{-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE TemplateHaskell #-} -------------------------------------------------------------------------------- -- | -- Module : Data.Geometry.Polygon.Convex -- Copyright : (C) Frank Staals -- License : see the LICENSE file -- Maintainer : Frank Staals -- -- Convex Polygons -- -------------------------------------------------------------------------------- module Data.Geometry.Polygon.Convex( ConvexPolygon(..), simplePolygon , merge , lowerTangent, lowerTangent' , upperTangent, upperTangent' , extremes , maxInDirection , leftTangent, rightTangent , minkowskiSum , bottomMost ) where import Control.DeepSeq import Control.Lens hiding ((:<), (:>)) import Data.CircularSeq (CSeq) import qualified Data.CircularSeq as C import Data.Ext import qualified Data.Foldable as F import Data.Function (on) import Data.Geometry.Box (IsBoxable(..)) import Data.Geometry.LineSegment import Data.Geometry.Point import Data.Geometry.Polygon.Core (fromPoints, SimplePolygon, outerBoundary) import Data.Geometry.Polygon.Extremes(cmpExtreme) import Data.Geometry.Properties import Data.Geometry.Transformation import Data.Geometry.Vector import Data.List.NonEmpty (NonEmpty(..)) import qualified Data.List.NonEmpty as NonEmpty import Data.Maybe (fromJust) import Data.Ord (comparing) import Data.Semigroup.Foldable (Foldable1(..)) import Data.Sequence (viewl,viewr, ViewL(..), ViewR(..)) import qualified Data.Sequence as S import Data.Util -- import Data.Geometry.Ipe -- import Debug.Trace -------------------------------------------------------------------------------- -- | Data Type representing a convex polygon newtype ConvexPolygon p r = ConvexPolygon {_simplePolygon :: SimplePolygon p r } deriving (Show,Eq,NFData) makeLenses ''ConvexPolygon instance PointFunctor (ConvexPolygon p) where pmap f (ConvexPolygon p) = ConvexPolygon $ pmap f p -- | Polygons are per definition 2 dimensional type instance Dimension (ConvexPolygon p r) = 2 type instance NumType (ConvexPolygon p r) = r instance Fractional r => IsTransformable (ConvexPolygon p r) where transformBy = transformPointFunctor instance IsBoxable (ConvexPolygon p r) where boundingBox = boundingBox . _simplePolygon -- convexPolygon :: SimplePolygon p r -> Maybe (ConvexPolygon p r) -- convexPolygon p = if isConvex p then Just p else Nothing -- isConvex :: SimplePolygon p r -> Bool -- isConvex p = let ch = convexHull $ p^.vertices -- in p^.vertices.size == ch^.simplePolygon.vertices.size -- mainWith inFile outFile = do -- ePage <- readSinglePageFile inFile -- case ePage of -- Left err -> error "" -- err -- Right (page :: IpePage Rational) -> case page^..content.traverse._withAttrs _IpePath _asSimplePolygon.core of -- [] -> error "No points found" -- polies@(_:_) -> do -- -- let out = [asIpe drawTriangulation dt, asIpe drawTree' emst] -- -- print $ length $ edges' dt -- -- print $ toPlaneGraph (Proxy :: Proxy DT) dt -- -- writeIpeFile outFile . singlePageFromContent $ out -- -- mapM_ (print . extremesNaive (v2 1 0)) polies -- pure $ map (flip rightTangent (Point2 80 528)) polies -- | Finds the extreme points, minimum and maximum, in a given direction -- -- pre: The input polygon is strictly convex. -- -- running time: \(O(\log n)\) -- -- extremes :: (Num r, Ord r) => Vector 2 r -> ConvexPolygon p r -> (Point 2 r :+ p, Point 2 r :+ p) extremes u p = (maxInDirection ((-1) *^ u) p, maxInDirection u p) -- | Finds the extreme maximum point in the given direction. Based on -- http://geomalgorithms.com/a14-_extreme_pts.html -- -- -- pre: The input polygon is strictly convex. -- -- running time: \(O(\log^2 n)\) maxInDirection :: (Num r, Ord r) => Vector 2 r -> ConvexPolygon p r -> Point 2 r :+ p maxInDirection u = findMaxWith (cmpExtreme u) findMaxWith :: (Point 2 r :+ p -> Point 2 r :+ p -> Ordering) -> ConvexPolygon p r -> Point 2 r :+ p findMaxWith cmp p = findMaxStart . C.rightElements . getVertices $ p where p' >=. q = (p' `cmp` q) /= LT findMaxStart s@(viewl -> (a: a:=. c) of (True,False,_) -> findMax (ac |> c) (True,True,True) -> findMax (ac |> c) (True,True,False) -> findMax (c <| cb) (False,True,_) -> findMax (c <| cb) (False,False,False) -> findMax (ac |> c) (False,False,True) -> findMax (c <| cb) binSearch _ _ _ = error "maxInDirection, binSearch: empty chain" isLocalMax (viewr -> _ :> l) c (viewl -> r :< _) = c >=. l && c >=. r isLocalMax (viewr -> _ :> l) c _ = c >=. l isLocalMax _ c (viewl -> r :< _) = c >=. r isLocalMax _ _ _ = True -- the Edge from a to b is upwards w.r.t b if a is not larger than b isUpwards a (viewl -> b :< _) = (a `cmp` b) /= GT isUpwards _ _ = error "isUpwards: no edge endpoint" tangentCmp :: (Num r, Ord r) => Point 2 r -> Point 2 r :+ p -> Point 2 r :+ q -> Ordering tangentCmp o p q = case ccw o (p^.core) (q^.core) of CCW -> LT -- q is left of the line from o to p CoLinear -> EQ -- q is *on* the line from o to p CW -> GT -- q is right of the line from o to p -- | Given a convex polygon poly, and a point outside the polygon, find the -- left tangent of q and the polygon, i.e. the vertex v of the convex polygon -- s.t. the polygon lies completely to the right of the line from q to v. -- -- running time: \(O(\log^2 n)\). leftTangent :: (Ord r, Num r) => ConvexPolygon p r -> Point 2 r -> Point 2 r :+ p leftTangent poly q = findMaxWith (tangentCmp q) poly -- | Given a convex polygon poly, and a point outside the polygon, find the -- right tangent of q and the polygon, i.e. the vertex v of the convex polygon -- s.t. the polygon lies completely to the left of the line from q to v. -- -- running time: \(O(\log^2 n)\). rightTangent :: (Ord r, Num r) => ConvexPolygon p r -> Point 2 r -> Point 2 r :+ p rightTangent poly q = findMaxWith (flip $ tangentCmp q) poly -- * Merging Two convex Hulls -- | Rotating Right <-> rotate clockwise -- -- Merging two convex hulls, based on the paper: -- -- Two Algorithms for Constructing a Delaunay Triangulation -- Lee and Schachter -- International Journal of Computer and Information Sciences, Vol 9, No. 3, 1980 -- -- : (combined hull, lower tangent that was added, upper tangent thtat was -- added) -- pre: - lp and rp are disjoint, and there is a vertical line separating -- the two polygons. -- - The vertices of the polygons are given in clockwise order -- -- Running time: O(n+m), where n and m are the sizes of the two polygons respectively merge :: (Num r, Ord r) => ConvexPolygon p r -> ConvexPolygon p r -> (ConvexPolygon p r, LineSegment 2 p r, LineSegment 2 p r) merge lp rp = (ConvexPolygon . fromPoints $ r' ++ l', lt, ut) where lt@(ClosedLineSegment a b) = lowerTangent lp rp ut@(ClosedLineSegment c d) = upperTangent lp rp takeUntil p xs = let (xs',x:_) = break p xs in xs' ++ [x] rightElems = F.toList . C.rightElements takeAndRotate x y = takeUntil (coreEq x) . rightElems . rotateTo' y . getVertices r' = takeAndRotate b d rp l' = takeAndRotate c a lp rotateTo' :: Eq a => (a :+ b) -> CSeq (a :+ b) -> CSeq (a :+ b) rotateTo' x = fromJust . C.findRotateTo (coreEq x) coreEq :: Eq a => (a :+ b) -> (a :+ b) -> Bool coreEq = (==) `on` (^.core) -------------------------------------------------------------------------------- -- * Computing Tangents -- | Compute the lower tangent of the two polgyons -- -- pre: - polygons lp and rp have at least 1 vertex -- - lp and rp are disjoint, and there is a vertical line separating -- the two polygons. -- - The vertices of the polygons are given in clockwise order -- -- Running time: O(n+m), where n and m are the sizes of the two polygons respectively lowerTangent :: (Num r, Ord r) => ConvexPolygon p r -> ConvexPolygon p r -> LineSegment 2 p r lowerTangent lp rp = ClosedLineSegment l r where mkH f = NonEmpty.fromList . F.toList . f . getVertices lh = mkH (C.rightElements . rightMost) lp rh = mkH (C.leftElements . leftMost) rp (Two (l :+ _) (r :+ _)) = lowerTangent' lh rh -- | Compute the lower tangent of the two convex chains lp and rp -- -- pre: - the chains lp and rp have at least 1 vertex -- - lp and rp are disjoint, and there is a vertical line -- having lp on the left and rp on the right. -- - The vertices in the left-chain are given in clockwise order, (right to left) -- - The vertices in the right chain are given in counterclockwise order (left-to-right) -- -- The result returned is the two endpoints l and r of the tangents, -- and the remainders lc and rc of the chains (i.e.) such that the lower hull -- of both chains is: (reverse lc) ++ [l,h] ++ rc -- -- Running time: \(O(n+m)\), where n and m are the sizes of the two chains -- respectively lowerTangent' :: (Ord r, Num r, Foldable1 f) => f (Point 2 r :+ p) -> f (Point 2 r :+ p) -> Two ((Point 2 r :+ p) :+ [Point 2 r :+ p]) lowerTangent' l0 r0 = go (toNonEmpty l0) (toNonEmpty r0) where ne = NonEmpty.fromList isRight' [] _ _ = False isRight' (x:_) l r = ccw' l r x /= CCW go lh@(l:|ls) rh@(r:|rs) | isRight' rs l r = go lh (ne rs) | isRight' ls l r = go (ne ls) rh | otherwise = Two (l :+ ls) (r :+ rs) -- | Compute the upper tangent of the two polgyons -- -- pre: - polygons lp and rp have at least 1 vertex -- - lp and rp are disjoint, and there is a vertical line separating -- the two polygons. -- - The vertices of the polygons are given in clockwise order -- -- Running time: O(n+m), where n and m are the sizes of the two polygons respectively upperTangent :: (Num r, Ord r) => ConvexPolygon p r -> ConvexPolygon p r -> LineSegment 2 p r upperTangent lp rp = ClosedLineSegment l r where mkH f = NonEmpty.fromList . F.toList . f . getVertices lh = mkH (C.leftElements . rightMost) lp rh = mkH (C.rightElements . leftMost) rp (Two (l :+ _) (r :+ _)) = upperTangent' lh rh -- | Compute the upper tangent of the two convex chains lp and rp -- -- pre: - the chains lp and rp have at least 1 vertex -- - lp and rp are disjoint, and there is a vertical line -- having lp on the left and rp on the right. -- - The vertices in the left-chain are given in clockwise order, (right to left) -- - The vertices in the right chain are given in counterclockwise order (left-to-right) -- -- The result returned is the two endpoints l and r of the tangents, -- and the remainders lc and rc of the chains (i.e.) such that the