{-# LANGUAGE TypeFamilies #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE IncoherentInstances #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE TypeSynonymInstances #-} {-# LANGUAGE BangPatterns #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE NoMonomorphismRestriction #-} module Math.Geometry ( module Math.Geometry ,module Data.VectorSpace ) where import Data.VectorSpace import Data.List (foldl') type Vertex2 a = (a, a) vec2 :: Double -> Double -> Vertex2 Double vec2 x y = (x, y) vx :: VectorSpace (v, v) => (v, v) -> v vx (x, _) = x vy :: VectorSpace (v, v) => (v, v) -> v vy (_, y) = y type Edge2 a = (Vertex2 a, Vertex2 a) type LineSegment = Edge2 Double -- "intersects" is a commutative binary predicate on two shapes. class Intersectable s0 s1 where intersects :: s0 -> s1 -> Bool --instance Intersectable s0 s1 => Intersectable s1 s0 where -- intersects s1 s0 = intersects s0 s1 intersections e es = filter (intersects e) es -- A Boundary describes an inclusive lower bound with a corner point and an exclusive upper bound -- with a size. data Boundary = Boundary { boundary_corner :: Vertex2 Double, boundary_size :: Double } deriving (Show) -- Boundaries b0 and b1 intersect if the min extent of the intersection of b1 with (the plane +x -- including b0.p unioned with the plane +y including b0.p) is within b0. instance Intersectable Boundary Boundary where intersects b0 b1 = let c = union_min_extent_planes_of b0 in if b1 `intersects` c then let (Boundary p _) = intersection b1 c in intersects b0 p else False -- The union of the min extent planes of a boundary is represented by the min extent newtype MinExtentPlanes = MinExtentPlanes (Vertex2 Double) union_min_extent_planes_of (Boundary p _) = MinExtentPlanes p -- A boundary intersects the min extent planes if the far extent extent of the boundary is within -- the range defined by the min extent planes. -- The comparison is > and not >= since the far extent is the point just beyond the boundary. Which -- needs to be just inside the planes in order for the boundary to be inside the planes. instance Intersectable Boundary MinExtentPlanes where intersects b (MinExtentPlanes (min_x, min_y)) = let (_, (b_max_x, b_max_y)) = boundary_extents b in (b_max_x > min_x) && (b_max_y > min_y) intersection :: Boundary -> MinExtentPlanes -> Boundary intersection (Boundary p size) (MinExtentPlanes min_p) = Boundary (ext_max min_p p) size instance Intersectable Boundary LineSegment where intersects b l@(p0, p1) = -- If any point of the line segment is contained in the boundary then the line segment intersects the -- element. intersects b p0 || intersects b p1 -- If niether point is in the element the line segment could still intersect the boundary. The line -- segment must, in this case, intersect an edge of the boundary. || any (intersects l) (boundary_edges b) --The equations for line intersection are pulled from -- http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/ -- without much thought. instance Intersectable LineSegment LineSegment where intersects (p0a, p0b) (p1a, p1b) = let x1 = vx p0a y1 = vy p0a x2 = vx p0b y2 = vy p0b x3 = vx p1a y3 = vy p1a x4 = vx p1b y4 = vy p1b div = (y4 - y3)*(x2 - x1) - (x4 - x3)*(y2 - y1) in if div < 1e-9 then False else let t0n = (x4 - x3)*(y1 - y3) - (y4 - y3)*(x1 - x3) t0 = t0n / div t1n = (x2 - x1)*(y1 - y3) - (y2 - y1)*(x1 - x3) t1 = t1n / div in t0 > 0.0 && t0 < 1.0 && t1 > 0.0 && t1 < 1.0 union_boundaries :: Boundary -> Boundary -> Boundary union_boundaries b0 b1 = let (min0, max0) = boundary_extents b0 (min1, max1) = boundary_extents b1 p = ext_min min0 min1 ext = ext_max max0 max1 (w,h) = ext ^-^ p size = max w h in Boundary p size ext_min (x0,y0) (x1,y1) = (min x0 x1, min y0 y1) ext_max (x0,y0) (x1,y1) = (max x0 x1, max y0 y1) --instance Show Boundary where -- show (Boundary p size) = show p ++ " -> " ++ show size instance Intersectable Boundary (Vertex2 Double) where intersects bounds (px, py) = let (x, y) = boundary_corner bounds s = boundary_size bounds in px < (x + s) && px >= x && py < (y + s) && py >= y class HasBoundary s where boundary_points :: s -> [Vertex2 Double] boundary_edges :: s -> [Edge2 Double] boundary_edges s = let ps@(p0 : ps') = boundary_points s in zip ps (ps' ++ [p0]) boundary_extents :: s -> (Vertex2 Double, Vertex2 Double) boundary_extents s = let (p0 : ps) = boundary_points s initial_min_extent = p0 initial_max_extent = p0 union_extents ((min_x, min_y), (max_x,max_y)) (x, y) = let min_x' = min min_x x min_y' = min min_y y max_x' = max max_x x max_y' = max max_y y in ((min_x', min_y'), (max_x', max_y')) in foldl' union_extents (initial_min_extent, initial_max_extent) ps boundary_square :: s -> Boundary boundary_square s = let (min_extent, max_extent) = boundary_extents s width = fst max_extent - fst min_extent height = snd max_extent - snd min_extent size = max width height in Boundary (fst min_extent, snd min_extent) size -- A boundary cleary has itself as it's boundary. instance HasBoundary Boundary where boundary_points (Boundary p s) = [ p , p ^+^ (0, s) , p ^+^ (s, s) , p ^+^ (s, 0) ] boundary_extents (Boundary p s) = (p, p ^+^ (s,s)) boundary_square b = b