelap = map (metric ey . snd) eδsp
ζ = case filter (>0) corelap of
[] -> const 0
nzrelap
-> let cre₁ = 1/minimum nzrelap
cre₂ = maximum nzrelap
edgeFactor = sqrt ( (1 + cre₁)^2 + (1 + cre₂)^2 )
/ (sqrt (1 + cre₁^2) + sqrt (1 + cre₂^2))
in \case
0 -> 0
sq -> edgeFactor / (recip sq + 1)
instance Refinable ℝ where
refineShade' (Shade' cl el) (Shade' cr er)
= case (metricSq el 1, metricSq er 1) of
(0, _) -> return $ Shade' cr er
(_, 0) -> return $ Shade' cl el
(ql,qr) | ql>0, qr>0
-> let [rl,rr] = sqrt . recip <$> [ql,qr]
b = maximum $ zipWith (-) [cl,cr] [rl,rr]
t = minimum $ zipWith (+) [cl,cr] [rl,rr]
in guard (b<t) >>
let cm = (b+t)/2
rm = (t-b)/2
in return $ Shade' cm (projector $ recip rm)
-- convolveShade' (Shade' y₀ ey) (Shade' δ₀ eδ)
-- = case (metricSq ey 1, metricSq eδ 1) of
-- (wy,wδ) | wy>0, wδ>0
-- -> Shade' (y₀.+~^δ₀)
-- ( projector . recip
-- $ recip (sqrt wy) + recip (sqrt wδ) )
-- (_ , _) -> Shade' y₀ zeroV
instance (Refinable a, Refinable b) => Refinable (a,b)
intersectShade's :: ∀ y . Refinable y => NonEmpty (Shade' y) -> Option (Shade' y)
intersectShade's (sh:|shs) = Hask.foldrM refineShade' sh shs
type DifferentialEqn x y = Shade (x,y) -> Shade' (LocalLinear x y)
filterDEqnSolution_loc :: ∀ x y . (WithField ℝ Manifold x, Refinable y)
=> DifferentialEqn x y -> ((x, Shade' y), NonEmpty (x, Shade' y))
-> Option (Shade' y)
filterDEqnSolution_loc f ((x, shy@(Shade' y expay)), neighbours) = yc
where jShade@(Shade' j₀ jExpa) = f shxy
[shxy] = pointsCovers [ (xs, ys')
| (xs, Shade' ys yse) <- (x,shy):NE.toList neighbours
, δy <- eigenCoSpan' yse
, ys' <- [ys.+~^δy, ys.-~^δy] ]
[Shade' _ expax] = pointsCover's $ x : (fst<$>NE.toList neighbours)
marginδs :: NonEmpty (Needle x, (Needle y, Metric y))
marginδs = [ (δxm, (δym, expany))
| (xn, Shade' yn expany) <- neighbours
, let (Option (Just δxm)) = xn.-~.x
(Option (Just δym)) = yn.-~.y
]
back2Centre :: (Needle x, (Needle y, Metric y)) -> Shade' y
back2Centre (δx, (δym, expany))
= convolveShade'
(Shade' y expany)
(Shade' δyb $ applyLinMapMetric jExpa δx')
where δyb = δym ^-^ (j₀ $ δx)
δx' = toDualWith expax δx
yc :: Option (Shade' y)
yc = intersectShade's $ back2Centre <$> marginδs
xSpan = eigenCoSpan' expax
-- Formerly, this was the signature of what has now become 'traverseTwigsWithEnvirons'.
-- The simple list-yielding version (see rev. b4a427d59ec82889bab2fde39225b14a57b694df
-- may well be more efficient than this version via a traversal.
twigsWithEnvirons :: ∀ x. WithField ℝ Manifold x
=> ShadeTree x -> [((Int, ShadeTree x), [(Int, ShadeTree x)])]
twigsWithEnvirons = execWriter . traverseTwigsWithEnvirons (writer . (snd.fst&&&pure))
data OuterMaybeT f a = OuterNothing | OuterJust (f a) deriving (Hask.Functor)
instance (Hask.Applicative f) => Hask.Applicative (OuterMaybeT f) where
pure = OuterJust . pure
OuterJust fs <*> OuterJust xs = OuterJust $ fs <*> xs
_ <*> _ = OuterNothing
traverseTwigsWithEnvirons :: ∀ x f .
