-------------------------------------------------------------------------------- -- | -- Module : Data.Geometry.Polygon -- Copyright : (C) Frank Staals -- License : see the LICENSE file -- Maintainer : Frank Staals -- -- A Polygon data type and some basic functions to interact with them. -- -------------------------------------------------------------------------------- module Data.Geometry.Polygon where import Algorithms.Geometry.LinearProgramming.LP2DRIC import Algorithms.Geometry.LinearProgramming.Types import Control.DeepSeq import Control.Lens hiding (Simple) import Control.Monad.Random.Class import Data.Bifoldable import Data.Bifunctor import Data.Bitraversable import qualified Data.CircularSeq as C import Data.Ext import qualified Data.Foldable as F import Data.Geometry.Boundary import Data.Geometry.Box import Data.Geometry.Line import Data.Geometry.HalfSpace(rightOf) import Data.Geometry.LineSegment import Data.Geometry.Point import Data.Geometry.Properties import Data.Geometry.Transformation import Data.Geometry.Triangle (Triangle(..), inTriangle) import Data.Geometry.Vector import qualified Data.List as List import Data.List.NonEmpty (NonEmpty(..)) import qualified Data.List.NonEmpty as NonEmpty import Data.Maybe (mapMaybe, catMaybes) import Data.Ord (comparing) import Data.Semigroup (sconcat) import Data.Semigroup.Foldable import qualified Data.Sequence as Seq import Data.Util import Data.Vinyl.CoRec (asA) -------------------------------------------------------------------------------- -- * Polygons {- $setup >>> :{ -- import qualified Data.CircularSeq as C let simplePoly :: SimplePolygon () Rational simplePoly = SimplePolygon . C.fromList . map ext $ [ point2 0 0 , point2 10 0 , point2 10 10 , point2 5 15 , point2 1 11 ] :} -} -- | We distinguish between simple polygons (without holes) and Polygons with holes. data PolygonType = Simple | Multi data Polygon (t :: PolygonType) p r where SimplePolygon :: C.CSeq (Point 2 r :+ p) -> Polygon Simple p r MultiPolygon :: C.CSeq (Point 2 r :+ p) -> [Polygon Simple p r] -> Polygon Multi p r instance Bifunctor (Polygon t) where bimap = bimapDefault instance Bifoldable (Polygon t) where bifoldMap = bifoldMapDefault instance Bitraversable (Polygon t) where bitraverse f g p = case p of SimplePolygon vs -> SimplePolygon <$> bitraverseVertices f g vs MultiPolygon vs hs -> MultiPolygon <$> bitraverseVertices f g vs <*> traverse (bitraverse f g) hs instance (NFData p, NFData r) => NFData (Polygon t p r) where rnf (SimplePolygon vs) = rnf vs rnf (MultiPolygon vs hs) = rnf (vs,hs) bitraverseVertices :: (Applicative f, Traversable t) => (p -> f q) -> (r -> f s) -> t (Point 2 r :+ p) -> f (t (Point 2 s :+ q)) bitraverseVertices f g = traverse (bitraverse (traverse g) f) type SimplePolygon = Polygon Simple type MultiPolygon = Polygon Multi -- | Either a simple or multipolygon type SomePolygon p r = Either (Polygon Simple p r) (Polygon Multi p r) type instance Dimension (SomePolygon p r) = 2 type instance NumType (SomePolygon p r) = r -- | Polygons are per definition 2 dimensional type instance Dimension (Polygon t p r) = 2 type instance NumType (Polygon t p r) = r instance (Show p, Show r) => Show (Polygon t p r) where show (SimplePolygon vs) = "SimplePolygon " <> show vs show (MultiPolygon vs hs) = "MultiPolygon " <> show vs <> " " <> show hs instance (Eq p, Eq r) => Eq (Polygon t p r) where (SimplePolygon vs) == (SimplePolygon vs') = vs == vs' (MultiPolygon vs hs) == (MultiPolygon vs' hs') = vs == vs' && hs == hs' instance PointFunctor (Polygon t p) where pmap f (SimplePolygon vs) = SimplePolygon (fmap (first f) vs) pmap f (MultiPolygon vs hs) = MultiPolygon (fmap (first f) vs) (map (pmap f) hs) instance Fractional r => IsTransformable (Polygon t p r) where transformBy = transformPointFunctor instance IsBoxable (Polygon t p r) where boundingBox = boundingBoxList' . toListOf (outerBoundary.traverse.core) type instance IntersectionOf (Line 2 r) (Boundary (Polygon t p r)) = '[Seq.Seq (Either (Point 2 r) (LineSegment 2 () r))] type instance IntersectionOf (Point 2 r) (Polygon t p r) = [NoIntersection, Point 2 r] instance (Fractional r, Ord r) => (Point 2 r) `IsIntersectableWith` (Polygon t p r) where nonEmptyIntersection = defaultNonEmptyIntersection q `intersects` pg = q `inPolygon` pg /= Outside q `intersect` pg | q `intersects` pg = coRec q | otherwise = coRec NoIntersection -- instance IsIntersectableWith (Line 2 r) (Boundary (Polygon t p r)) where -- nonEmptyIntersection _ _ (CoRec xs) = null xs -- l `intersect` (Boundary (SimplePolygon vs)) = -- undefined -- l `intersect` (Boundary (MultiPolygon vs hs)) = coRec . -- Seq.sortBy f . Seq.fromList -- . concatMap (unpack . (l `intersect`) . Boundary) -- $ SimplePolygon vs : hs -- where -- unpack (CoRec x) = x -- f = undefined -- * Functions on Polygons outerBoundary :: forall t p r. Lens' (Polygon t p r) (C.CSeq (Point 2 r :+ p)) outerBoundary = lens g s where g :: Polygon t p r -> C.CSeq (Point 2 r :+ p) g (SimplePolygon vs) = vs g (MultiPolygon vs _) = vs s :: Polygon t p r -> C.CSeq (Point 2 r :+ p) -> Polygon t p r s (SimplePolygon _) vs = SimplePolygon vs s (MultiPolygon _ hs) vs = MultiPolygon vs hs polygonHoles :: forall p r. Lens' (Polygon Multi p r) [Polygon Simple p r] polygonHoles = lens g s where g :: Polygon Multi p r -> [Polygon Simple p r] g (MultiPolygon _ hs) = hs s :: Polygon Multi p r -> [Polygon Simple p r] -> Polygon Multi p r s (MultiPolygon vs _) = MultiPolygon vs -- | Access the i^th vertex on the outer boundary outerVertex :: Int -> Lens' (Polygon t p r) (Point 2 r :+ p) outerVertex i = outerBoundary.C.item i -- running time: \(O(\log i)\) outerBoundaryEdge :: Int -> Polygon t p r -> LineSegment 2 p r outerBoundaryEdge i p = let u = p^.outerVertex i v = p^.outerVertex (i+1) in LineSegment (Closed u) (Open v) -- | Get all holes in a polygon holeList :: Polygon t p r -> [Polygon Simple p r] holeList (SimplePolygon _) = [] holeList (MultiPolygon _ hs) = hs -- | The vertices in the polygon. No guarantees are given on the order in which -- they appear! polygonVertices :: Polygon t p r -> NonEmpty.NonEmpty (Point 2 r :+ p) polygonVertices (SimplePolygon vs) = toNonEmpty vs polygonVertices (MultiPolygon vs hs) = sconcat $ toNonEmpty vs NonEmpty.:| map polygonVertices hs -- | Creates a simple polygon from the given list of vertices. -- -- pre: the input list constains no repeated vertices. fromPoints :: [Point 2 r :+ p] -> SimplePolygon p r fromPoints = SimplePolygon . C.fromList -- | The edges along the outer boundary of the polygon. The edges are half open. -- -- running time: \(O(n)\) outerBoundaryEdges :: Polygon t p r -> C.CSeq (LineSegment 2 p r) outerBoundaryEdges = toEdges . (^.outerBoundary) -- | Lists all edges. The edges on the outer boundary are given before the ones -- on the holes. However, no other guarantees are given on the order. -- -- running time: \(O(n)\) listEdges :: Polygon t p r -> [LineSegment 2 p r] listEdges pg = let f = F.toList . outerBoundaryEdges in f pg <> concatMap f (holeList pg) -- | Pairs every vertex with its incident edges. The first one is its -- predecessor edge, the second one its successor edge. -- -- >>> mapM_ print . polygonVertices $ withIncidentEdges simplePoly -- Point2 [0 % 1,0 % 1] :+ SP LineSegment (Closed (Point2 [1 % 1,11 % 1] :+ ())) (Closed (Point2 [0 % 1,0 % 1] :+ ())) LineSegment (Closed (Point2 [0 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,0 % 1] :+ ())) -- Point2 [10 % 1,0 % 1] :+ SP LineSegment (Closed (Point2 [0 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,0 % 1] :+ ())) LineSegment (Closed (Point2 [10 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,10 % 1] :+ ())) -- Point2 [10 % 1,10 % 1] :+ SP LineSegment (Closed (Point2 [10 % 1,0 % 1] :+ ())) (Closed (Point2 [10 % 1,10 % 1] :+ ())) LineSegment (Closed (Point2 [10 % 1,10 % 1] :+ ())) (Closed (Point2 [5 % 1,15 % 1] :+ ())) -- Point2 [5 % 1,15 % 1] :+ SP LineSegment (Closed (Point2 [10 % 1,10 % 1] :+ ())) (Closed (Point2 [5 % 1,15 % 1] :+ ())) LineSegment (Closed (Point2 [5 % 1,15 % 1] :+ ())) (Closed (Point2 [1 % 1,11 % 1] :+ ())) -- Point2 [1 % 1,11 % 1] :+ SP LineSegment (Closed (Point2 [5 % 1,15 % 1] :+ ())) (Closed (Point2 [1 % 1,11 % 1] :+ ())) LineSegment (Closed (Point2 [1 % 1,11 % 1] :+ ())) (Closed (Point2 [0 % 1,0 % 1] :+ ())) withIncidentEdges :: Polygon t p r -> Polygon t (Two (LineSegment 2 p r)) r withIncidentEdges (SimplePolygon vs) = SimplePolygon $ C.zip3LWith f (C.rotateL vs) vs (C.rotateR vs) where f p c n = c&extra .~ SP (ClosedLineSegment p c) (ClosedLineSegment c n) withIncidentEdges (MultiPolygon vs hs) = MultiPolygon vs' hs' where (SimplePolygon vs') = withIncidentEdges $ SimplePolygon vs hs' = map withIncidentEdges hs -- -- | Gets the i^th edge on the outer boundary of the polygon, that is the edge ---- with vertices i and i+1 with respect to the current focus. All indices -- -- modulo n. -- -- -- | Given the vertices of the polygon. Produce a list of edges. The edges are -- half-open. toEdges :: C.CSeq (Point 2 r :+ p) -> C.CSeq (LineSegment 2 p r) toEdges vs = C.zipLWith (\p q -> LineSegment (Closed p) (Open q)) vs (C.rotateR vs) -- let vs' = F.toList vs in -- C.fromList $ zipWith (\p q -> LineSegment (Closed p) (Open q)) vs' (tail vs' ++ vs') -- | Test if q lies on the boundary of the polygon. Running time: O(n) -- -- >>> point2 1 1 `onBoundary` simplePoly -- False -- >>> point2 0 0 `onBoundary` simplePoly -- True -- >>> point2 10 0 `onBoundary` simplePoly -- True -- >>> point2 5 13 `onBoundary` simplePoly -- False -- >>> point2 5 10 `onBoundary` simplePoly -- False -- >>> point2 10 5 `onBoundary` simplePoly -- True -- >>> point2 20 5 `onBoundary` simplePoly -- False -- -- TODO: testcases multipolygon onBoundary :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool q `onBoundary` pg = any (q `onSegment`) es where out = SimplePolygon $ pg^.outerBoundary es = concatMap (F.toList . outerBoundaryEdges) $ out : holeList pg -- | Check if a point lies inside a polygon, on the boundary, or outside of the polygon. -- Running time: O(n). -- -- >>> point2 1 1 `inPolygon` simplePoly -- Inside -- >>> point2 0 0 `inPolygon` simplePoly -- OnBoundary -- >>> point2 10 0 `inPolygon` simplePoly -- OnBoundary -- >>> point2 5 13 `inPolygon` simplePoly -- Inside -- >>> point2 5 10 `inPolygon` simplePoly -- Inside -- >>> point2 10 5 `inPolygon` simplePoly -- OnBoundary -- >>> point2 20 5 `inPolygon` simplePoly -- Outside -- -- TODO: Add some testcases with multiPolygons -- TODO: Add some more onBoundary testcases inPolygon :: forall t p r. (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> PointLocationResult q `inPolygon` pg | q `onBoundary` pg = OnBoundary | odd kl && odd kr && not (any (q `inHole`) hs) = Inside | otherwise = Outside where l = horizontalLine $ q^.yCoord -- Given a line segment, compute the intersection point (if a point) with the -- line l intersectionPoint = asA @(Point 2 r) . (`intersect` l) -- Count the number of intersections that the horizontal line through q -- maxes with the polygon, that are strictly to the left and strictly to -- the right of q. If these numbers are both odd the point lies within the polygon. -- -- -- note that: - by the asA (Point 2 r) we ignore horizontal segments (as desired) -- - by the filtering, we effectively limit l to an open-half line, starting -- at the (open) point q. -- - by using half-open segments as edges we avoid double counting -- intersections that coincide with vertices. -- - If the point is outside, and on the same height as the -- minimum or maximum coordinate of the polygon. The number of -- intersections to the left or right may be one. Thus -- incorrectly classifying the point as inside. To avoid this, -- we count both the points to the left *and* to the right of -- p. Only if both are odd the point is inside. so that if -- the point is outside, and on the same y-coordinate as one -- of the extermal vertices (one ofth) -- -- See http://geomalgorithms.com/a03-_inclusion.html for more information. SP kl kr = count (\p -> (p^.xCoord) `compare` (q^.xCoord)) . mapMaybe intersectionPoint . F.toList . outerBoundaryEdges $ pg -- For multi polygons we have to test if we do not lie in a hole . inHole = insidePolygon hs = holeList pg count :: (a -> Ordering) -> [a] -> SP Int Int count f = foldr (\x (SP lts gts) -> case f x of LT -> SP (lts + 1) gts EQ -> SP lts gts GT -> SP lts (gts + 1)) (SP 0 0) -- | Test if a point lies strictly inside the polgyon. insidePolygon :: (Fractional r, Ord r) => Point 2 r -> Polygon t p r -> Bool q `insidePolygon` pg = q `inPolygon` pg == Inside -- testQ = map (`inPolygon` testPoly) [ point2 1 1 -- Inside -- , point2 0 0 -- OnBoundary -- , point2 5 14 -- Inside -- , point2 5 10 -- Inside -- , point2 10 5 -- OnBoundary -- , point2 20 5 -- Outside -- ] -- testPoly :: SimplePolygon () Rational -- testPoly = SimplePolygon . C.fromList . map ext $ [ point2 0 0 -- , point2 10 0 -- , point2 10 10 -- , point2 5 15 -- , point2 1 11 -- ] -- | Compute the area of a polygon area :: Fractional r => Polygon t p r -> r area poly@(SimplePolygon _) = abs $ signedArea poly area (MultiPolygon vs hs) = area (SimplePolygon vs) - sum [area h | h <- hs] -- | Compute the signed area of a simple polygon. The the vertices are in -- clockwise order, the signed area will be negative, if the verices are given -- in counter clockwise order, the area will be positive. signedArea :: Fractional r => SimplePolygon p r -> r signedArea poly = x / 2 where x = sum [ p^.core.xCoord * q^.core.yCoord - q^.core.xCoord * p^.core.yCoord | LineSegment' p q <- F.toList $ outerBoundaryEdges poly ] -- | Compute the centroid of a simple polygon. centroid :: Fractional r => SimplePolygon p r -> Point 2 r centroid poly = Point $ sum' xs ^/ (6 * signedArea poly) where xs = [ (toVec p ^+^ toVec q) ^* (p^.xCoord * q^.yCoord - q^.xCoord * p^.yCoord) | LineSegment' (p :+ _) (q :+ _) <- F.toList $ outerBoundaryEdges poly ] sum' = F.foldl' (^+^) zero -- | Pick a point that is inside the polygon. -- -- (note: if the polygon is degenerate; i.e. has <3 vertices, we report a -- vertex of the polygon instead.) -- -- pre: the polygon is given in CCW order -- -- running time: \(O(n)\) pickPoint :: (Ord r, Fractional r) => Polygon p t r -> Point 2 r pickPoint pg | isTriangle pg = centroid . SimplePolygon $ pg^.outerBoundary | otherwise = let LineSegment' (p :+ _) (q :+ _) = findDiagonal pg in p .+^ (0.5 *^ (q .-. p)) -- | Test if the polygon is a triangle -- -- running time: \(O(1)\) isTriangle :: Polygon p t r -> Bool isTriangle = \case SimplePolygon vs -> go vs MultiPolygon vs [] -> go vs MultiPolygon _ _ -> False where go vs = case toNonEmpty vs of (_ :| [_,_]) -> True _ -> False -- | Find a diagonal of the polygon. -- -- pre: the polygon is given in CCW order -- -- running time: \(O(n)\) findDiagonal :: (Ord r, Fractional r) => Polygon t p r -> LineSegment 2 p r findDiagonal pg = List.head . catMaybes . F.toList $ diags -- note that a diagonal is guaranteed to exist, so the usage of head is safe. where vs = pg^.outerBoundary diags = C.zip3LWith f (C.rotateL vs) vs (C.rotateR vs) f u v w = case ccw (u^.core) (v^.core) (w^.core) of CCW -> Just $ findDiag u v w -- v is a convex vertex, so find a diagonal -- (either uw) or from v to a point inside the -- triangle CW -> Nothing -- v is a reflex vertex CoLinear -> Nothing -- colinear vertex!? -- we test if