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243 lines
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243 lines
21 KiB
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<span class="decl"><span class="nottickedoff">never executed</span> <span class="tickonlytrue">always true</span> <span class="tickonlyfalse">always false</span></span>
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<pre>
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<span class="lineno"> 1 </span>{-# LANGUAGE FlexibleInstances #-}
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<span class="lineno"> 2 </span>{-# OPTIONS_GHC -Wno-orphans #-}
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<span class="lineno"> 3 </span>{-|
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<span class="lineno"> 4 </span>Module : Reanimate.Math.Common
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<span class="lineno"> 5 </span>Copyright : Written by David Himmelstrup
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<span class="lineno"> 6 </span>License : Unlicense
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<span class="lineno"> 7 </span>Maintainer : lemmih@gmail.com
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<span class="lineno"> 8 </span>Stability : experimental
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<span class="lineno"> 9 </span>Portability : POSIX
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<span class="lineno"> 10 </span>
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<span class="lineno"> 11 </span>Low-level primitives related to computational geometry.
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<span class="lineno"> 12 </span>
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<span class="lineno"> 13 </span>-}
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<span class="lineno"> 14 </span>module Reanimate.Math.Common
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<span class="lineno"> 15 </span> ( -- * Ring
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<span class="lineno"> 16 </span> Ring(..)
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<span class="lineno"> 17 </span> , ringSize -- :: Ring a -> Int
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<span class="lineno"> 18 </span> , ringAccess -- :: Ring a -> Int -> V2 a
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<span class="lineno"> 19 </span> , ringClamp -- :: Ring a -> Int -> Int
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<span class="lineno"> 20 </span> , ringUnpack -- :: Ring a -> Vector (V2 a)
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<span class="lineno"> 21 </span> , ringPack -- :: Vector (V2 a) -> Ring a
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<span class="lineno"> 22 </span> , ringMap -- :: (V2 a -> V2 b) -> Ring a -> Ring b
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<span class="lineno"> 23 </span> , ringRayIntersect -- :: Ring Rational -> (Int, Int) -> (Int,Int) -> Maybe (V2 Rational)
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<span class="lineno"> 24 </span> -- * Math
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<span class="lineno"> 25 </span> , area -- :: Fractional a => V2 a -> V2 a -> V2 a -> a
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<span class="lineno"> 26 </span> , area2X -- :: Fractional a => V2 a -> V2 a -> V2 a -> a
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<span class="lineno"> 27 </span> , isLeftTurn -- :: (Num a, Ord a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 28 </span> , isLeftTurnOrLinear -- :: (Num a, Ord a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 29 </span> , isRightTurn -- :: (Num a, Ord a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 30 </span> , isRightTurnOrLinear -- :: (Num a, Ord a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 31 </span> , direction -- :: Num a => V2 a -> V2 a -> V2 a -> a
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<span class="lineno"> 32 </span> , isInside -- :: (Fractional a, Ord a) => V2 a -> V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 33 </span> , isInsideStrict -- :: (Fractional a, Ord a) => V2 a -> V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 34 </span> , barycentricCoords -- :: Fractional a => V2 a -> V2 a -> V2 a -> V2 a -> (a, a, a)
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<span class="lineno"> 35 </span> , rayIntersect -- :: (Fractional a, Ord a) => (V2 a,V2 a) -> (V2 a,V2 a) -> Maybe (V2 a)
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<span class="lineno"> 36 </span> , isBetween -- :: (Ord a, Fractional a) => V2 a -> (V2 a, V2 a) -> Bool
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<span class="lineno"> 37 </span> , lineIntersect -- :: (Ord a, Fractional a) => (V2 a, V2 a) -> (V2 a, V2 a) -> Maybe (V2 a)
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<span class="lineno"> 38 </span> , distSquared -- :: (Fractional a) => V2 a -> V2 a -> a
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<span class="lineno"> 39 </span> , approxDist -- :: (Real a, Fractional a) => V2 a -> V2 a -> a
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<span class="lineno"> 40 </span> , distance' -- :: (Real a, Fractional a) => V2 a -> V2 a -> Double
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<span class="lineno"> 41 </span> , triangleAngles -- :: V2 Double -> V2 Double -> V2 Double -> (Double, Double, Double)
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<span class="lineno"> 42 </span> , Epsilon(..)
