272 lines
10 KiB
Haskell
272 lines
10 KiB
Haskell
{-# LANGUAGE BangPatterns #-}
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--{-# LANGUAGE TupleSections #-}
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{- Testing for and finding intersection points. -}
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module Geometry.Intersect where
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import Geometry.Data
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import Geometry.Vector
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import Geometry.LHS
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import Control.Applicative
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import Data.List
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import Data.Maybe
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-- | If two lines intersect, return 'Just' that point.
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intersectLineLine' :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE intersectLineLine' #-}
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intersectLineLine' (V2 x1 y1) (V2 x2 y2) (V2 x3 y3) (V2 x4 y4)
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| den == 0 = Nothing
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| otherwise = Just $ V2 (x1 + (x2-x1)*t'/den) (y1 + (y2-y1)*t'/den)
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where
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den = (x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)
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t' = (x1-x3)*(y3-y4) - (y1-y3)*(x3-x4)
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intersectSegSegErrorTest :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE intersectSegSegErrorTest #-}
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intersectSegSegErrorTest a b c d = case intersectSegSeg a b c d of
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Nothing | intersectSegSegFullTest a b c d ->
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error $ "intersectSegSeg did not intersect"++show a ++ show b ++ show c++ show d
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Just x | not $ intersectSegSegFullTest a b c d ->
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error $ "intersectSegSeg did intersect"++show a ++ show b ++ show c++ show d
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++ " at " ++ show x
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m -> m
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-- | If two segments intersect, return 'Just' that point.
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intersectSegSeg :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE intersectSegSeg #-}
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intersectSegSeg (V2 x1 y1) (V2 x2 y2) (V2 x3 y3) (V2 x4 y4)
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| V2 x1 y1 == V2 x2 y2 || V2 x3 y3 == V2 x4 y4 = Nothing
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| den == 0 = Nothing
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| den > 0 && (t' < 0 || u' < 0 || t' > den || u' > den)
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= Nothing
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| den < 0 && (t' > 0 || u' > 0 || t' < den || u' < den)
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= Nothing
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| otherwise = Just $ V2 (x1 + (x2-x1)*t'/den) (y1 + (y2-y1)*t'/den)
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where
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den = (x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)
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t' = (x1-x3)*(y3-y4) - (y1-y3)*(x3-x4)
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u' = (y1-y2)*(x1-x3) - (x1-x2)*(y1-y3)
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-- | Intended to intersect a segment with a half-line-segment, ie a segment
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-- extending infinitely in one direction.
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intersectSegLineFrom :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE intersectSegLineFrom #-}
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intersectSegLineFrom (V2 x1 y1) (V2 x2 y2) (V2 x3 y3) (V2 x4 y4)
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| den == 0 = Nothing
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| den > 0 && ( t' < 0 || u' < 0 || t' > den )
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= Nothing
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| den < 0 && ( t' > 0 || u' > 0 || t' < den )
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= Nothing
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| otherwise = Just $ V2 (x1 + (x2-x1)*t'/den) (y1 + (y2-y1)*t'/den)
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where
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den = (x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)
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t' = (x1-x3)*(y3-y4) - (y1-y3)*(x3-x4)
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u' = (y1-y2)*(x1-x3) - (x1-x2)*(y1-y3)
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-- | Similar to 'intersectSegLineFrom'', but this version is probably not correct...
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intersectSegLineext :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE intersectSegLineext #-}
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intersectSegLineext (V2 x1 y1) (V2 x2 y2) (V2 x3 y3) (V2 x4 y4)
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| den == 0 = Nothing
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| den > 0 && ( t' < 0 || u' < den || t' > den )
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= Nothing
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| den < 0 && ( t' > 0 || u' > - den || t' < den )
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= Nothing
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| otherwise = Just $ V2 (x1 + (x2-x1)*t'/den) (y1 + (y2-y1)*t'/den)
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where
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den = (x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)
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t' = (x1-x3)*(y3-y4) - (y1-y3)*(x3-x4)
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u' = (y1-y2)*(x1-x3) - (x1-x2)*(y1-y3)
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-- | Intersect a segment with a line.
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intersectSegLine :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE intersectSegLine #-}
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intersectSegLine (V2 x1 y1) (V2 x2 y2) (V2 x3 y3) (V2 x4 y4)
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| den == 0 = Nothing
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| den > 0 && (t' < 0 || t' > den)
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= Nothing
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| den < 0 && (t' > 0 || t' < den)
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= Nothing
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| otherwise = Just $ V2 (x1 + (x2-x1)*t'/den) ( y1 + (y2-y1)*t'/den)
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where
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den = (x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)
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t' = (x1-x3)*(y3-y4) - (y1-y3)*(x3-x4)
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--u' = (y1-y2)*(x1-x3) - (x1-x2)*(y1-y3)
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-- | A test that should align with Just values from intersectSegSeg.