(WithField ℝ Manifold x, Hask.Applicative f)
=> ( ((Int, ShadeTree x), [(Int, ShadeTree x)]) -> f (ShadeTree x))
-> ShadeTree x -> f (ShadeTree x)
traverseTwigsWithEnvirons f = fst . go [] . (0,)
where go :: [(Int, ShadeTree x)] -> (Int, ShadeTree x)
-> (f (ShadeTree x), Bool)
go _ (i₀, DisjointBranches nlvs djbs) = ( fmap (DisjointBranches nlvs)
. Hask.traverse (fst . go [])
$ NE.zip ioffs djbs
, False )
where ioffs = NE.scanl (\i -> (+i) . nLeaves) i₀ djbs
go envi ct@(i₀, (OverlappingBranches nlvs rob@(Shade robc _) brs))
= ( case descentResult of
OuterNothing -> f
$ purgeRemotes
(ct, Hask.foldMap (\(io,te)
-> first (+io) <$> twigProximæ robc te) envi)
OuterJust dR -> fmap (OverlappingBranches nlvs rob . NE.fromList) dR
, False )
where descentResult = traverseDirectionChoices tdc $ NE.toList brs
tdc (io, (vy, ty)) alts = case go envi'' (i₀+io, ty) of
(_, True) -> OuterNothing
(down, _) -> OuterJust down
where envi'' = filter (snd >>> trunks >>> \(Shade ce _:_)
-> let Option (Just δyenv) = ce.-~.robc
qq = vy<.>^δyenv
in qq > -1 && qq < 5
) envi'
++ map ((+i₀)***snd) alts
envi' = approach =<< envi
approach (i₀e, apt@(OverlappingBranches _ (Shade envc _) _))
= first (+i₀e) <$> twigsaveTrim hither apt
where Option (Just δxenv) = robc .-~. envc
hither (DBranch bdir (Hourglass bdc₁ bdc₂))
| bdir<.>^δxenv > 0 = [(0 , bdc₁)]
| otherwise = [(nLeaves bdc₁, bdc₂)]
approach q = [q]
go envi plvs@(i₀, (PlainLeaves _))
= (f $ purgeRemotes (plvs, envi), True)
twigProximæ :: x -> ShadeTree x -> [(Int, ShadeTree x)]
twigProximæ x₀ (DisjointBranches _ djbs)
= Hask.foldMap (\(i₀,st) -> first (+i₀) <$> twigProximæ x₀ st)
$ NE.zip ioffs djbs
where ioffs = NE.scanl (\i -> (+i) . nLeaves) 0 djbs
twigProximæ x₀ ct@(OverlappingBranches _ (Shade xb qb) brs)
= twigsaveTrim hither ct
where Option (Just δxb) = x₀ .-~. xb
hither (DBranch bdir (Hourglass bdc₁ bdc₂))
| bdir<.>^δxb > 0 = twigProximæ x₀ bdc₁
| otherwise = first (+nLeaves bdc₁)
<$> twigProximæ x₀ bdc₂
twigProximæ _ plainLeaves = [(0, plainLeaves)]
twigsaveTrim :: (DBranch x -> [(Int,ShadeTree x)])
-> ShadeTree x -> [(Int,ShadeTree x)]
twigsaveTrim f ct@(OverlappingBranches _ _ dbs)
= case Hask.mapM (\(i₀,dbr) -> noLeaf $ first(+i₀)<$>f dbr)
$ NE.zip ioffs dbs of
Just pqe -> Hask.fold pqe
_ -> [(0,ct)]
where noLeaf [(_,PlainLeaves _)] = empty
noLeaf bqs = pure bqs
ioffs = NE.scanl (\i -> (+i) . sum . fmap nLeaves . toList) 0 dbs
purgeRemotes :: ((Int,ShadeTree x), [(Int,ShadeTree x)])
-> ((Int,ShadeTree x), [(Int,ShadeTree x)])
purgeRemotes = id -- See 7d1f3a4 for the implementation; this didn't work reliable.
completeTopShading :: (WithField ℝ Manifold x, WithField ℝ Manifold y)
=> x`Shaded`y -> [Shade' (x,y)]
completeTopShading (PlainLeaves plvs)
= pointsShade's $ (_topological &&& _untopological) <$> plvs
completeTopShading (DisjointBranches _ bqs)
= take 1 . completeTopShading =<< NE.toList bqs
completeTopShading t = pointsCover's . map (_topological &&& _untopological) $ onlyLeaves t
flexTopShading :: ∀ x y f . ( WithField ℝ Manifold x, WithField ℝ Manifold y
, Applicative f (->) (->) )
=> (Shade' (x,y) -> f (x, (Shade' y, LocalLinear x y)))
-> x`Shaded`y -> f (x`Shaded`y)
flexTopShading f tr = seq (assert_onlyToplevDisjoint tr)
$ recst (completeTopShading tr) tr
where recst qsh@(_:_) (DisjointBranches n bqs)
= undefined -- DisjointBranches n $ NE.zipWith (recst . (:[])) (NE.fromList qsh) bqs
recst [sha@(Shade' (_,yc₀) expa₀)] t = fmap fts $ f sha
where expa'₀ = recipMetric' expa₀
j₀ :: LocalLinear x y
Option (Just j₀) = covariance expa'₀
(_,expay₀) = factoriseMetric expa₀
fts (xc, (Shade' yc expay, jtg)) = unsafeFmapLeaves applδj t
where Option (Just δyc) = yc.-~.yc₀
tfm = imitateMetricSpanChange expay₀ (recipMetric' expay)
applδj (WithAny y x)
= WithAny (yc₀ .+~^ ((tfm$δy) ^+^ (jtg$δx) ^+^ δyc)) x
where Option (Just δx) = x.-~.xc
Option (Just δy) = y.-~.(yc₀.+~^(j₀$δx))
assert_onlyToplevDisjoint, assert_connected :: x`Shaded`y -> ()
assert_onlyToplevDisjoint (DisjointBranches _ dp) = rnf (assert_connected<$>dp)
assert_onlyToplevDisjoint t = assert_connected t
assert_connected (OverlappingBranches _ _ dp)
= rnf (Hask.foldMap assert_connected<$>dp)
assert_connected (PlainLeaves _) = ()
flexTwigsShading :: ∀ x y f . ( WithField ℝ Manifold x, WithField ℝ Manifold y
, Hask.Applicative f )
=> (Shade' (x,y) -> f (x, (Shade' y, LocalLinear x y)))
-> x`Shaded`y -> f (x`Shaded`y)
flexTwigsShading f = traverseTwigsWithEnvirons locFlex
where locFlex :: ∀ μ . ((Int, x`Shaded`y), μ) -> f (x`Shaded`y)
locFlex ((_,lsh), _) = flexTopShading f lsh
-- simplexFaces :: forall n x . Simplex (S n) x -> Triangulation n x
-- simplexFaces (Simplex p (ZeroSimplex q)) = TriangVertices $ Arr.fromList [p, q]
-- simplexFaces splx = carpent splx $ TriangVertices ps
-- where ps = Arr.fromList $ p : splxVertices qs
-- where carpent (ZeroSimplex (Simplex p qs@(Simplex _ _))
-- | Triangulation es <- simplexFaces qs = TriangSkeleton $ Simplex p <$> es
newtype BaryCoords n = BaryCoords { getBaryCoordsTail :: FreeVect n ℝ }
instance (KnownNat n) => AffineSpace (BaryCoords n) where
type Diff (BaryCoords n) = FreeVect n ℝ
BaryCoords v .-. BaryCoords w = v ^-^ w
BaryCoords v .+^ w = BaryCoords $ v ^+^ w
instance (KnownNat n) => Semimanifold (BaryCoords n) where
type Needle (BaryCoords n) = FreeVect n ℝ
fromInterior = id
toInterior = pure
translateP = Tagged (.+~^)
(.+~^) = (.+^)
instance (KnownNat n) => PseudoAffine (BaryCoords n) where
(.-~.) = pure .: (.-.)