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<span class="lineno"> 43 </span> ) where
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<span class="lineno"> 44 </span>
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<span class="lineno"> 45 </span>import Data.Vector (Vector)
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<span class="lineno"> 46 </span>import qualified Data.Vector as V
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<span class="lineno"> 47 </span>import Linear.Matrix (det33)
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<span class="lineno"> 48 </span>import Linear.Metric
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<span class="lineno"> 49 </span>import Linear.V2
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<span class="lineno"> 50 </span>import Linear.V3
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<span class="lineno"> 51 </span>import Linear.Vector
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<span class="lineno"> 52 </span>import Linear.Epsilon
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<span class="lineno"> 53 </span>
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<span class="lineno"> 54 </span>instance Epsilon Rational where
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<span class="lineno"> 55 </span> <span class="decl"><span class="nottickedoff">nearZero r = r==0</span></span>
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<span class="lineno"> 56 </span>
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<span class="lineno"> 57 </span>-- | Circular collection of pairs.
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<span class="lineno"> 58 </span>newtype Ring a = Ring (Vector (V2 a))
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<span class="lineno"> 59 </span>
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<span class="lineno"> 60 </span>-- | Number of elements in the ring.
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<span class="lineno"> 61 </span>ringSize :: Ring a -> Int
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<span class="lineno"> 62 </span><span class="decl"><span class="nottickedoff">ringSize (Ring v) = length v</span></span>
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<span class="lineno"> 63 </span>
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<span class="lineno"> 64 </span>-- | Safe method for accessing elements in the ring.
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<span class="lineno"> 65 </span>ringAccess :: Ring a -> Int -> V2 a
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<span class="lineno"> 66 </span><span class="decl"><span class="nottickedoff">ringAccess (Ring v) i = v V.! mod i (length v)</span></span>
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<span class="lineno"> 67 </span>
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<span class="lineno"> 68 </span>-- | Clamp index to within the usable range for the ring.
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<span class="lineno"> 69 </span>ringClamp :: Ring a -> Int -> Int
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<span class="lineno"> 70 </span><span class="decl"><span class="nottickedoff">ringClamp (Ring v) i = mod i (length v)</span></span>
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<span class="lineno"> 71 </span>
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<span class="lineno"> 72 </span>-- | Convert ring to a vector.
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<span class="lineno"> 73 </span>ringUnpack :: Ring a -> Vector (V2 a)
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<span class="lineno"> 74 </span><span class="decl"><span class="nottickedoff">ringUnpack (Ring v) = v</span></span>
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<span class="lineno"> 75 </span>
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<span class="lineno"> 76 </span>-- | Convert vector to a ring.
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<span class="lineno"> 77 </span>ringPack :: Vector (V2 a) -> Ring a
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<span class="lineno"> 78 </span><span class="decl"><span class="nottickedoff">ringPack = Ring</span></span>
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<span class="lineno"> 79 </span>
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<span class="lineno"> 80 </span>-- | Map each element of a ring.
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<span class="lineno"> 81 </span>ringMap :: (V2 a -> V2 b) -> Ring a -> Ring b
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<span class="lineno"> 82 </span><span class="decl"><span class="nottickedoff">ringMap fn (Ring v) = Ring (V.map fn v)</span></span>
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<span class="lineno"> 83 </span>
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<span class="lineno"> 84 </span>-- | Compute the intersection of two pairs of nodes in the ring.
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<span class="lineno"> 85 </span>ringRayIntersect :: Ring Rational -> (Int, Int) -> (Int,Int) -> Maybe (V2 Rational)
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<span class="lineno"> 86 </span><span class="decl"><span class="nottickedoff">ringRayIntersect p (a,b) (c,d) =</span>
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<span class="lineno"> 87 </span><span class="spaces"> </span><span class="nottickedoff">rayIntersect (ringAccess p a, ringAccess p b) (ringAccess p c, ringAccess p d)</span></span>
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<span class="lineno"> 88 </span>
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<span class="lineno"> 89 </span>-- | Compute area of triangle.
