hgeometry-0.12.0.0: src/Data/Geometry/LineSegment/Internal.hs
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE UndecidableInstances #-}
--------------------------------------------------------------------------------
-- |
-- Module : Data.Geometry.LineSegment.Internal
-- Copyright : (C) Frank Staals
-- License : see the LICENSE file
-- Maintainer : Frank Staals
--
-- Line segment data type and some basic functions on line segments
--
--------------------------------------------------------------------------------
module Data.Geometry.LineSegment.Internal
( LineSegment(LineSegment, LineSegment', ClosedLineSegment, OpenLineSegment)
, endPoints
, _SubLine
, module Data.Geometry.Interval
, toLineSegment
, onSegment, onSegment2
, orderedEndPoints
, segmentLength
, sqSegmentLength
, sqDistanceToSeg, sqDistanceToSegArg
, flipSegment
, interpolate
, validSegment
, sampleLineSegment
) where
import Control.Arrow ((&&&))
import Control.DeepSeq
import Control.Lens
import Control.Monad.Random
import Data.Ext
import qualified Data.Foldable as F
import Data.Geometry.Box.Internal
import Data.Geometry.Interval hiding (width, midPoint)
import Data.Geometry.Line.Internal
import Data.Geometry.Point
import Data.Geometry.Properties
import Data.Geometry.SubLine
import Data.Geometry.Transformation
import Data.Geometry.Vector
import Data.Ord (comparing)
import Data.Vinyl
import Data.Vinyl.CoRec
import GHC.TypeLits
import Test.QuickCheck (Arbitrary(..), suchThatMap)
import Text.Read
--------------------------------------------------------------------------------
-- * d-dimensional LineSegments
-- | Line segments. LineSegments have a start and end point, both of which may
-- contain additional data of type p. We can think of a Line-Segment being defined as
--
--
-- >>> data LineSegment d p r = LineSegment (EndPoint (Point d r :+ p)) (EndPoint (Point d r :+ p))
--
-- it is assumed that the two endpoints of the line segment are disjoint. This is not checked.
newtype LineSegment d p r = GLineSegment { _unLineSeg :: Interval p (Point d r) }
makeLenses ''LineSegment
pattern LineSegment :: EndPoint (Point d r :+ p)
-> EndPoint (Point d r :+ p)
-> LineSegment d p r
pattern LineSegment s t = GLineSegment (Interval s t)
{-# COMPLETE LineSegment #-}
-- | Gets the start and end point, but forgetting if they are open or closed.
pattern LineSegment' :: Point d r :+ p
-> Point d r :+ p
-> LineSegment d p r
pattern LineSegment' s t <- ((^.start) &&& (^.end) -> (s,t))
{-# COMPLETE LineSegment' #-}
pattern ClosedLineSegment :: Point d r :+ p -> Point d r :+ p -> LineSegment d p r
pattern ClosedLineSegment s t = GLineSegment (ClosedInterval s t)
{-# COMPLETE ClosedLineSegment #-}
pattern OpenLineSegment :: Point d r :+ p -> Point d r :+ p -> LineSegment d p r
pattern OpenLineSegment s t = GLineSegment (OpenInterval s t)
{-# COMPLETE OpenLineSegment #-}
type instance Dimension (LineSegment d p r) = d
type instance NumType (LineSegment d p r) = r
instance HasStart (LineSegment d p r) where
type StartCore (LineSegment d p r) = Point d r
type StartExtra (LineSegment d p r) = p
start = unLineSeg.start
instance HasEnd (LineSegment d p r) where
type EndCore (LineSegment d p r) = Point d r
type EndExtra (LineSegment d p r) = p
end = unLineSeg.end
instance (Arbitrary r, Arbitrary p, Eq r, Arity d) => Arbitrary (LineSegment d p r) where
arbitrary = suchThatMap ((,) <$> arbitrary <*> arbitrary)
(uncurry validSegment)
deriving instance (Arity d, NFData r, NFData p) => NFData (LineSegment d p r)
sampleLineSegment :: (Arity d, RandomGen g, Random r) => Rand g (LineSegment d () r)
sampleLineSegment = do
a <- ext <$> getRandom
a' <- getRandom
b <- ext <$> getRandom
b' <- getRandom
pure $ LineSegment (if a' then Open a else Closed a) (if b' then Open b else Closed b)
{- HLINT ignore endPoints -}
-- | Traversal to access the endpoints. Note that this traversal
-- allows you to change more or less everything, even the dimension
-- and the numeric type used, but it preservers if the segment is open
-- or closed.