getBaryCoords :: BaryCoords n -> ℝ ^ S n
getBaryCoords (BaryCoords (FreeVect bcs)) = FreeVect $ (1 - Arr.sum bcs) `Arr.cons` bcs
getBaryCoords' :: BaryCoords n -> [ℝ]
getBaryCoords' (BaryCoords (FreeVect bcs)) = 1 - Arr.sum bcs : Arr.toList bcs
getBaryCoord :: BaryCoords n -> Int -> ℝ
getBaryCoord (BaryCoords (FreeVect bcs)) 0 = 1 - Arr.sum bcs
getBaryCoord (BaryCoords (FreeVect bcs)) i = case bcs Arr.!? i of
Just a -> a
_ -> 0
mkBaryCoords :: KnownNat n => ℝ ^ S n -> BaryCoords n
mkBaryCoords (FreeVect bcs) = BaryCoords $ FreeVect (Arr.tail bcs) ^/ Arr.sum bcs
mkBaryCoords' :: KnownNat n => [ℝ] -> Option (BaryCoords n)
mkBaryCoords' bcs = fmap (BaryCoords . (^/sum bcs)) . freeVector . Arr.fromList $ tail bcs
newtype ISimplex n x = ISimplex { iSimplexBCCordEmbed :: Embedding (->) (BaryCoords n) x }
data TriangBuilder n x where
TriangVerticesSt :: [x] -> TriangBuilder Z x
TriangBuilder :: Triangulation (S n) x
-> [x]
-> [(Simplex n x, [x] -> Option x)]
-> TriangBuilder (S n) x
-- startTriangulation :: forall n x . (KnownNat n, WithField ℝ Manifold x)
-- => ISimplex n x -> TriangBuilder n x
-- startTriangulation ispl@(ISimplex emb) = startWith $ fromISimplex ispl
-- where startWith (ZeroSimplex p) = TriangVerticesSt [p]
-- startWith s@(Simplex _ _)
-- = TriangBuilder (Triangulation [s])
-- (splxVertices s)
-- [ (s', expandInDir j)
-- | j<-[0..n]
-- | s' <- getTriangulation $ simplexFaces s ]
-- where expandInDir j xs = case sortBy (comparing snd) $ filter ((> -1) . snd) xs_bc of
-- ((x, q) : _) | q<0 -> pure x
-- _ -> empty
-- where xs_bc = map (\x -> (x, getBaryCoord (emb >-$ x) j)) xs
-- (Tagged n) = theNatN :: Tagged n Int
-- extendTriangulation :: forall n x . (KnownNat n, WithField ℝ Manifold x)
-- => [x] -> TriangBuilder n x -> TriangBuilder n x
-- extendTriangulation xs (TriangBuilder tr tb te) = foldr tryex (TriangBuilder tr tb []) te
-- where tryex (bspl, expd) (TriangBuilder (Triangulation tr') tb' te')
-- | Option (Just fav) <- expd xs
-- = let snew = Simplex fav bspl
-- in TriangBuilder (Triangulation $ snew:tr') (fav:tb') undefined
bottomExtendSuitability :: (KnownNat n, WithField ℝ Manifold x)
=> ISimplex (S n) x -> x -> ℝ
bottomExtendSuitability (ISimplex emb) x = case getBaryCoord (emb >-$ x) 0 of
0 -> 0
r -> - recip r
optimalBottomExtension :: (KnownNat n, WithField ℝ Manifold x)
=> ISimplex (S n) x -> [x] -> Option Int
optimalBottomExtension s xs
= case filter ((>0).snd)
$ zipWith ((. bottomExtendSuitability s) . (,)) [0..] xs of
[] -> empty
qs -> pure . fst . maximumBy (comparing snd) $ qs
simplexPlane :: forall n x . (KnownNat n, WithField ℝ Manifold x)
=> Metric x -> Simplex n x -> Embedding (Linear ℝ) (FreeVect n ℝ) (Needle x)
simplexPlane m s = embedding
where bc = simplexBarycenter s
spread = init . map ((.-~.bc) >>> \(Option (Just v)) -> v) $ splxVertices s
embedding = case spanHilbertSubspace m spread of
(Option (Just e)) -> e
_ -> error "Trying to obtain simplexPlane from zero-volume\
\ simplex (which cannot span sufficient basis vectors)."
