butterfly curve

Temple Fay discovered this complicated curve in 1989. It can be defined either parametrically or as a polar equation; I did it the former way.

One application I thought of for this is object motion in games: I tried it out by writing this little Canvas game, where the comets follow the curve's trajectory. The differences in plot density along the curve create natural-looking comet tails.

	{-# LANGUAGE NoMonomorphismRestriction #-}
	import Data.Bits ((.|.), shiftL)
	import Control.Arrow ((***), (&&&))
	import Graphics.UI.SDL as SDL
	import Graphics.UI.SDL.Primitives (pixel)
	(xres, yres, zz) = (1050, 1050, round *** round)
	main = withInit [InitVideo] $ do
	w <- setVideoMode xres yres 32 [NoFrame]
	enableEvent SDLMouseMotion False
	setCaption "Butterfly Curve" "Butterfly Curve"
	fillRect w (Just $ Rect 0 0 xres yres) $ Pixel 0
	plot w $ center $ scale curve
	loop w []
	where
	scale = map ((150 *) *** (* 150))
	center = map (((xres / 2) +) *** (+ ((yres + 190) / 2)))
	curve = map (f &&& (negate . g)) ts
	where
	ts = [-999, -998.99.. 999]
	f t = sin t * (e ** cos t - 2 * cos (4*t) - h (t / 12))
	g t = cos t * (e ** cos t - 2 * cos (4*t) - h (t / 12))
	h = foldr1 (.) $ replicate 5 sin
	e = exp 1
	plot w = mapM_ (f . zz)
	where
	f (x,y) = pixel w (fromIntegral x) (fromIntegral y) $ Pixel rgb
	where
	rgb = rgb' .|. shiftL (255 `div` (max x y `div` min x y)) 8
	rgb' = rgb'' .|. shiftL (255 `div` (xres `div` x)) 16
	rgb'' = 0xFF .|. shiftL (255 `div` (yres `div` y)) 24
	loop w ps = do
	delay 128
	event <- pollEvent
	case event of
	KeyUp (Keysym SDLK_ESCAPE _ _) -> return ()
	_ -> loop w ps

view raw fay_butterfly_curve.hs hosted with ❤ by GitHub