endPoints :: Traversal (LineSegment d p r) (LineSegment d' q s)
(Point d r :+ p) (Point d' s :+ q)
endPoints = \f (LineSegment p q) -> LineSegment <$> traverse f p
<*> traverse f q
_SubLine :: (Num r, Arity d) => Iso' (LineSegment d p r) (SubLine d p r r)
_SubLine = iso segment2SubLine subLineToSegment
{-# INLINE _SubLine #-}
segment2SubLine :: (Num r, Arity d)
=> LineSegment d p r -> SubLine d p r r
segment2SubLine ss = SubLine (Line p (q .-. p)) (Interval s e)
where
p = ss^.start.core
q = ss^.end.core
(Interval a b) = ss^.unLineSeg
s = a&unEndPoint.core .~ 0
e = b&unEndPoint.core .~ 1
{- HLINT ignore subLineToSegment -}
subLineToSegment :: (Num r, Arity d) => SubLine d p r r -> LineSegment d p r
subLineToSegment sl = let Interval s' e' = (fixEndPoints sl)^.subRange
s = s'&unEndPoint %~ (^.extra)
e = e'&unEndPoint %~ (^.extra)
in LineSegment s e
instance (Num r, Arity d) => HasSupportingLine (LineSegment d p r) where
supportingLine s = lineThrough (s^.start.core) (s^.end.core)
instance (Show r, Show p, Arity d) => Show (LineSegment d p r) where
showsPrec d (LineSegment p' q') = case (p',q') of
(Closed p, Closed q) -> f "ClosedLineSegment" p q
(Open p, Open q) -> f "OpenLineSegment" p q
(p,q) -> f "LineSegment" p q
where
app_prec = 10
f :: (Show a, Show b) => String -> a -> b -> String -> String
f cn p q = showParen (d > app_prec) $
showString cn . showString " "
. showsPrec (app_prec+1) p
. showString " "
. showsPrec (app_prec+1) q
instance (Read r, Read p, Arity d) => Read (LineSegment d p r) where
readPrec = parens $ (prec app_prec $ do
Ident "ClosedLineSegment" <- lexP
p <- step readPrec
q <- step readPrec
return (ClosedLineSegment p q))
+++
(prec app_prec $ do
Ident "OpenLineSegment" <- lexP
p <- step readPrec
q <- step readPrec
return (OpenLineSegment p q))
+++
(prec app_prec $ do
Ident "LineSegment" <- lexP
p <- step readPrec
q <- step readPrec
return (LineSegment p q))
where app_prec = 10
deriving instance (Eq r, Eq p, Arity d) => Eq (LineSegment d p r)
-- deriving instance (Ord r, Ord p, Arity d) => Ord (LineSegment d p r)
deriving instance Arity d => Functor (LineSegment d p)
instance PointFunctor (LineSegment d p) where
pmap f ~(LineSegment s e) = LineSegment (s&unEndPoint.core %~ f)
(e&unEndPoint.core %~ f)
instance Arity d => IsBoxable (LineSegment d p r) where
boundingBox l = boundingBox (l^.start.core) <> boundingBox (l^.end.core)
instance (Fractional r, Arity d, Arity (d + 1)) => IsTransformable (LineSegment d p r) where
transformBy = transformPointFunctor
instance Arity d => Bifunctor (LineSegment d) where
bimap f g (GLineSegment i) = GLineSegment $ bimap f (fmap g) i
-- ** Converting between Lines and LineSegments
-- | Directly convert a line into a line segment.