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intersectSegSegFullTest
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:: Point2
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-> Point2
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-> Point2
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-> Point2
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-> Bool
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{-# INLINE intersectSegSegFullTest #-}
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intersectSegSegFullTest x y z w
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= f x y z w && f z w x y && x /= y && z /= w
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&& normalizeV (x -.- y) /= normalizeV (z -.- w)
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&& normalizeV (y -.- x) /= normalizeV (z -.- w)
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where
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f a b c d = ( not (isRHS a b c) && not (isLHS a b d) )
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|| ( not (isLHS a b c) && not (isRHS a b d) )
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-- | It is not always necessary to find a point of intersection, sometimes a
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-- test may suffice.
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intersectSegSegTest
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:: Point2
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-> Point2
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-> Point2
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-> Point2
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-> Bool
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{-# INLINE intersectSegSegTest #-}
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intersectSegSegTest x y z w
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= f x y z w && f z w x y && x /= y && z /= w
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where
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f a b c d = ( not (isRHS a b c) && not (isLHS a b d) )
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|| ( not (isLHS a b c) && not (isRHS a b d) )
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intersectSegSegPreTest
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:: Point2
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-> Point2
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-> Point2
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-> Point2
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-> Maybe Point2
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{-# INLINE intersectSegSegPreTest #-}
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intersectSegSegPreTest a b c d
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| intersectSegSegTest a b c d = myIntersectSegSeg a b c d
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| otherwise = Nothing
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-- | Due to floating point issues, 'intersectSegSeg'' is not always
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-- accurate---'myIntersectSegSeg'
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-- fixes at least some of
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-- the problem cases by transforming the points into rationals and then doing the
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-- intersection.
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-- This version is, probably, slower---both testing and benchmarking should be
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-- done.
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myIntersectSegSeg
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:: Point2
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-> Point2
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-> Point2
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-> Point2
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-> Maybe Point2
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{-# INLINE myIntersectSegSeg #-}
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myIntersectSegSeg a@(V2 ax ay) b@(V2 bx by) c@(V2 cx cy) d@(V2 dx dy) = case ratIntersectLineLine a b c d of
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Nothing -> Nothing
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Just (V2 x y) -> if inbetween x && inbetween' y
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then Just (V2 x y)
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else Nothing
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where
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inbetween x = ((ax <= x && x <= bx) || (bx <= x && x <= ax))
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&& ((cx <= x && x <= dx) || (dx <= x && x <= cx))
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inbetween' y = ((ay <= y && y <= by) || (by <= y && y <= ay))
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&& ((cy <= y && y <= dy) || (dy <= y && y <= cy))
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-- | Polymorphic intersection of fractional line points.
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myIntersectLineLine :: (Eq a,Fractional a) => V2 a -> V2 a -> V2 a -> V2 a -> Maybe (V2 a)
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{-# INLINE myIntersectLineLine #-}
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myIntersectLineLine a@(V2 ax _) b c@(V2 cx _) d
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| isNothing (linGrad a b) = V2 ax <$> axisInt (c *-* V2 ax 0) (d *-* V2 ax 0)
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| isNothing (linGrad c d) = V2 cx <$> axisInt (a *-* V2 cx 0) (b *-* V2 cx 0)
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| otherwise
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= case linGrad a b ^-^ linGrad c d of
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Just 0 -> Nothing
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_ -> liftA2 V2 newx ((linGrad a b ^*^ newx) ^+^ axisInt a b)
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where
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(^-^) = liftA2 (-)
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(^+^) = liftA2 (+)
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(^/^) = liftA2 (/)
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(^*^) = liftA2 (*)
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newx = (axisInt c d ^-^ axisInt a b) ^/^ (linGrad a b ^-^ linGrad c d)
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(*-*) (V2 ax' ay) (V2 bx by) = V2 (ax'-bx) (ay-by)
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-- | Transforms floating points to rationals then performs line intersection.
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ratIntersectLineLine :: Point2 -> Point2 -> Point2 -> Point2 -> Maybe Point2
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{-# INLINE ratIntersectLineLine #-}
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ratIntersectLineLine a b c d = toNumPoint2
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<$> myIntersectLineLine (toRatPoint2 a) (toRatPoint2 b) (toRatPoint2 c) (toRatPoint2 d)
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where
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toRatPoint2 (V2 x y) = V2 (toRational x) (toRational y)
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toNumPoint2 (V2 x y) = V2 (fromRational x) (fromRational y)
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{- | Round the floats within a 'Point2' to the nearest integer.