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<span class="lineno"> 90 </span>area :: Fractional a => V2 a -> V2 a -> V2 a -> a
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<span class="lineno"> 91 </span><span class="decl"><span class="nottickedoff">area a b c = 1/2 * area2X a b c</span></span>
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<span class="lineno"> 92 </span>
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<span class="lineno"> 93 </span>-- | Compute 2x area of triangle. This avoids a division.
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<span class="lineno"> 94 </span>area2X :: Fractional a => V2 a -> V2 a -> V2 a -> a
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<span class="lineno"> 95 </span><span class="decl"><span class="nottickedoff">area2X (V2 a1 a2) (V2 b1 b2) (V2 c1 c2) =</span>
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<span class="lineno"> 96 </span><span class="spaces"> </span><span class="nottickedoff">det33 (V3 (V3 a1 a2 1)</span>
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<span class="lineno"> 97 </span><span class="spaces"> </span><span class="nottickedoff">(V3 b1 b2 1)</span>
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<span class="lineno"> 98 </span><span class="spaces"> </span><span class="nottickedoff">(V3 c1 c2 1))</span></span>
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<span class="lineno"> 99 </span>
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<span class="lineno"> 100 </span>compareEpsZero :: (Ord a, Fractional a, Epsilon a) => a -> Ordering
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<span class="lineno"> 101 </span><span class="decl"><span class="nottickedoff">compareEpsZero val</span>
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<span class="lineno"> 102 </span><span class="spaces"> </span><span class="nottickedoff">| nearZero val = EQ</span>
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<span class="lineno"> 103 </span><span class="spaces"> </span><span class="nottickedoff">| otherwise = compare val 0</span></span>
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<span class="lineno"> 104 </span>
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<span class="lineno"> 105 </span>{-# INLINE isLeftTurn #-}
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<span class="lineno"> 106 </span>-- | Return @True@ iff the line from @p1@ to @p2@ makes a left-turn to @p3@.
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<span class="lineno"> 107 </span>isLeftTurn :: (Fractional a, Ord a, Epsilon a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 108 </span><span class="decl"><span class="nottickedoff">isLeftTurn p1 p2 p3 =</span>
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<span class="lineno"> 109 </span><span class="spaces"> </span><span class="nottickedoff">case compareEpsZero (direction p1 p2 p3) of</span>
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<span class="lineno"> 110 </span><span class="spaces"> </span><span class="nottickedoff">LT -> True</span>
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<span class="lineno"> 111 </span><span class="spaces"> </span><span class="nottickedoff">EQ -> False -- colinear</span>
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<span class="lineno"> 112 </span><span class="spaces"> </span><span class="nottickedoff">GT -> False</span></span>
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<span class="lineno"> 113 </span>
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<span class="lineno"> 114 </span>{-# INLINE isLeftTurnOrLinear #-}
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<span class="lineno"> 115 </span>-- | Return @True@ iff the line from @p1@ to @p2@ does not make a right-turn to @p3@.
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<span class="lineno"> 116 </span>isLeftTurnOrLinear :: (Fractional a, Ord a, Epsilon a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 117 </span><span class="decl"><span class="nottickedoff">isLeftTurnOrLinear p1 p2 p3 =</span>
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<span class="lineno"> 118 </span><span class="spaces"> </span><span class="nottickedoff">case compareEpsZero (direction p1 p2 p3) of</span>
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<span class="lineno"> 119 </span><span class="spaces"> </span><span class="nottickedoff">LT -> True</span>
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<span class="lineno"> 120 </span><span class="spaces"> </span><span class="nottickedoff">EQ -> True -- colinear</span>
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<span class="lineno"> 121 </span><span class="spaces"> </span><span class="nottickedoff">GT -> False</span></span>
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<span class="lineno"> 122 </span>
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<span class="lineno"> 123 </span>{-# INLINE isRightTurn #-}
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<span class="lineno"> 124 </span>-- | Return @True@ iff the line from @p1@ to @p2@ makes a right-turn to @p3@.
