cga/Algebra/Vector.hs

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{-# OPTIONS_HADDOCK ignore-exports #-}
{-# LANGUAGE ViewPatterns #-}
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module Algebra.Vector where
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import Control.Applicative
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import Control.Arrow ((***))
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import Data.List (sortBy)
import Diagrams.Coordinates
import Diagrams.TwoD.Types
import GHC.Float
import MyPrelude
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data Alignment = CW
| CCW
| CL
deriving (Eq)
-- |Convert two dimensions such as (xmin, xmax) and (ymin, ymax)
-- to proper square coordinates, as in:
-- ((xmin, ymin), (xmax, ymax))
dimToSquare :: (Double, Double) -- ^ x dimension
-> (Double, Double) -- ^ y dimension
-> ((Double, Double), (Double, Double)) -- ^ square describing those dimensions
dimToSquare (x1, x2) (y1, y2) = ((x1, y1), (x2, y2))
-- |Checks whether the Point is in a given Square.
inRange :: ((Double, Double), (Double, Double)) -- ^ the square: ((xmin, ymin), (xmax, ymax))
-> P2 Double -- ^ Coordinate
-> Bool -- ^ result
inRange ((xmin, ymin), (xmax, ymax)) (coords -> x :& y)
= x >= min xmin xmax
&& x <= max xmin xmax
&& y >= min ymin ymax
&& y <= max ymin ymax
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-- |Get the angle between two vectors.
getAngle :: V2 Double -> V2 Double -> Double
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getAngle a b =
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acos
. flip (/) (vecLength a * vecLength b)
. scalarProd a
$ b
-- |Get the length of a vector.
vecLength :: V2 Double -> Double
vecLength v = sqrt (x^(2 :: Int) + y^(2 :: Int))
where
(x, y) = unr2 v
-- |Compute the scalar product of two vectors.
scalarProd :: V2 Double -> V2 Double -> Double
scalarProd (V2 a1 a2) (V2 b1 b2) = a1 * b1 + a2 * b2
-- |Multiply a scalar with a vector.
scalarMul :: Double -> V2 Double -> V2 Double
scalarMul d (V2 a b) = V2 (a * d) (b * d)
-- |Construct a vector that points to a point from the origin.
pt2Vec :: P2 Double -> V2 Double
pt2Vec = r2 . unp2
-- |Give the point which is at the coordinates the vector
-- points to from the origin.
vec2Pt :: V2 Double -> P2 Double
vec2Pt = p2 . unr2
-- |Construct a vector between two points.
vp2 :: P2 Double -- ^ vector origin
-> P2 Double -- ^ vector points here
-> V2 Double
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vp2 a b = pt2Vec b - pt2Vec a
-- |Computes the determinant of 3 points.
det :: P2 Double -> P2 Double -> P2 Double -> Double
det (coords -> ax :& ay) (coords -> bx :& by) (coords -> cx :& cy) =
(bx - ax) * (cy - ay) - (by - ay) * (cx - ax)
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-- |Get the point where two lines intesect, if any. Excludes the
-- case of end-points intersecting.
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intersectSeg :: (P2 Double, P2 Double) -> (P2 Double, P2 Double) -> Maybe (P2 Double)
intersectSeg (a, b) (c, d) = case intersectSegSeg a b c d of
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Just x -> if x `notElem` [a,b,c,d] then Just a else Nothing
Nothing -> Nothing
-- |Get the orientation of 3 points which can either be
-- * clock-wise
-- * counter-clock-wise
-- * collinear
getOrient :: P2 Double -> P2 Double -> P2 Double -> Alignment
getOrient a b c = case compare (det a b c) 0 of
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LT -> CW
GT -> CCW
EQ -> CL
--- |Checks if 3 points a,b,c do not build a clockwise triangle by
--- connecting a-b-c. This is done by computing the determinant and
--- checking the algebraic sign.
notcw :: P2 Double -> P2 Double -> P2 Double -> Bool
notcw a b c = case getOrient a b c of
CW -> False
_ -> True
--- |Checks if 3 points a,b,c do build a clockwise triangle by
--- connecting a-b-c. This is done by computing the determinant and
--- checking the algebraic sign.
cw :: P2 Double -> P2 Double -> P2 Double -> Bool
cw a b c = not . notcw a b $ c
-- |Sort X and Y coordinates lexicographically.