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__Examples__
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Rounding jumps after intervals of .5:
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>>> roundPoint (0.5,0.5001)
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(0.0,1.0)
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but is symmetric around 0:
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>>> roundPoint2 (0.5,-0.5)
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(0.0,0.0)
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-}
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roundPoint2 :: Point2 -> Point2
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roundPoint2 (V2 x y) = V2 (fromIntegral (round x :: Int)) (fromIntegral (round y :: Int))
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-- | Given two points, finds the linear gradient if it is non-infinite.
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linGrad :: (Eq a,Fractional a) => V2 a -> V2 a -> Maybe a
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{-# INLINE linGrad #-}
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linGrad (V2 x y) (V2 a b)
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| x-a == 0 = Nothing
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| otherwise = Just $ (y-b)/(x-a)
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-- | Given two points, finds the intersection with the y axis if it exists.
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axisInt :: (Eq a,Fractional a) => V2 a -> V2 a -> Maybe a
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{-# INLINE axisInt #-}
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axisInt p (V2 a b) = (\lg -> b - (a*lg)) <$> linGrad p (V2 a b)
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-- | Placeholder, undefined.
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intersectSegsSeg :: [Point2] -> Point2 -> Point2 -> Maybe Point2
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intersectSegsSeg = undefined
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-- | Placeholder: should intersect a segment with a bezier curve.
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intersectSegBezquad :: Point2 -> Point2 -> Point2 -> Point2 -> Point2 -> [Point2]
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intersectSegBezquad = undefined
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-- | finds one (if any) of the points of intersection between a segment and a
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-- polygon.
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-- Can almost certainly be optimised.
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intersectSegPolyFirst :: Point2 -> Point2 -> [Point2] -> Maybe Point2
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intersectSegPolyFirst a b xs = foldr (<|>) Nothing $ zipWith lineColl xs (tail xs ++ [head xs])
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where
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lineColl = intersectSegSeg a b
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-- | Given a line and a point return the point on the line closest to the
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-- point.
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closestPointOnLine
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:: Point2 -- ^ First line point.
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-> Point2 -- ^ Second line point.
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-> Point2 -- ^ Point not on line.
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-> Point2
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{-# INLINE closestPointOnLine #-}
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closestPointOnLine !a !b !p = a +.+ u *.* (b -.- a)
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where u = closestPointOnLineParam a b p
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-- | Given a line and a point return a value corresponding to how far along the
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-- line the point is.
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closestPointOnLineParam
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:: Point2 -- ^ First line point.
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-> Point2 -- ^ Second line point.
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-> Point2 -- ^ Point not on line.
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-> Float
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{-# INLINE closestPointOnLineParam #-}
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closestPointOnLineParam !a !b !p
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= (p -.- a) `dotV` (b -.- a) / (b -.- a) `dotV` (b -.- a)
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-- | Given a segment and external point, find the closest point on the segment.
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-- clamps to the end of the segment
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closestPointOnSeg :: Point2 -> Point2 -> Point2 -> Point2
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closestPointOnSeg segP1 segP2 p
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| closestPointOnLineParam segP1 segP2 p <= 0 = segP1
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| closestPointOnLineParam segP1 segP2 p >= 1 = segP2
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| otherwise = closestPointOnLine segP1 segP2 p
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-- | Given a segment and external point, find the closest point on the segment.
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-- does not return closest points beyond the segment
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orthogonalPointOnSeg :: Point2 -> Point2 -> Point2 -> Maybe Point2
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orthogonalPointOnSeg a b p
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| param < 0 || param > 1 = Nothing
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| otherwise = Just $ a +.+ param *.* normalizeV (b -.- a)
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where
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param = closestPointOnLineParam a b p
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inSegArea :: Point2 -> Point2 -> Point2 -> Bool
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inSegArea a b c = param >= 0 && param <= dotV (b -.- a) (b -.- a)
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where
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param = dotV (b -.- a) (c -.- a)
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intersectCircSeg :: Point2 -> Float -> Point2 -> Point2 -> [Point2]
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intersectCircSeg c r a b
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| y < 0 = []
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| otherwise = nub $ filter (inSegArea a b) [ d -.- v, d +.+ v ]
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where
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d = closestPointOnLine a b c
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x = dist d c
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y = r ** 2 - x ** 2
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z = sqrt y
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v = z *.* normalizeV (b -.- a)
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intersectCircSegFirst :: Point2 -> Float -> Point2 -> Point2 -> Maybe Point2
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intersectCircSegFirst c r a = listToMaybe . intersectCircSeg c r a
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