leavesBarycenter :: WithField ℝ Manifold x => NonEmpty x -> x
leavesBarycenter (x :| xs) = x .+~^ sumV [x'–x | x'<-xs] ^/ (n+1)
where n = fromIntegral $ length xs
x' – x = case x'.-~.x of {Option(Just v)->v}
-- simplexShade :: forall x n . (KnownNat n, WithField ℝ Manifold x)
simplexBarycenter :: forall x n . (KnownNat n, WithField ℝ Manifold x) => Simplex n x -> x
simplexBarycenter = bc
where bc (ZS x) = x
bc (x :<| xs') = x .+~^ sumV [x'–x | x'<-splxVertices xs'] ^/ (n+1)
Tagged n = theNatN :: Tagged n ℝ
x' – x = case x'.-~.x of {Option(Just v)->v}
toISimplex :: forall x n . (KnownNat n, WithField ℝ Manifold x)
=> Metric x -> Simplex n x -> ISimplex n x
toISimplex m s = ISimplex $ fromEmbedProject fromBrc toBrc
where bc = simplexBarycenter s
(Embedding emb (DenseLinear prj))
= simplexPlane m s
(r₀:rs) = [ prj HMat.#> asPackedVector v
| x <- splxVertices s, let (Option (Just v)) = x.-~.bc ]
tmat = HMat.inv $ HMat.fromColumns [ r - r₀ | r<-rs ]
toBrc x = case x.-~.bc of
Option (Just v) -> let rx = prj HMat.#> asPackedVector v - r₀
in finalise $ tmat HMat.#> rx
finalise v = case freeVector $ HMat.toList v of
Option (Just bv) -> BaryCoords bv
fromBrc bccs = bc .+~^ (emb $ v)
where v = linearCombo $ (fromPackedVector r₀, b₀) : zip (fromPackedVector<$>rs) bs
(b₀:bs) = getBaryCoords' bccs
fromISimplex :: forall x n . (KnownNat n, WithField ℝ Manifold x)
=> ISimplex n x -> Simplex n x
fromISimplex (ISimplex emb) = s
where (Option (Just s))
= makeSimplex' [ emb $-> jOnly
| j <- [0..n]
, let (Option (Just jOnly)) = mkBaryCoords' [ if k==j then 1 else 0
| k<-[0..n] ]
]
(Tagged n) = theNatN :: Tagged n Int
iSimplexSideViews :: ∀ n x . KnownNat n => ISimplex n x -> [ISimplex n x]
iSimplexSideViews = \(ISimplex is)
-> take (n+1) $ [ISimplex $ rot j is | j<-[0..] ]
where rot j (Embedding emb proj)
= Embedding ( emb . mkBaryCoords . freeRotate j . getBaryCoords )
( mkBaryCoords . freeRotate (n-j) . getBaryCoords . proj )
(Tagged n) = theNatN :: Tagged n Int
type FullTriang t n x = TriangT t n x
(State (Map.Map (SimplexIT t n x) (ISimplex n x)))
type TriangBuild t n x = TriangT t (S n) x
( State (Map.Map (SimplexIT t n x) (Metric x, ISimplex (S n) x) ))
doTriangBuild :: KnownNat n => (∀ t . TriangBuild t n x ()) -> [Simplex (S n) x]
doTriangBuild t = runIdentity (fst <$>
doTriangT (unliftInTriangT (`evalStateT`mempty) t >> simplexITList >>= mapM lookSimplex))
singleFullSimplex :: ∀ t n x . (KnownNat n, WithField ℝ Manifold x)
=> ISimplex n x -> FullTriang t n x (SimplexIT t n x)
singleFullSimplex is = do
frame <- disjointSimplex (fromISimplex is)
lift . modify' $ Map.insert frame is
return frame
fullOpenSimplex :: ∀ t n x . (KnownNat n, WithField ℝ Manifold x)
=> Metric x -> Simplex (S n) x -> TriangBuild t n x [SimplexIT t n x]
fullOpenSimplex m s = do
let is = toISimplex m s
frame <- disjointSimplex (fromISimplex is)
fsides <- toList <$> lookSplxFacesIT frame
lift . forM (zip fsides $ iSimplexSideViews is)
$ \(fside,is') -> modify' $ Map.insert fside (m,is')
return fsides
hypotheticalSimplexScore :: ∀ t n n' x . (KnownNat n', WithField ℝ Manifold x, n~S n')
=> SimplexIT t Z x
-> SimplexIT t n x
-> TriangBuild t n x ( Option Double )
hypotheticalSimplexScore p b = do
altViews :: [(SimplexIT t Z x, SimplexIT t n x)] <- do
pSups <- lookSupersimplicesIT p
nOpts <- forM pSups $ \psup -> fmap (fmap $ \((bq,_p), _b') -> (bq,psup))
$ distinctSimplices b psup
return $ catOptions nOpts
scores <- forM ((p,b) :| altViews) $ \(p',b') -> do
x <- lookVertexIT p'
q <- lift $ Map.lookup b' <$> get
return $ case q of
Just(_,is) | s<-bottomExtendSuitability is x, s>0
-> pure s
_ -> empty
return . fmap sum $ Hask.sequence scores
spanSemiOpenSimplex :: ∀ t n n' x . (KnownNat n', WithField ℝ Manifold x, n~S n')
=> SimplexIT t Z x -- ^ Tip of the desired simplex.
-> SimplexIT t n x -- ^ Base of the desired simplex.