uw is a diagonal by figuring out if there is a vertex -- strictly inside the triangle t. If there is no such vertex then uw must -- be a diagonal (i.e. uw intersects the polygon boundary iff there is a -- vtx inside t). If there are vertices inside the triangle, we find the -- one z furthest from the line(segment) uw. It then follows that vz is a -- diagonal. Indeed this is pretty much the argument used to prove that any -- polygon can be triangulated. See BKOS Chapter 3 for details. findDiag u v w = let t = Triangle u v w uw = ClosedLineSegment u w in maybe uw (ClosedLineSegment v) . safeMaximumOn (distTo $ supportingLine uw) . filter (\(z :+ _) -> z `inTriangle` t == Inside) . F.toList . polygonVertices $ pg distTo l (z :+ _) = sqDistanceTo z l safeMaximumOn :: Ord b => (a -> b) -> [a] -> Maybe a safeMaximumOn f = \case [] -> Nothing xs -> Just $ List.maximumBy (comparing f) xs -- | Test if the outer boundary of the polygon is in clockwise or counter -- clockwise order. -- -- running time: \(O(n)\) -- isCounterClockwise :: (Eq r, Fractional r) => Polygon t p r -> Bool isCounterClockwise = (\x -> x == abs x) . signedArea . fromPoints . F.toList . (^.outerBoundary) -- | Orient the outer boundary to clockwise order toClockwiseOrder :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r toClockwiseOrder p | isCounterClockwise p = reverseOuterBoundary p | otherwise = p -- | Orient the outer boundary to counter clockwise order toCounterClockWiseOrder :: (Eq r, Fractional r) => Polygon t p r -> Polygon t p r toCounterClockWiseOrder p | not $ isCounterClockwise p = reverseOuterBoundary p | otherwise = p reverseOuterBoundary :: Polygon t p r -> Polygon t p r reverseOuterBoundary p = p&outerBoundary %~ C.reverseDirection -- | Convert a Polygon to a simple polygon by forgetting about any holes. asSimplePolygon :: Polygon t p r -> SimplePolygon p r asSimplePolygon poly@(SimplePolygon _) = poly asSimplePolygon (MultiPolygon vs _) = SimplePolygon vs -- | Comparison that compares which point is 'larger' in the direction given by -- the vector u. cmpExtreme :: (Num r, Ord r) => Vector 2 r -> Point 2 r :+ p -> Point 2 r :+ q -> Ordering cmpExtreme u p q = u `dot` (p^.core .-. q^.core) `compare` 0 -- | Finds the extreme points, minimum and maximum, in a given direction -- -- running time: \(O(n)\) extremesLinear :: (Ord r, Num r) => Vector 2 r -> Polygon t p r -> (Point 2 r :+ p, Point 2 r :+ p) extremesLinear u p = let vs = p^.outerBoundary f = cmpExtreme u in (F.minimumBy f vs, F.maximumBy f vs) -- | assigns unique integer numbers to all vertices. Numbers start from 0, and -- are increasing along the outer boundary. The vertices of holes -- will be numbered last, in the same order. -- -- >>> numberVertices simplePoly -- SimplePolygon CSeq [Point2 [0 % 1,0 % 1] :+ SP 0 (),Point2 [10 % 1,0 % 1] :+ SP 1 (),Point2 [10 % 1,10 % 1] :+ SP 2 (),Point2 [5 % 1,15 % 1] :+ SP 3 (),Point2 [1 % 1,11 % 1] :+ SP 4 ()] numberVertices :: Polygon t p r -> Polygon t (SP Int p) r numberVertices = snd . bimapAccumL (\a p -> (a+1,SP a p)) (\a r -> (a,r)) 0 -- TODO: Make sure that this does not have the same issues as foldl vs foldl' -------------------------------------------------------------------------------- -- | Test if a Simple polygon is star-shaped. Returns a point in the kernel -- (i.e. from which the entire polygon is visible), if it exists. -- -- -- \(O(n)\) expected time isStarShaped :: (MonadRandom m, Ord r, Fractional r) => SimplePolygon p r -> m (Maybe (Point 2 r)) isStarShaped (toClockwiseOrder -> pg) = solveBoundedLinearProgram $ LinearProgram c (F.toList hs) where c = pg^.outerVertex 1.core.vector -- the first vertex is the intersection point of the two supporting lines -- bounding it, so the first two edges bound the shape in this sirection hs = fmap (rightOf . supportingLine) . outerBoundaryEdges $ pg