toLineSegment :: (Monoid p, Num r, Arity d) => Line d r -> LineSegment d p r
toLineSegment (Line p v) = ClosedLineSegment (p :+ mempty)
(p .+^ v :+ mempty)
-- *** Intersecting LineSegments
type instance IntersectionOf (Point d r) (LineSegment d p r) = [ NoIntersection
, Point d r
]
type instance IntersectionOf (LineSegment 2 p r) (LineSegment 2 p r) = [ NoIntersection
, Point 2 r
, LineSegment 2 p r
]
type instance IntersectionOf (LineSegment 2 p r) (Line 2 r) = [ NoIntersection
, Point 2 r
, LineSegment 2 p r
]
instance {-# OVERLAPPING #-} (Ord r, Num r)
=> Point 2 r `IsIntersectableWith` LineSegment 2 p r where
nonEmptyIntersection = defaultNonEmptyIntersection
intersects = onSegment2
p `intersect` seg | p `intersects` seg = coRec p
| otherwise = coRec NoIntersection
instance {-# OVERLAPPABLE #-} (Ord r, Fractional r, Arity d)
=> Point d r `IsIntersectableWith` LineSegment d p r where
nonEmptyIntersection = defaultNonEmptyIntersection
intersects = onSegment
p `intersect` seg | p `intersects` seg = coRec p
| otherwise = coRec NoIntersection
-- | Test if a point lies on a line segment.
--
-- As a user, you should typically just use 'intersects' instead.
onSegment :: (Ord r, Fractional r, Arity d) => Point d r -> LineSegment d p r -> Bool
p `onSegment` (LineSegment up vp) =
maybe False inRange' (scalarMultiple (p .-. u) (v .-. u))
where
u = up^.unEndPoint.core
v = vp^.unEndPoint.core
atMostUpperBound = if isClosed vp then (<= 1) else (< 1)
atLeastLowerBound = if isClosed up then (0 <=) else (0 <)
inRange' x = atLeastLowerBound x && atMostUpperBound x
-- the type of test we use for the 2D version might actually also
-- work in higher dimensions that might allow us to drop the
-- Fractional constraint
instance (Ord r, Fractional r) =>
LineSegment 2 p r `IsIntersectableWith` LineSegment 2 p r where
nonEmptyIntersection = defaultNonEmptyIntersection
a `intersect` b = match ((a^._SubLine) `intersect` (b^._SubLine)) $
H coRec
:& H coRec
:& H (coRec . subLineToSegment)
:& RNil
instance (Ord r, Fractional r) =>
LineSegment 2 p r `IsIntersectableWith` Line 2 r where
nonEmptyIntersection = defaultNonEmptyIntersection
s `intersect` l = let ubSL = s^._SubLine.re _unBounded.to dropExtra
in match (ubSL `intersect` fromLine l) $
H coRec
:& H coRec
:& H (const (coRec s))
:& RNil
-- * Functions on LineSegments
-- | Test if a point lies on a line segment.
--
-- >>> (Point2 1 0) `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 2 0 :+ ()))
-- True
-- >>> (Point2 1 1) `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 2 0 :+ ()))
-- False
-- >>> (Point2 5 0) `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 2 0 :+ ()))
-- False
-- >>> (Point2 (-1) 0) `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 2 0 :+ ()))
-- False
-- >>> (Point2 1 1) `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 3 3 :+ ()))
-- True
-- >>> (Point2 2 0) `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 2 0 :+ ()))
-- True
-- >>> origin `onSegment2` (ClosedLineSegment (origin :+ ()) (Point2 2 0 :+ ()))
-- True
onSegment2 :: (Ord r, Num r)
=> Point 2 r -> LineSegment 2 p r -> Bool
p `onSegment2` s@(LineSegment u v) = case ccw' (ext p) (u^.unEndPoint) (v^.unEndPoint) of
CoLinear -> let su = p `onSide` lu
sv = p `onSide` lv
in su /= sv
&& ((su == OnLine) `implies` isClosed u)
&& ((sv == OnLine) `implies` isClosed v)
_ -> False
where
(Line _ w) = perpendicularTo $ supportingLine s
lu = Line (u^.unEndPoint.core) w
lv = Line (v^.unEndPoint.core) w
a `implies` b = b || not a
-- | The left and right end point (or left below right if they have equal x-coords)
orderedEndPoints :: Ord r => LineSegment 2 p r -> (Point 2 r :+ p, Point 2 r :+ p)
orderedEndPoints s = if pc <= qc then (p, q) else (q,p)
where
p@(pc :+ _) = s^.start
q@(qc :+ _) = s^.end
-- | Length of the line segment
segmentLength :: (Arity d, Floating r) => LineSegment d p r -> r
segmentLength ~(LineSegment' p q) = distanceA (p^.core) (q^.core)
sqSegmentLength :: (Arity d, Num r) => LineSegment d p r -> r
sqSegmentLength ~(LineSegment' p q) = qdA (p^.core) (q^.core)
-- | Squared distance from the point to the Segment s. The same remark as for
-- the 'sqDistanceToSegArg' applies here.