-> TriangBuild t n x [SimplexIT t n x]
-- ^ Return the exposed faces of the new simplices.
spanSemiOpenSimplex p b = do
m <- lift $ fst <$> (Map.!b) <$> get
neighbours <- filterM isAdjacent =<< lookSupersimplicesIT p
let bs = b:|neighbours
frame <- webinateTriang p b
backSplx <- lookSimplex frame
let iSplx = toISimplex m backSplx
fsides <- toList <$> lookSplxFacesIT frame
let sviews = filter (not . (`elem`bs) . fst) $ zip fsides (iSimplexSideViews iSplx)
lift . forM sviews $ \(fside,is') -> modify' $ Map.insert fside (m,is')
lift . Hask.forM_ bs $ \fside -> modify' $ Map.delete fside
return $ fst <$> sviews
where isAdjacent = fmap (isJust . getOption) . sharedBoundary b
multiextendTriang :: ∀ t n n' x . (KnownNat n', WithField ℝ Manifold x, n~S n')
=> [SimplexIT t Z x] -> TriangBuild t n x ()
multiextendTriang vs = do
ps <- mapM lookVertexIT vs
sides <- lift $ Map.toList <$> get
forM_ sides $ \(f,(m,s)) ->
case optimalBottomExtension s ps of
Option (Just c) -> spanSemiOpenSimplex (vs !! c) f
_ -> return []
-- | BUGGY: this does connect the supplied triangulations, but it doesn't choose
-- the right boundary simplices yet. Probable cause: inconsistent internal
-- numbering of the subsimplices.
autoglueTriangulation :: ∀ t n n' n'' x
. (KnownNat n'', WithField ℝ Manifold x, n~S n', n'~S n'')
=> (∀ t' . TriangBuild t' n' x ()) -> TriangBuild t n' x ()
autoglueTriangulation tb = do
mbBounds <- Map.toList <$> lift get
mps <- pointsOfSurf mbBounds
WriterT gbBounds <- liftInTriangT $ mixinTriangulation tb'
lift . forM_ gbBounds $ \(i,ms) -> do
modify' $ Map.insert i ms
gps <- pointsOfSurf gbBounds
autoglue mps gbBounds
autoglue gps mbBounds
where tb' :: ∀ s . TriangT s n x Identity
(WriterT (Metric x, ISimplex n x) [] (SimplexIT s n' x))
tb' = unliftInTriangT (`evalStateT`mempty) $
tb >> (WriterT . Map.toList) <$> lift get
pointsOfSurf s = fnubConcatMap Hask.toList <$> forM s (lookSplxVerticesIT . fst)
autoglue :: [SimplexIT t Z x] -> [(SimplexIT t n' x, (Metric x, ISimplex n x))]
-> TriangBuild t n' x ()
autoglue vs sides = do
forM_ sides $ \(f,_) -> do
possibs <- forM vs $ \p -> fmap(p,) <$> hypotheticalSimplexScore p f
case catOptions possibs of
[] -> return ()
qs -> do
spanSemiOpenSimplex (fst `id` maximumBy (comparing $ snd) qs) f
return ()
data AutoTriang n x where
AutoTriang :: { getAutoTriang :: ∀ t . TriangBuild t n x () } -> AutoTriang (S n) x
instance (KnownNat n, WithField ℝ Manifold x) => Semigroup (AutoTriang (S (S n)) x) where
(<>) = autoTriangMappend
autoTriangMappend :: ∀ n n' n'' x . ( KnownNat n'', n ~ S n', n' ~ S n''
, WithField ℝ Manifold x )
=> AutoTriang n x -> AutoTriang n x -> AutoTriang n x
AutoTriang a `autoTriangMappend` AutoTriang b = AutoTriang c
where c :: ∀ t . TriangBuild t n' x ()
c = a >> autoglueTriangulation b
elementaryTriang :: ∀ n n' x . (KnownNat n', n~S n', WithField ℝ EuclidSpace x)
=> Simplex n x -> AutoTriang n x
elementaryTriang t = AutoTriang (fullOpenSimplex m t >> return ())
where m = euclideanMetric t
breakdownAutoTriang :: ∀ n n' x . (KnownNat n', n ~ S n') => AutoTriang n x -> [Simplex n x]
breakdownAutoTriang (AutoTriang t) = doTriangBuild t
-- where tr :: Triangulation n x
-- outfc :: Map.Map (SimplexIT t n' x) (Metric x, ISimplex n x)
-- (((), tr), outfc) = runState (doTriangT tb') mempty
-- tb' :: ∀ t' . TriangT t' n x
-- ( State ( Map.Map (SimplexIT t' n' x)
-- (Metric x, ISimplex n x) ) ) ()
-- tb' = tb
-- primitiveTriangulation :: forall x n . (KnownNat n,WithField ℝ Manifold x)
-- => [x] -> Triangulation n x
-- primitiveTriangulation xs = head $ build <$> buildOpts
-- where build :: ([x], [x]) -> Triangulation n x
-- build (mainVerts, sideVerts) = Triangulation [mainSplx]
-- where (Option (Just mainSplx)) = makeSimplex mainVerts
-- -- mainFaces = Map.fromAscList . zip [0..] . getTriangulation
-- -- $ simplexFaces mainSplx
-- buildOpts = partitionsOfFstLength n xs
-- (Tagged n) = theNatN :: Tagged n Int
partitionsOfFstLength :: Int -> [a] -> [([a],[a])]
partitionsOfFstLength 0 l = [([],l)]
partitionsOfFstLength n [] = []
partitionsOfFstLength n (x:xs) = ( first (x:) <$> partitionsOfFstLength (n-1) xs )
++ ( second (x:) <$> partitionsOfFstLength n xs )
splxVertices :: Simplex n x -> [x]
splxVertices (ZS x) = [x]
splxVertices (x :<| s') = x : splxVertices s'
-- triangulate :: forall x n . (KnownNat n, WithField ℝ Manifold x)
-- => ShadeTree x -> Triangulation n x
-- triangulate (DisjointBranches _ brs)
