convex hulls

This code demonstrates the Graham scan, an O(n log n) method for finding the convex hull (smallest enclosing polygon) of a planar point set. It's a great example of exploiting order: it works by sorting the set by angle about a known extreme point, allowing the hull points to be found in linear time.

A variation on the algorithm, noted by A.M. Andrew, sorts the set lexicographically and finds the upper and lower hull chains separately. This 'monotone chain' version is often preferred, since it's easier to do robustly.

	{-# LANGUAGE NoMonomorphismRestriction #-}
	import Control.Arrow ((***))
	import Data.List (maximumBy, delete, sort, sortBy, unfoldr)
	import Data.Ord (comparing)
	import Graphics.UI.SDL as SDL
	import Graphics.UI.SDL.Primitives (filledCircle, line)
	import System.Random.Mersenne.Pure64 (newPureMT, randomDouble)
	res = 250
	main = withInit [InitVideo] $ do
	w <- setVideoMode res res 32 []
	ps <- randPoints
	enableEvent SDLMouseMotion False
	setCaption "Graham Scan" "Graham Scan"
	fillRect w (Just $ Rect 0 0 res res) $ Pixel 0
	limn w $ map (round *** round) $ hull ps
	plot w ps
	pause
	plot w ps = do
	mapM_ (f . (round *** round)) ps
	SDL.flip w
	where
	f (x,y) = filledCircle w x y 1 $ Pixel 0xFFFFFFFF
	limn w ps = f $ ps ++ [head ps]
	where
	f [_] = return ()
	f ((a,b):(x,y):ps) = do
	line w a b x y $ Pixel 0xFF0000FF
	f $ (x,y) : ps
	pause = do
	delay 128
	e <- pollEvent
	case e of
	KeyUp (Keysym SDLK_ESCAPE _ _) -> return ()
	_ -> pause
	hull qs = go (drop 2 ps) $ reverse $ take 2 ps
	where
	o = bottomRightP qs
	ps = o : sortBy (ccw o) (delete o qs)
	go [] qs = qs
	go (p:ps) s@(a:b:qs)
	| ccw a b p /= GT = go (p:ps) $ b:qs
	| otherwise = go ps $ p:s
	ccw (ax, ay) (bx, by) (cx, cy)
	| d < 0 = LT
	| d > 0 = GT
	| True = EQ
	where
	d = (bx - ax) * (cy - ay) - (by - ay) * (cx - ax)
	bottomRightP = maximumBy (comparing snd) . sort
	randPoints = fmap f newPureMT
	where
	f = uncurry zip . splitAt 20 . g
	g = map (* res) . unfoldr (Just . randomDouble)

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