sqDistanceToSeg :: (Arity d, Fractional r, Ord r) => Point d r -> LineSegment d p r -> r
sqDistanceToSeg p = fst . sqDistanceToSegArg p
-- | Squared distance from the point to the Segment s, and the point on s
-- realizing it. Note that if the segment is *open*, the closest point
-- returned may be one of the (open) end points, even though technically the
-- end point does not lie on the segment. (The true closest point then lies
-- arbitrarily close to the end point).
sqDistanceToSegArg :: (Arity d, Fractional r, Ord r)
=> Point d r -> LineSegment d p r -> (r, Point d r)
sqDistanceToSegArg p s = let m = sqDistanceToArg p (supportingLine s)
xs = m : map (\(q :+ _) -> (qdA p q, q)) [s^.start, s^.end]
in F.minimumBy (comparing fst)
. filter (flip onSegment s . snd) $ xs
-- | flips the start and end point of the segment
flipSegment :: LineSegment d p r -> LineSegment d p r
flipSegment s = let p = s^.start
q = s^.end
in (s&start .~ q)&end .~ p
-- testSeg :: LineSegment 2 () Rational
-- testSeg = LineSegment (Open $ ext origin) (Closed $ ext (Point2 10 0))
-- horL' :: Line 2 Rational
-- horL' = horizontalLine 0
-- testI = testSeg `intersect` horL'
-- ff = bimap (fmap Val) (const ())
-- ss' = let (LineSegment p q) = testSeg in
-- LineSegment (p&unEndPoint %~ ff)
-- (q&unEndPoint %~ ff)
-- ss'' = ss'^._SubLine
-- | Linearly interpolate the two endpoints with a value in the range [0,1]
--
-- >>> interpolate 0.5 $ ClosedLineSegment (ext $ origin) (ext $ Point2 10.0 10.0)
-- Point2 5.0 5.0
-- >>> interpolate 0.1 $ ClosedLineSegment (ext $ origin) (ext $ Point2 10.0 10.0)
-- Point2 1.0 1.0
-- >>> interpolate 0 $ ClosedLineSegment (ext $ origin) (ext $ Point2 10.0 10.0)
-- Point2 0.0 0.0
-- >>> interpolate 1 $ ClosedLineSegment (ext $ origin) (ext $ Point2 10.0 10.0)
-- Point2 10.0 10.0
interpolate :: (Fractional r, Arity d) => r -> LineSegment d p r -> Point d r
interpolate t (LineSegment' p q) = Point $ (asV p ^* (1-t)) ^+^ (asV q ^* t)
where
asV = (^.core.vector)
-- | smart constructor that creates a valid segment, i.e. it validates
-- that the endpoints are disjoint.
validSegment :: (Eq r, Arity d)
=> EndPoint (Point d r :+ p) -> EndPoint (Point d r :+ p)
-> Maybe (LineSegment d p r)
validSegment u v = let s = LineSegment u v
in if s^.start.core /= s^.end.core then Just s else Nothing