-- = Triangulation $ Hask.foldMap (getTriangulation . triangulate) brs
-- triangulate (PlainLeaves xs) = primitiveTriangulation xs
-- triangBranches :: WithField ℝ Manifold x
-- => ShadeTree x -> Branchwise x (Triangulation x) n
-- triangBranches _ = undefined
--
-- tringComplete :: WithField ℝ Manifold x
-- => Triangulation x (n-1) -> Triangulation x n -> Triangulation x n
-- tringComplete (Triangulation trr) (Triangulation tr) = undefined
-- where
-- bbSimplices = Map.fromList [(i, Left s) | s <- tr | i <- [0::Int ..] ]
-- bbVertices = [(i, splxVertices s) | s <- tr | i <- [0::Int ..] ]
--
-- |
-- @
-- 'SimpleTree' x ≅ Maybe (x, 'Trees' x)
-- @
type SimpleTree = GenericTree Maybe []
-- |
-- @
-- 'Trees' x ≅ [(x, 'Trees' x)]
-- @
type Trees = GenericTree [] []
-- |
-- @
-- 'NonEmptyTree' x ≅ (x, 'Trees' x)
-- @
type NonEmptyTree = GenericTree NonEmpty []
newtype GenericTree c b x = GenericTree { treeBranches :: c (x,GenericTree b b x) }
deriving (Generic, Hask.Functor, Hask.Foldable, Hask.Traversable)
instance (NFData x, Hask.Foldable c, Hask.Foldable b) => NFData (GenericTree c b x) where
rnf (GenericTree t) = rnf $ toList t
instance (Hask.MonadPlus c) => Semigroup (GenericTree c b x) where
GenericTree b1 <> GenericTree b2 = GenericTree $ Hask.mplus b1 b2
instance (Hask.MonadPlus c) => Monoid (GenericTree c b x) where
mempty = GenericTree Hask.mzero
mappend = (<>)
deriving instance Show (c (x, GenericTree b b x)) => Show (GenericTree c b x)
-- | Imitate the specialised 'ShadeTree' structure with a simpler, generic tree.
onlyNodes :: WithField ℝ Manifold x => ShadeTree x -> Trees x
onlyNodes (PlainLeaves []) = GenericTree []
onlyNodes (PlainLeaves ps) = let (ctr,_) = pseudoECM $ NE.fromList ps
in GenericTree [ (ctr, GenericTree $ (,mempty) <$> ps) ]
onlyNodes (DisjointBranches _ brs) = Hask.foldMap onlyNodes brs
onlyNodes (OverlappingBranches _ (Shade ctr _) brs)
= GenericTree [ (ctr, Hask.foldMap (Hask.foldMap onlyNodes) brs) ]
-- | Left (and, typically, also right) inverse of 'fromLeafNodes'.
onlyLeaves :: WithField ℝ Manifold x => ShadeTree x -> [x]
onlyLeaves tree = dismantle tree []
where dismantle (PlainLeaves xs) = (xs++)
dismantle (OverlappingBranches _ _ brs)
= foldr ((.) . dismantle) id $ Hask.foldMap (Hask.toList) brs
dismantle (DisjointBranches _ brs) = foldr ((.) . dismantle) id $ NE.toList brs
data Sawbones x = Sawbones { sawnTrunk1, sawnTrunk2 :: [x]->[x]
, sawdust1, sawdust2 :: [x] }
instance Semigroup (Sawbones x) where
Sawbones st11 st12 sd11 sd12 <> Sawbones st21 st22 sd21 sd22
= Sawbones (st11.st21) (st12.st22) (sd11<>sd21) (sd12<>sd22)
instance Monoid (Sawbones x) where
mempty = Sawbones id id [] []
mappend = (<>)
chainsaw :: WithField ℝ Manifold x => Cutplane x -> ShadeTree x -> Sawbones x
chainsaw cpln (PlainLeaves xs) = Sawbones (sd1++) (sd2++) sd2 sd1
where (sd1,sd2) = partition (\x -> sideOfCut cpln x == Option(Just PositiveHalfSphere)) xs
chainsaw cpln (DisjointBranches _ brs) = Hask.foldMap (chainsaw cpln) brs
chainsaw cpln (OverlappingBranches _ (Shade _ bexpa) brs) = Sawbones t1 t2 d1 d2
where (Sawbones t1 t2 subD1 subD2)
= Hask.foldMap (Hask.foldMap (chainsaw cpln) . boughContents) brs
[d1,d2] = map (foldl' go [] . foci) [subD1, subD2]
where go d' (dp,dqs) = case fathomCD dp of
Option (Just dpCD) | not $ any (shelter dpCD) dqs
-> dp:d' -- dp is close enough to cut plane to make dust.
_ -> d' -- some dq is actually closer than the cut plane => discard dp.
where shelter dpCutDist dq = case ptsDist dp dq of
Option (Just d) -> d < abs dpCutDist
_ -> False
ptsDist = fmap (metric $ recipMetric bexpa) .: (.-~.)
fathomCD = fathomCutDistance cpln bexpa
type DList x = [x]->[x]
data DustyEdges x = DustyEdges { sawChunk :: DList x, chunkDust :: DBranches' x [x] }
instance Semigroup (DustyEdges x) where
DustyEdges c1 d1 <> DustyEdges c2 d2 = DustyEdges (c1.c2) (d1<>d2)
data Sawboneses x = SingleCut (Sawbones x)
| Sawboneses (DBranches' x (DustyEdges x))
deriving (Generic)
instance Semigroup (Sawboneses x) where
SingleCut c <> SingleCut d = SingleCut $ c<>d
Sawboneses c <> Sawboneses d = Sawboneses $ c<>d
-- | Saw a tree into the domains covered by the respective branches of another tree.
sShSaw :: WithField ℝ Manifold x
=> ShadeTree x -- ^ “Reference tree”, defines the cut regions.
-- Must be at least one level of 'OverlappingBranches' deep.
-> ShadeTree x -- ^ Tree to take the actual contents from.