sortedXY :: [P2 Double] -> [P2 Double]
sortedXY = fmap p2 . sortLex . fmap unp2
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-- |Sort Y and X coordinates lexicographically.
sortedYX :: [P2 Double] -> [P2 Double]
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sortedYX = fmap p2 . sortLexSwapped . fmap unp2
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-- |Sort all points according to their X-coordinates only.
sortedX :: [P2 Double] -> [P2 Double]
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sortedX xs =
fmap p2
. sortBy (\(a1, _) (a2, _) -> compare a1 a2)
$ fmap unp2 xs
-- |Sort all points according to their Y-coordinates only.
sortedY :: [P2 Double] -> [P2 Double]
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sortedY xs =
fmap p2
. sortBy (\(_, b1) (_, b2) -> compare b1 b2)
$ fmap unp2 xs
-- |Apply a function on the coordinates of a point.
onPT :: ((Double, Double) -> (Double, Double)) -> P2 Double -> P2 Double
onPT f = p2 . f . unp2
-- |Compare the y-coordinate of two points.
ptCmpY :: P2 Double -> P2 Double -> Ordering
ptCmpY (coords -> _ :& y1) (coords -> _ :& y2) =
compare y1 y2
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-- |Compare the x-coordinate of two points.
ptCmpX :: P2 Double -> P2 Double -> Ordering
ptCmpX (coords -> x1 :& _) (coords -> x2 :& _) =
compare x1 x2
posInfPT :: P2 Double
posInfPT = p2 (read "Infinity", read "Infinity")
negInfPT :: P2 Double
negInfPT = p2 (negate . read $ "Infinity", negate . read $ "Infinity")
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-- | Given an infinite line which intersects P1 and P2,
-- let P4 be the point on the line that is closest to P3.
--
-- Return an indication of where on the line P4 is relative to P1 and P2.
--
-- @
-- if P4 == P1 then 0
-- if P4 == P2 then 1
-- if P4 is halfway between P1 and P2 then 0.5
-- @
--
-- @
-- |
-- P1
-- |
-- P4 +---- P3
-- |
-- P2
-- |
-- @
--
{-# INLINE closestPointOnLineParam #-}
closestPointOnLineParam
:: P2 Double -- ^ `P1`
-> P2 Double -- ^ `P2`
-> P2 Double -- ^ `P3`
-> Double
closestPointOnLineParam p1 p2 p3
= pt2Vec (p3 - p1) `scalarProd` pt2Vec (p2 - p1)
/ pt2Vec (p2 - p1) `scalarProd` pt2Vec (p2 - p1)
-- | Given four points specifying two lines, get the point where the two lines
-- cross, if any. Note that the lines extend off to infinity, so the
-- intersection point might not line between either of the two pairs of points.
--
-- @
-- \\ /
-- P1 P4
-- \\ /
-- +
-- / \\
-- P3 P2
-- / \\
-- @
--
intersectLineLine
:: P2 Double -- ^ `P1`
-> P2 Double -- ^ `P2`
-> P2 Double -- ^ `P3`
-> P2 Double -- ^ `P4`
-> Maybe (P2 Double)
intersectLineLine (coords -> x1 :& y1)
(coords -> x2 :& y2)
(coords -> x3 :& y3)
(coords -> x4 :& y4)
= let dx12 = x1 - x2
dx34 = x3 - x4
dy12 = y1 - y2
dy34 = y3 - y4
den = dx12 * dy34 - dy12 * dx34
in if den == 0
then Nothing
else let
det12 = x1*y2 - y1*x2
det34 = x3*y4 - y3*x4
numx = det12 * dx34 - dx12 * det34
numy = det12 * dy34 - dy12 * det34
in Just $ p2 (numx / den, numy / den)
-- | Get the point where a segment @P1-P2@ crosses another segement @P3-P4@,
-- if any.
intersectSegSeg
:: P2 Double -- ^ `P1`
-> P2 Double -- ^ `P2`
-> P2 Double -- ^ `P3`
-> P2 Double -- ^ `P4`
-> Maybe (P2 Double)
intersectSegSeg p1 p2 p3 p4
-- TODO: merge closest point checks with intersection, reuse subterms.
| Just p0 <- intersectLineLine p1 p2 p3 p4
, t12 <- closestPointOnLineParam p1 p2 p0
, t23 <- closestPointOnLineParam p3 p4 p0
, t12 >= 0 && t12 <= 1
, t23 >= 0 && t23 <= 1
= Just p0
| otherwise
= Nothing