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<span class="lineno"> 125 </span>isRightTurn :: (Fractional a, Ord a, Epsilon a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 126 </span><span class="decl"><span class="nottickedoff">isRightTurn a b c = not (isLeftTurnOrLinear a b c)</span></span>
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<span class="lineno"> 127 </span>
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<span class="lineno"> 128 </span>{-# INLINE isRightTurnOrLinear #-}
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<span class="lineno"> 129 </span>-- | Return @True@ iff the line from @p1@ to @p2@ does not make a left-turn to @p3@.
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<span class="lineno"> 130 </span>isRightTurnOrLinear :: (Fractional a, Ord a, Epsilon a) => V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 131 </span><span class="decl"><span class="nottickedoff">isRightTurnOrLinear a b c = not (isLeftTurn a b c)</span></span>
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<span class="lineno"> 132 </span>
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<span class="lineno"> 133 </span>{-# INLINE direction #-}
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<span class="lineno"> 134 </span>-- | Compute the change in direction in a line between the three points.
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<span class="lineno"> 135 </span>direction :: Num a => V2 a -> V2 a -> V2 a -> a
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<span class="lineno"> 136 </span><span class="decl"><span class="nottickedoff">direction p1 p2 p3 = crossZ (p3-p1) (p2-p1)</span></span>
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<span class="lineno"> 137 </span>
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<span class="lineno"> 138 </span>{-# INLINE isInside #-}
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<span class="lineno"> 139 </span>-- | Returns @True@ if the fourth argument is inside the triangle or
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<span class="lineno"> 140 </span>-- on the border.
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<span class="lineno"> 141 </span>isInside :: (Fractional a, Ord a) => V2 a -> V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 142 </span><span class="decl"><span class="nottickedoff">isInside a b c d =</span>
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<span class="lineno"> 143 </span><span class="spaces"> </span><span class="nottickedoff">s >= 0 && s <= 1 && t >= 0 && t <= 1 && i >= 0 && i <= 1</span>
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<span class="lineno"> 144 </span><span class="spaces"> </span><span class="nottickedoff">where</span>
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<span class="lineno"> 145 </span><span class="spaces"> </span><span class="nottickedoff">(s, t, i) = barycentricCoords a b c d</span></span>
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<span class="lineno"> 146 </span>
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<span class="lineno"> 147 </span>{-# INLINE isInsideStrict #-}
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<span class="lineno"> 148 </span>-- | Returns @True@ iff the fourth argument is inside the triangle.
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<span class="lineno"> 149 </span>isInsideStrict :: (Fractional a, Ord a) => V2 a -> V2 a -> V2 a -> V2 a -> Bool
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<span class="lineno"> 150 </span><span class="decl"><span class="nottickedoff">isInsideStrict a b c d =</span>
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<span class="lineno"> 151 </span><span class="spaces"> </span><span class="nottickedoff">s > 0 && s < 1 && t > 0 && t < 1 && i > 0 && i < 1</span>
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<span class="lineno"> 152 </span><span class="spaces"> </span><span class="nottickedoff">where</span>
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<span class="lineno"> 153 </span><span class="spaces"> </span><span class="nottickedoff">(s, t, i) = barycentricCoords a b c d</span></span>
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<span class="lineno"> 154 </span>
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<span class="lineno"> 155 </span>{-# INLINE barycentricCoords #-}
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<span class="lineno"> 156 </span>-- | Compute relative coordinates inside the triangle. Invariant: @a+b+c=1@
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<span class="lineno"> 157 </span>barycentricCoords :: Fractional a => V2 a -> V2 a -> V2 a -> V2 a -> (a, a, a)
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<span class="lineno"> 158 </span><span class="decl"><span class="nottickedoff">barycentricCoords (V2 x1 y1) (V2 x2 y2) (V2 x3 y3) (V2 x y) =</span>