-> Sawboneses x -- ^ All points within each region, plus those from the
-- boundaries of each neighbouring region.
sShSaw (OverlappingBranches _ (Shade sh _) (DBranch dir _ :| [])) src
= SingleCut $ chainsaw (Cutplane sh $ stiefel1Project dir) src
sShSaw (OverlappingBranches _ (Shade cctr _) cbrs) (PlainLeaves xs)
= Sawboneses . DBranches $ NE.fromList ngbsAdded
where brsEmpty = fmap (\(DBranch dir _)-> DBranch dir mempty) cbrs
srcDistrib = sShIdPartition' cctr xs brsEmpty
ngbsAdded = fmap (\(DBranch dir (Hourglass u l), othrs)
-> let [allOthr,allOthr']
= map (DBranches . NE.fromList)
[othrs, fmap (\(DBranch d' o)
->DBranch(negateV d') o) othrs]
in DBranch dir $ Hourglass (DustyEdges (u++) allOthr)
(DustyEdges (l++) allOthr')
) $ foci (NE.toList srcDistrib)
sShSaw cuts@(OverlappingBranches _ (Shade sh _) cbrs)
(OverlappingBranches _ (Shade _ bexpa) brs)
= Sawboneses . DBranches $ ftr'd
where Option (Just (Sawboneses (DBranches recursed)))
= Hask.foldMap (Hask.foldMap (pure . sShSaw cuts) . boughContents) brs
ftr'd = fmap (\(DBranch dir1 ds) -> DBranch dir1 $ fmap (
\(DustyEdges bk (DBranches dds))
-> DustyEdges bk . DBranches $ fmap (obsFilter dir1) dds
) ds ) recursed
obsFilter dir1 (DBranch dir2 (Hourglass pd2 md2))
= DBranch dir2 $ Hourglass pd2' md2'
where cpln cpSgn = Cutplane sh . stiefel1Project $ dir1 ^+^ cpSgn*^dir2
[pd2', md2'] = zipWith (occl . cpln) [-1, 1] [pd2, md2]
occl cpl = foldl' go [] . foci
where go d' (dp,dqs) = case fathomCD dp of
Option (Just dpCD) | not $ any (shelter dpCD) dqs
-> dp:d'
_ -> d'
where shelter dpCutDist dq = case ptsDist dp dq of
Option (Just d) -> d < abs dpCutDist
_ -> False
ptsDist = fmap (metric $ recipMetric bexpa) .: (.-~.)
fathomCD = fathomCutDistance cpl bexpa
sShSaw _ _ = error "`sShSaw` is not supposed to cut anything else but `OverlappingBranches`"
-- | Essentially the same as @(x,y)@, but not considered as a product topology.
-- The 'Semimanifold' etc. instances just copy the topology of @x@, ignoring @y@.
data x`WithAny`y
= WithAny { _untopological :: y
, _topological :: !x }
deriving (Hask.Functor, Show, Generic)
instance (NFData x, NFData y) => NFData (WithAny x y)
instance (Semimanifold x) => Semimanifold (x`WithAny`y) where
type Needle (WithAny x y) = Needle x
type Interior (WithAny x y) = Interior x `WithAny` y
WithAny y x .+~^ δx = WithAny y $ x.+~^δx
fromInterior (WithAny y x) = WithAny y $ fromInterior x
toInterior (WithAny y x) = fmap (WithAny y) $ toInterior x
translateP = tpWD
where tpWD :: ∀ x y . Semimanifold x => Tagged (WithAny x y)
(Interior x`WithAny`y -> Needle x -> Interior x`WithAny`y)
tpWD = Tagged `id` \(WithAny y x) δx -> WithAny y $ tpx x δx
where Tagged tpx = translateP :: Tagged x (Interior x -> Needle x -> Interior x)
instance (PseudoAffine x) => PseudoAffine (x`WithAny`y) where
WithAny _ x .-~. WithAny _ ξ = x.-~.ξ
instance (AffineSpace x) => AffineSpace (x`WithAny`y) where
type Diff (WithAny x y) = Diff x
WithAny _ x .-. WithAny _ ξ = x.-.ξ
WithAny y x .+^ δx = WithAny y $ x.+^δx
instance (VectorSpace x, Monoid y) => VectorSpace (x`WithAny`y) where
type Scalar (WithAny x y) = Scalar x
μ *^ WithAny y x = WithAny y $ μ*^x
instance (AdditiveGroup x, Monoid y) => AdditiveGroup (x`WithAny`y) where
zeroV = WithAny mempty zeroV
negateV (WithAny y x) = WithAny y $ negateV x
WithAny y x ^+^ WithAny υ ξ = WithAny (mappend y υ) (x^+^ξ)
instance (AdditiveGroup x) => Hask.Applicative (WithAny x) where
pure x = WithAny x zeroV
WithAny f x <*> WithAny t ξ = WithAny (f t) (x^+^ξ)
instance (AdditiveGroup x) => Hask.Monad (WithAny x) where
return x = WithAny x zeroV
WithAny y x >>= f = WithAny r $ x^+^q
where WithAny r q = f y
shadeWithAny :: y -> Shade x -> Shade (x`WithAny`y)
shadeWithAny y (Shade x xe) = Shade (WithAny y x) xe
shadeWithoutAnything :: Shade (x`WithAny`y) -> Shade x
shadeWithoutAnything (Shade (WithAny _ b) e) = Shade b e
constShaded :: y -> ShadeTree x -> x`Shaded`y
constShaded y = unsafeFmapTree (WithAny y<$>) id (shadeWithAny y)
stripShadedUntopological :: x`Shaded`y -> ShadeTree x
stripShadedUntopological = unsafeFmapTree (fmap _topological) id shadeWithoutAnything
-- | This is to 'ShadeTree' as 'Data.Map.Map' is to 'Data.Set.Set'.
