MarchingSquares.hs

-- Implicit CAD. Copyright (C) 2011, Christopher Olah (chris@colah.ca)
-- Released under the GNU GPL, see LICENSE

module Graphics.Implicit.Export.MarchingSquares (getContour) where

import Graphics.Implicit.Definitions
import Control.Parallel.Strategies (using, parList, rdeepseq)
import Debug.Trace

-- | getContour gets a polyline describe the edge of your 2D
--  object. It's really the only function in this file you need
--  to care about from an external perspective.

getContour :: ℝ2 -> ℝ2 -> ℝ2 -> Obj2 -> [Polyline]
getContour (x1, y1) (x2, y2) (dx, dy) obj = 
	let
		-- How many steps will we take on each axis?
		nx = fromIntegral $ ceiling $ (x2 - x1) / dx
		ny = fromIntegral $ ceiling $ (y2 - y1) / dy
		-- Divide it up and compute the polylines
		linesOnGrid :: [[[Polyline]]]
		linesOnGrid = [[getSquareLineSegs 
		           (x1 + (x2 - x1)*mx/nx,     y1 + (y2 - y1)*my/ny)
		           (x1 + (x2 - x1)*(mx+1)/nx, y1 + (y2 - y1)*(my+1)/ny)
		           obj
		     | mx <- [0.. nx-1] ] | my <- [0..ny-1] ]
		-- Cleanup, cleanup, everybody cleanup!
		-- (We connect multilines, delete redundant vertices on them, etc)
		multilines = (filter polylineNotNull) $ (map reducePolyline) $ orderLinesDC $ linesOnGrid
	in
		multilines

getContour2 :: ℝ2 -> ℝ2 -> ℝ2 -> Obj2 -> [Polyline]
getContour2 (x1, y1) (x2, y2) (dx, dy) obj = 
	let
		-- How many steps will we take on each axis?
		nx = fromIntegral $ ceiling $ (x2 - x1) / dx
		ny = fromIntegral $ ceiling $ (y2 - y1) / dy
		-- Grid mapping funcs
		fromGrid (mx, my) = (x1 + (x2 - x1)*mx/nx, y1 + (y2 - y1)*my/ny)
		toGrid (x,y) =(\a-> traceShow a a) (floor $ nx*(x-x1)/(x2-x1), floor $ ny*(y-y1)/(y2-y1) ) :: (ℕ, ℕ)
		-- Evalueate obj on a grid, in parallel.
		valsOnGrid :: [[ℝ]]
		valsOnGrid = [[ obj (fromGrid (mx, my)) | mx <- [0.. nx-1] ] | my <- [0..ny-1] ]
		              `using` parList rdeepseq
		-- A faster version of the obj. Sort of like memoization, but done in advance, in parallel.
		preEvaledObj p = valsOnGrid !! my !! mx where (mx,my) = toGrid p
		-- Divide it up and compute the polylines
		linesOnGrid :: [[[Polyline]]]
		linesOnGrid = [[getSquareLineSegs (fromGrid (mx, my)) (fromGrid (mx+1, my+1)) preEvaledObj
		     | mx <- [0.. nx-1] ] | my <- [0..ny-1] ]
		-- Cleanup, cleanup, everybody cleanup!
		-- (We connect multilines, delete redundant vertices on them, etc)
		multilines = (filter polylineNotNull) $ (map reducePolyline) $ orderLinesDC $ linesOnGrid
	in
		multilines
		

-- | This function gives line segmensts to divde negative interior
--  regions and positive exterior ones inside a square, based on its 
--  values at its vertices.
--  It is based on the linearly-interpolated marching squares algorithm.

getSquareLineSegs :: ℝ2 -> ℝ2 -> Obj2 -> [Polyline]
getSquareLineSegs (x1, y1) (x2, y2) obj = 
	let 
		(x,y) = (x1, y1)

		-- Let's evlauate obj at a few points...
		x1y1 = obj (x1, y1)
		x2y1 = obj (x2, y1)
		x1y2 = obj (x1, y2)
		x2y2 = obj (x2, y2)
		c = obj ((x1+x2)/2, (y1+y2)/2)

		dx = x2 - x1
		dy = y2 - y1

		-- linearly interpolated midpoints on the relevant axis
		--             midy2
		--      _________*__________
		--     |                    |
		--     |                    |
		--     |                    |
		--midx1*                    * midx2
		--     |                    |
		--     |                    |
		--     |                    |
		--     -----------*----------
		--              midy1