upper hull -- of both chains is: (reverse lc) ++ [l,h] ++ rc -- -- Running time: \(O(n+m)\), where n and m are the sizes of the two chains -- respectively upperTangent' :: (Ord r, Num r, Foldable1 f) => f (Point 2 r :+ p) -> f (Point 2 r :+ p) -> Two ((Point 2 r :+ p) :+ [Point 2 r :+ p]) upperTangent' l0 r0 = go (toNonEmpty l0) (toNonEmpty r0) where ne = NonEmpty.fromList isLeft' [] _ _ = False isLeft' (x:_) l r = ccw' l r x /= CW go lh@(l:|ls) rh@(r:|rs) | isLeft' rs l r = go lh (ne rs) | isLeft' ls l r = go (ne ls) rh | otherwise = Two (l :+ ls) (r :+ rs) -------------------------------------------------------------------------------- -- | Computes the Minkowski sum of the two input polygons with $n$ and $m$ -- vertices respectively. -- -- pre: input polygons are in CCW order. -- -- running time: \(O(n+m)\). minkowskiSum :: (Ord r, Num r) => ConvexPolygon p r -> ConvexPolygon q r -> ConvexPolygon (p,q) r minkowskiSum p q = ConvexPolygon . fromPoints $ merge' (f p) (f q) where f p' = let xs@(S.viewl -> (v :< _)) = C.asSeq . bottomMost . getVertices $ p' in F.toList $ xs |> v (v :+ ve) .+. (w :+ we) = v .+^ (toVec w) :+ (ve,we) cmpAngle v v' w w' = ccwCmpAround (ext $ origin) (ext . Point $ v' .-. v) (ext . Point $ w' .-. w) merge' [_] [_] = [] merge' vs@[v] (w:ws) = v .+. w : merge' vs ws merge' (v:vs) ws@[w] = v .+. w : merge' vs ws merge' (v:v':vs) (w:w':ws) = v .+. w : case cmpAngle (v^.core) (v'^.core) (w^.core) (w'^.core) of LT -> merge' (v':vs) (w:w':ws) GT -> merge' (v:v':vs) (w':ws) EQ -> merge' (v':vs) (w':ws) merge' _ _ = error $ "minkowskiSum: Should not happen" -------------------------------------------------------------------------------- -- | Rotate to the rightmost point (rightmost and topmost in case of ties) rightMost :: Ord r => CSeq (Point 2 r :+ p) -> CSeq (Point 2 r :+ p) rightMost xs = let m = F.maximumBy (comparing (^.core)) xs in rotateTo' m xs -- | Rotate to the leftmost point (and bottommost in case of ties) leftMost :: Ord r => CSeq (Point 2 r :+ p) -> CSeq (Point 2 r :+ p) leftMost xs = let m = F.minimumBy (comparing (^.core)) xs in rotateTo' m xs -- | Rotate to the bottommost point (and leftmost in case of ties) bottomMost :: Ord r => CSeq (Point 2 r :+ p) -> CSeq (Point 2 r :+ p) bottomMost xs = let f p = (p^.core.yCoord,p^.core.xCoord) m = F.minimumBy (comparing f) xs in rotateTo' m xs -- | Helper to get the vertices of a convex polygon getVertices :: ConvexPolygon p r -> CSeq (Point 2 r :+ p) getVertices = view (simplePolygon.outerBoundary) -- -- | rotate right while p 'current' 'rightNeibhour' is true -- rotateRWhile :: (a -> a -> Bool) -> C.CList a -> C.CList a -- rotateRWhile p lst -- | C.isEmpty lst = lst -- | otherwise = go lst -- where -- go xs = let cur = focus xs -- xs' = C.rotR xs -- nxt = focus' xs' -- in if p cur nxt then go xs' else xs -- test1 :: Num r => ConvexPolygon () r -- test1 = ConvexPolygon . fromPoints . map ext . reverse $ [origin, Point2 1 4, Point2 5 6, Point2 10 3] -- test2 :: Num r => ConvexPolygon () r -- test2 = ConvexPolygon . fromPoints . map ext . reverse $ [Point2 11 6, Point2 10 10, Point2 15 18, Point2 12 5] -- testA :: Num r => ConvexPolygon () r -- testA = ConvexPolygon . fromPoints . map ext $ [origin, Point2 5 1, Point2 2 2] -- testB :: Num r => ConvexPolygon () r -- testB = ConvexPolygon . fromPoints . map ext $ [origin, Point2 5 3, Point2 (-2) 2, Point2 (-2) 1]