type x`Shaded`y = ShadeTree (x`WithAny`y)
stiWithDensity :: (WithField ℝ Manifold x, WithField ℝ LinearManifold y)
=> x`Shaded`y -> x -> Cℝay y
stiWithDensity (PlainLeaves lvs)
| [locShape@(Shade baryc expa)] <- pointsShades $ _topological <$> lvs
= let nlvs = fromIntegral $ length lvs :: ℝ
indiShapes = [(Shade p expa, y) | WithAny y p <- lvs]
in \x -> let lcCoeffs = [ occlusion psh x | (psh, _) <- indiShapes ]
dens = sum lcCoeffs
in mkCone dens . linearCombo . zip (snd<$>indiShapes)
$ (/dens)<$>lcCoeffs
stiWithDensity (DisjointBranches _ lvs)
= \x -> foldr1 qGather $ (`stiWithDensity`x)<$>lvs
where qGather (Cℝay 0 _) o = o
qGather o _ = o
stiWithDensity (OverlappingBranches n (Shade (WithAny _ bc) extend) brs) = ovbSWD
where ovbSWD x = case x .-~. bc of
Option (Just v)
| dist² <- metricSq ε v
, dist² < 9
, att <- exp(1/(dist²-9)+1/9)
-> qGather att $ fmap ($x) downPrepared
_ -> coneTip
ε = recipMetric extend
downPrepared = dp =<< brs
where dp (DBranch _ (Hourglass up dn))
= fmap stiWithDensity $ up:|[dn]
qGather att contribs = mkCone (att*dens)
$ linearCombo [(v, d/dens) | Cℝay d v <- NE.toList contribs]
where dens = sum (hParamCℝay <$> contribs)
stiAsIntervalMapping :: (x ~ ℝ, y ~ ℝ)
=> x`Shaded`y -> [(x, ((y, Diff y), Linear ℝ x y))]
stiAsIntervalMapping = twigsWithEnvirons >=> pure.snd.fst >=> completeTopShading >=> pure.
\(Shade' (xloc, yloc) shd)
-> ( xloc, ( (yloc, recip $ metric shd (0,1))
, case covariance (recipMetric' shd) of
{Option(Just j)->j} ) )
smoothInterpolate :: (WithField ℝ Manifold x, WithField ℝ LinearManifold y)
=> NonEmpty (x,y) -> x -> y
smoothInterpolate l = \x ->
case ltr x of
Cℝay 0 _ -> defy
Cℝay _ y -> y
where defy = linearCombo [(y, 1/n) | WithAny y _ <- l']
n = fromIntegral $ length l'
l' = (uncurry WithAny . swap) <$> NE.toList l
ltr = stiWithDensity $ fromLeafPoints l'
spanShading :: ∀ x y . (WithField ℝ Manifold x, WithField ℝ Manifold y)
=> (Shade x -> Shade y) -> ShadeTree x -> x`Shaded`y
spanShading f = unsafeFmapTree addYs id addYSh
where addYs :: NonEmpty x -> NonEmpty (x`WithAny`y)
addYs l = foldr (NE.<|) (fmap ( WithAny ymid) l )
(fmap (`WithAny`xmid) yexamp)
where [xsh@(Shade xmid _)] = pointsCovers $ toList l
Shade ymid yexpa = f xsh
yexamp = [ ymid .+~^ σ*^δy
| δy <- eigenSpan yexpa, σ <- [-1,1] ]
addYSh :: Shade x -> Shade (x`WithAny`y)
addYSh xsh = shadeWithAny (_shadeCtr $ f xsh) xsh
coneTip :: (AdditiveGroup v) => Cℝay v
coneTip = Cℝay 0 zeroV
mkCone :: AdditiveGroup v => ℝ -> v -> Cℝay v
mkCone 0 _ = coneTip
mkCone h v = Cℝay h v
foci :: [a] -> [(a,[a])]
foci [] = []
foci (x:xs) = (x,xs) : fmap (second (x:)) (foci xs)
fociNE :: NonEmpty a -> NonEmpty (a,[a])
fociNE (x:|xs) = (x,xs) :| fmap (second (x:)) (foci xs)
(.:) :: (c->d) -> (a->b->c) -> a->b->d
(.:) = (.) . (.)
catOptions :: [Option a] -> [a]
catOptions = catMaybes . map getOption
class HasFlatView f where
type FlatView f x
flatView :: f x -> FlatView f x
superFlatView :: f x -> [[x]]
instance HasFlatView Sawbones where
type FlatView Sawbones x = [([x],[[x]])]
flatView (Sawbones t1 t2 d1 d2) = [(t1[],[d1]), (t2[],[d2])]
superFlatView = foldMap go . flatView
where go (t,ds) = t : ds
instance HasFlatView Sawboneses where
type FlatView Sawboneses x = [([x],[[x]])]
flatView (SingleCut (Sawbones t1 t2 d1 d2)) = [(t1[],[d1]), (t2[],[d2])]
flatView (Sawboneses (DBranches bs)) =
[ (m[], NE.toList ds >>= \(DBranch _ (Hourglass u' l')) -> [u',l'])
| (DBranch _ (Hourglass u l)) <- NE.toList bs
, (DustyEdges m (DBranches ds)) <- [u,l]
]
superFlatView = foldMap go . flatView
where go (t,ds) = t : ds
extractJust :: (a->Maybe b) -> [a] -> (Maybe b, [a])
extractJust f [] = (Nothing,[])
extractJust f (x:xs) | Just r <- f x = (Just r, xs)
| otherwise = second (x:) $ extractJust f xs