		midx1 = (x,                       y + dy*x1y1/(x1y1-x1y2))
		midx2 = (x + dx,                  y + dy*x2y1/(x2y1-x2y2))
		midy1 = (x + dx*x1y1/(x1y1-x2y1), y )
		midy2 = (x + dx*x1y2/(x1y2-x2y2), y + dy)
		notPointLine (p1:p2:[]) = p1 /= p2
	in filter (notPointLine) $ case (x1y2 <= 0, x2y2 <= 0,
	                                 x1y1 <= 0, x2y1 <= 0) of
		-- Yes, there's some symetries that could reduce the amount of code...
		-- But I don't think they're worth exploiting...
		(True,  True, 
		 True,  True)  -> []
		(False, False,
		 False, False) -> []
		(True,  True, 
		 False, False) -> [[midx1, midx2]]
		(False, False,
		 True,  True)  -> [[midx1, midx2]]
		(False, True, 
		 False, True)  -> [[midy1, midy2]]
		(True,  False,
		 True,  False) -> [[midy1, midy2]]
		(True,  False,
		 False, False) -> [[midx1, midy2]]
		(False, True, 
		 True,  True)  -> [[midx1, midy2]]
		(True,  True, 
		 False, True)  -> [[midx1, midy1]]
		(False, False,
		 True,  False) -> [[midx1, midy1]]
		(True,  True, 
		 True,  False) -> [[midx2, midy1]]
		(False, False,
		 False, True)  -> [[midx2, midy1]]
		(True,  False,
		 True,  True)  -> [[midx2, midy2]]
		(False, True, 
		 False, False) -> [[midx2, midy2]]
		(True,  False,
		 False, True)  -> if c > 0
			then [[midx1, midy2], [midx2, midy1]]
			else [[midx1, midy1], [midx2, midy2]]
		(False, True, 
		 True,  False) -> if c <= 0
			then [[midx1, midy2], [midx2, midy1]]
			else [[midx1, midy1], [midx2, midy2]]



-- $ Functions for cleaning up the polylines
-- Many have multiple implementations as efficiency experiments.
-- At some point, we'll get rid of the redundant ones....


orderLines :: [Polyline] -> [Polyline]
orderLines [] = []
orderLines (present:remaining) =
	let
		findNext ((p3:ps):segs) = if p3 == last present then (Just (p3:ps), segs) else
			if last ps == last present then (Just (reverse $ p3:ps), segs) else
			case findNext segs of (res1,res2) -> (res1,(p3:ps):res2)
		findNext [] = (Nothing, [])
	in
		case findNext remaining of
			(Nothing, _) -> present:(orderLines remaining)
			(Just match, others) -> orderLines $ (present ++ tail match): others

reducePolyline ((x1,y1):(x2,y2):(x3,y3):others) = 
	if (x1,y1) == (x2,y2) then reducePolyline ((x2,y2):(x3,y3):others) else
	if abs ( (y2-y1)/(x2-x1) - (y3-y1)/(x3-x1) ) < 0.0001 
	   || ( (x2-x1) == 0 && (x3-x1) == 0 && (y2-y1)*(y3-y1) > 0)
	then reducePolyline ((x1,y1):(x3,y3):others)
	else (x1,y1) : reducePolyline ((x2,y2):(x3,y3):others)
reducePolyline ((x1,y1):(x2,y2):others) = 
	if (x1,y1) == (x2,y2) then reducePolyline ((x2,y2):others) else (x1,y1):(x2,y2):others
reducePolyline l = l


orderLinesDC :: [[[Polyline]]] -> [Polyline]
orderLinesDC segs =
	let
		halve :: [a] -> ([a], [a])
		halve l = splitAt (div (length l) 2) l
		splitOrder segs = case (\(x,y) -> (halve x, halve y)) . unzip . map (halve) $ segs of
			((a,b),(c,d)) -> orderLinesDC a ++ orderLinesDC b ++ orderLinesDC c ++ orderLinesDC d
	in
		if (length segs < 5 || length (head segs) < 5 ) then concat $ concat segs else
		case (\(x,y) -> (halve x, halve y)) $ unzip $ map (halve) segs of
			((a,b),(c,d)) ->orderLines $ 
				orderLinesDC a ++ orderLinesDC b ++ orderLinesDC c ++ orderLinesDC d

{-
orderLinesP :: [[[Polyline]]] -> [Polyline]
orderLinesP segs =
	let
		halve l = splitAt (div (length l) 2) l
		splitOrder segs = case (\(x,y) -> (halve x, halve y)) $ unzip $ map (halve) segs of
			((a,b),(c,d)) -> orderLinesDC a ++ orderLinesDC b ++ orderLinesDC c ++ orderLinesDC d
		-- force is frome real world haskell
		force xs = go xs `pseq` ()
		    where go (_:xs) = go xs
		          go [] = 1
	in
		if (length segs < 5 || length (head segs) < 5 ) then concat $ concat segs else
		case (\(x,y) -> (halve x, halve y)) $ unzip $ map (halve) segs of
			((a,b),(c,d)) -> orderLines $ 
				let
					a' = orderLinesP a
					b' = orderLinesP b
					c' = orderLinesP c
					d' = orderLinesP d
				in (force a' `par` force b' `par` force c' `par` force d') `pseq` 
					(a' ++ b' ++ c' ++ d')
-}


polylineNotNull (a:l) = not (null l)
polylineNotNull [] = False