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<span class="lineno"> 159 </span><span class="spaces"> </span><span class="nottickedoff">(lam1, lam2, lam3)</span>
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<span class="lineno"> 160 </span><span class="spaces"> </span><span class="nottickedoff">where</span>
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<span class="lineno"> 161 </span><span class="spaces"> </span><span class="nottickedoff">lam1 = ((y2-y3)*(x-x3) + (x3 - x2)*(y-y3)) /</span>
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<span class="lineno"> 162 </span><span class="spaces"> </span><span class="nottickedoff">((y2-y3)*(x1-x3) + (x3-x2)*(y1-y3))</span>
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<span class="lineno"> 163 </span><span class="spaces"> </span><span class="nottickedoff">lam2 = ((y3-y1)*(x-x3) + (x1-x3)*(y-y3)) /</span>
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<span class="lineno"> 164 </span><span class="spaces"> </span><span class="nottickedoff">((y2-y3)*(x1-x3) + (x3-x2)*(y1-y3))</span>
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<span class="lineno"> 165 </span><span class="spaces"> </span><span class="nottickedoff">lam3 = 1 - lam1 - lam2</span></span>
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<span class="lineno"> 166 </span>
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<span class="lineno"> 167 </span>
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<span class="lineno"> 168 </span>{-# INLINE rayIntersect #-}
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<span class="lineno"> 169 </span>-- | Compute intersection of two infinite lines.
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<span class="lineno"> 170 </span>rayIntersect :: (Fractional a, Ord a) => (V2 a,V2 a) -> (V2 a,V2 a) -> Maybe (V2 a)
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<span class="lineno"> 171 </span><span class="decl"><span class="nottickedoff">rayIntersect (V2 x1 y1,V2 x2 y2) (V2 x3 y3, V2 x4 y4)</span>
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<span class="lineno"> 172 </span><span class="spaces"> </span><span class="nottickedoff">| yBot == 0 = Nothing</span>
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<span class="lineno"> 173 </span><span class="spaces"> </span><span class="nottickedoff">| otherwise = Just $</span>
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<span class="lineno"> 174 </span><span class="spaces"> </span><span class="nottickedoff">V2 (xTop/xBot) (yTop/yBot)</span>
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<span class="lineno"> 175 </span><span class="spaces"> </span><span class="nottickedoff">where</span>
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<span class="lineno"> 176 </span><span class="spaces"> </span><span class="nottickedoff">xTop = (x1*y2 - y1*x2)*(x3-x4) - (x1 - x2)*(x3*y4-y3*x4)</span>
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<span class="lineno"> 177 </span><span class="spaces"> </span><span class="nottickedoff">xBot = (x1-x2)*(y3-y4)-(y1-y2)*(x3-x4)</span>
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<span class="lineno"> 178 </span><span class="spaces"> </span><span class="nottickedoff">yTop = (x1*y2 - y1*x2)*(y3-y4) - (y1-y2)*(x3*y4-y3*x4)</span>
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<span class="lineno"> 179 </span><span class="spaces"> </span><span class="nottickedoff">yBot = (x1-x2)*(y3-y4) - (y1-y2)*(x3-x4)</span></span>
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<span class="lineno"> 180 </span>
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<span class="lineno"> 181 </span>{-# INLINE isBetween #-}
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<span class="lineno"> 182 </span>-- | Returns @True@ iff a point is on a line segment.
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<span class="lineno"> 183 </span>isBetween :: (Ord a, Fractional a) => V2 a -> (V2 a, V2 a) -> Bool
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<span class="lineno"> 184 </span><span class="decl"><span class="nottickedoff">isBetween (V2 x y) (V2 x1 y1, V2 x2 y2) =</span>
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<span class="lineno"> 185 </span><span class="spaces"> </span><span class="nottickedoff">((y1 > y) /= (y2 > y) || y == y1 || y == y2) && -- y is between y1 and y2</span>
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<span class="lineno"> 186 </span><span class="spaces"> </span><span class="nottickedoff">((x1 > x) /= (x2 > x) || x == x1 || x == x2)</span></span>
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<span class="lineno"> 187 </span>
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<span class="lineno"> 188 </span>{-# INLINE lineIntersect #-}
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<span class="lineno"> 189 </span>-- | Compute intersection of two line segments.
