cga/Algorithms/PolygonIntersection.hs

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module Algorithms.PolygonIntersection where
import Algebra.Vector
import Control.Applicative
import Data.Dequeue (BankersDequeue)
import qualified Data.Dequeue as Q
import Data.List
import Data.Maybe
import Diagrams.TwoD.Types
import MyPrelude
import QueueEx
-- TODO: probably use a zipper.
-- |Describes a point on the convex hull of the polygon.
-- In addition to the point itself, both it's predecessor and
-- successor are saved for convenience.
data PolyPT =
PolyA {
id' :: P2 Double
, pre :: P2 Double
, suc :: P2 Double
}
| PolyB {
id' :: P2 Double
, pre :: P2 Double
, suc :: P2 Double
}
deriving (Show, Eq)
isPolyA :: PolyPT -> Bool
isPolyA PolyA {} = True
isPolyA _ = False
isPolyB :: PolyPT -> Bool
isPolyB = not . isPolyA
-- |Shift a list of sorted convex hull points of a polygon so that
-- the first element in the list is the one with the highest y-coordinate.
-- This is done in O(n).
sortLexPoly :: [P2 Double] -> [P2 Double]
sortLexPoly ps = maybe [] (`shiftM` ps) (elemIndex (yMax ps) ps)
where
yMax = foldl1 (\x y -> if ptCmpY x y == GT then x else y)
-- |Make a PolyPT list out of a regular list of points, so
-- the predecessor and successors are all saved.
mkPolyPTList :: (P2 Double -> P2 Double -> P2 Double -> PolyPT) -- ^ PolyA or PolyB function
-> [P2 Double] -- ^ polygon points
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-> [PolyPT]
mkPolyPTList f' pts@(x':y':_:_) =
f' x' (last pts) y' : go f' pts
where
go f (x:y:z:xs) = f y x z : go f (y:z:xs)
go f [x, y] = [f y x x']
go _ _ = []
mkPolyPTList _ _ = []
-- |Sort the points of two polygons according to their y-coordinates,
-- while saving the origin of that point. This is done in O(n).
sortLexPolys :: ([P2 Double], [P2 Double]) -> [PolyPT]
sortLexPolys (pA'@(_:_), pB'@(_:_)) =
queueToList $ go (Q.fromList . mkPolyPTList PolyA . sortLexPoly $ pA')
(Q.fromList . mkPolyPTList PolyB . sortLexPoly $ pB')
where
-- Start recursive algorithm, each polygon is represented by a Queue.
-- Traverse predecessor and successor and insert them in the right
-- order into the resulting queue.
-- We start at the max y-coordinates of both polygons.
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go :: BankersDequeue PolyPT -- polyA
-> BankersDequeue PolyPT -- polyB
-> BankersDequeue PolyPT -- sorted queue
go pA pB
-- Nothing to sort.
| Q.null pA && Q.null pB = Q.empty
-- Current point of polygon A is higher on the y-axis than the
-- current point of polygon B, so insert it into the resulting
-- queue and traverse the rest.
| ptCmpY (fromMaybe negInfPT (id' <$> Q.first pA))
(fromMaybe negInfPT (id' <$> Q.first pB)) == GT
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= Q.pushFront (go (maybeShift . snd . fromJust . Q.popFront $ pA) pB)
(fromJust . Q.first $ pA)
-- Same as above, except that the current point of polygon B
-- is higher.
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| otherwise = Q.pushFront (go pA (maybeShift . snd . fromJust . Q.popFront $ pB))
(fromJust . Q.first $ pB)
-- Compare the first and the last element of the queue according
-- to their y-coordinate and shift the queue (if necessary) so that
-- the element with the highest value is at the front.
maybeShift :: BankersDequeue PolyPT -> BankersDequeue PolyPT
maybeShift q = if ptCmpY (fromMaybe posInfPT (id' <$> Q.first q))
(fromMaybe negInfPT (id' <$> Q.last q)) == GT
then q
else shiftQueueRight q
sortLexPolys _ = []
-- |Get all points that intersect between both polygons. This is done
-- in O(n).
intersectionPoints :: [PolyPT] -> [P2 Double]
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intersectionPoints xs' = rmdups . go $ xs'
where
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go [] = []
go xs = (++) (segIntersections . scanLine $ xs)
(go (tail xs))
-- Get the scan line or in other words the
-- Segment pairs we are going to check for intersection.
scanLine :: [PolyPT] -> ([(P2 Double, P2 Double)], [(P2 Double, P2 Double)])
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scanLine sp@(_:_) = (,) (getSegment isPolyA) (getSegment isPolyB)
where
getSegment f = fromMaybe []
((\x -> [(id' x, suc x), (id' x, pre x)])
<$> (listToMaybe . filter f $ sp))
scanLine _ = ([], [])
-- Gets the actual intersections between the segments of
-- both polygons we currently examine. This is done in O(1)
-- since we have max 4 segments.
segIntersections :: ([(P2 Double, P2 Double)], [(P2 Double, P2 Double)]) -> [P2 Double]
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segIntersections (a@(_:_), b@(_:_)) =
catMaybes
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. fmap (\[(x1, y1), (x2, y2)] -> intersectSegSeg x1 y1 x2 y2)
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$ combinations a b
segIntersections _ = []
-- Gets all unique(!) combinations of two arrays. Both arrays
-- are max 2, so this is actually O(1) for this algorithm.
combinations :: [a] -> [a] -> [[a]]
combinations xs ys = concat . fmap (\y -> fmap (\x -> [y, x]) xs) $ ys