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<span class="lineno"> 190 </span>lineIntersect :: (Ord a, Fractional a) => (V2 a, V2 a) -> (V2 a, V2 a) -> Maybe (V2 a)
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<span class="lineno"> 191 </span><span class="decl"><span class="nottickedoff">lineIntersect a b =</span>
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<span class="lineno"> 192 </span><span class="spaces"> </span><span class="nottickedoff">case rayIntersect a b of</span>
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<span class="lineno"> 193 </span><span class="spaces"> </span><span class="nottickedoff">Just u</span>
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<span class="lineno"> 194 </span><span class="spaces"> </span><span class="nottickedoff">| isBetween u a && isBetween u b -> Just u</span>
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<span class="lineno"> 195 </span><span class="spaces"> </span><span class="nottickedoff">_ -> Nothing</span></span>
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<span class="lineno"> 196 </span>
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<span class="lineno"> 197 </span>-- circleIntersect :: (Ord a, Fractional a) => (V2 a, V2 a) -> (V2 a, V2 a) -> [V2 a]
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<span class="lineno"> 198 </span>
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<span class="lineno"> 199 </span>-- | Compute the square of the distance between two points.
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<span class="lineno"> 200 </span>distSquared :: (Num a) => V2 a -> V2 a -> a
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<span class="lineno"> 201 </span><span class="decl"><span class="nottickedoff">distSquared a b = quadrance (a ^-^ b)</span></span>
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<span class="lineno"> 202 </span>
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<span class="lineno"> 203 </span>-- | Approximate the distance between two points.
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<span class="lineno"> 204 </span>approxDist :: (Real a, Fractional a) => V2 a -> V2 a -> a
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<span class="lineno"> 205 </span><span class="decl"><span class="nottickedoff">approxDist a b = realToFrac (sqrt (realToFrac (distSquared a b) :: Double))</span></span>
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<span class="lineno"> 206 </span>
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<span class="lineno"> 207 </span>-- | Approximate the distance between two points.
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<span class="lineno"> 208 </span>distance' :: (Real a, Fractional a) => V2 a -> V2 a -> Double
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<span class="lineno"> 209 </span><span class="decl"><span class="nottickedoff">distance' a b = sqrt (realToFrac (distSquared a b))</span></span>
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<span class="lineno"> 210 </span>
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<span class="lineno"> 211 </span>-- sum of angles is always pi.
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<span class="lineno"> 212 </span>-- | Approximate the angles of a triangle.
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<span class="lineno"> 213 </span>triangleAngles :: V2 Double -> V2 Double -> V2 Double -> (Double, Double, Double)
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<span class="lineno"> 214 </span><span class="decl"><span class="nottickedoff">triangleAngles a b c =</span>
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<span class="lineno"> 215 </span><span class="spaces"> </span><span class="nottickedoff">(findAngle (b-a) (c-a)</span>
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<span class="lineno"> 216 </span><span class="spaces"> </span><span class="nottickedoff">,findAngle (c-b) (a-b)</span>
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<span class="lineno"> 217 </span><span class="spaces"> </span><span class="nottickedoff">,findAngle (a-c) (b-c))</span>
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<span class="lineno"> 218 </span><span class="spaces"> </span><span class="nottickedoff">where</span>
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<span class="lineno"> 219 </span><span class="spaces"> </span><span class="nottickedoff">findAngle v1 v2 = abs (atan2 (crossZ v1 v2) (dot v1 v2))</span></span>
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<span class="lineno"> 220 </span> -- findAngle v1 v2 = acos (dot v1 v2 / (norm v1 * norm v2))
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</pre>
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