------------------------------------------------------------------------ -- | -- Module : Math.Geometry.GridInternal -- Copyright : (c) Amy de Buitléir 2012 -- License : BSD-style -- Maintainer : amy@nualeargais.ie -- Stability : experimental -- Portability : portable -- -- A module containing private @Grid@ internals. Most developers should -- use @Grid@ instead. This module is subject to change without notice. -- ------------------------------------------------------------------------ {-# LANGUAGE UnicodeSyntax, MultiParamTypeClasses, FunctionalDependencies, TypeSynonymInstances, FlexibleInstances, FlexibleContexts #-} module Math.Geometry.GridInternal ( -- * Generic Grid(..), BoundedGrid(..), -- * Grids with triangular tiles TriTriGrid, triTriGrid, ParaTriGrid, paraTriGrid, -- * Grids with square tiles RectSquareGrid, rectSquareGrid, TorSquareGrid, torSquareGrid, -- * Grids with hexagonal tiles HexHexGrid, hexHexGrid, ParaHexGrid, paraHexGrid ) where import Data.Eq.Unicode ((≡), (≠)) import Data.Function (on) import Data.List (groupBy, nub, nubBy, sortBy) import Data.Ord (comparing) import Data.Ord.Unicode ((≤), (≥)) -- | A regular arrangement of tiles. -- Minimal complete definition: @indices@, @distance@ and @size@. class Eq x ⇒ Grid g s x | g → s, g → x where -- | Returns the indices of all tiles in a grid. indices ∷ g → [x] -- | @'distance' g a b@ returns the minimum number of moves required -- to get from the tile at index @a@ to the tile at index @b@ in -- grid @g@, moving between adjacent tiles at each step. (Two tiles -- are adjacent if they share an edge.) If @a@ or @b@ are not -- contained within @g@, the result is undefined. distance ∷ g → x → x → Int -- | @'minDistance' g bs a@ returns the minimum number of moves -- required to get from any of the tiles at indices @bs@ to the tile -- at index @a@ in grid @g@, moving between adjacent tiles at each -- step. (Two tiles are adjacent if they share an edge.) If @a@ or -- any of @bs@ are not contained within @g@, the result is -- undefined. minDistance ∷ g → [x] → x → Int minDistance g xs x = minimum . map (distance g x) $ xs -- | Returns the dimensions of the grid. -- For example, if @g@ is a 4x3 rectangular grid, @'size' g@ would -- return @(4, 3)@, while @'tileCount' g@ would return @12@. size ∷ g → s -- | @'neighbours' g x@ returns the indices of the tiles in the grid -- @g@ which are adjacent to the tile with index @x@. neighbours ∷ g → x → [x] neighbours g x = filter (\a → distance g x a ≡ 1 ) $ indices g -- | @'numNeighbours' g x@ returns the number of tiles in the grid -- @g@ which are adjacent to the tile with index @x@. numNeighbours ∷ g → x → Int numNeighbours g = length . neighbours g -- | @g `'contains'` x@ returns @True@ if the index @x@ is contained -- within the grid @g@, otherwise it returns false. contains ∷ g → x → Bool contains g x = x `elem` indices g -- | @'viewpoint' g x@ returns a list of pairs associating the index -- of each tile in @g@ with its distance to the tile with index @x@. -- If @x@ is not contained within @g@, the result is undefined. viewpoint ∷ g → x → [(x, Int)] viewpoint g p = map f (indices g) where f x = (x, distance g p x) -- | Returns the number of tiles in a grid. Compare with @'size'@. tileCount ∷ g → Int tileCount = length . indices -- | Returns @True@ if the number of tiles in a grid is zero, @False@ -- otherwise. empty ∷ g → Bool empty g = tileCount g ≡ 0 -- | Returns @False@ if the number of tiles in a grid is zero, @True@ -- otherwise. nonEmpty ∷ g → Bool nonEmpty = not . empty -- | A list of all edges in a grid, where the edges are represented by -- a pair of indices of adjacent tiles. edges ∷ g → [(x,x)] edges g = nubBy sameEdge $ concatMap (`adjacentEdges` g) $ indices g -- | @'isAdjacent' g a b@ returns @True@ if the tile at index @a@ is -- adjacent to the tile at index @b@ in @g@. (Two tiles are adjacent -- if they share an edge.) If @a@ or @b@ are not contained within -- @g@, the result is undefined. isAdjacent ∷ Grid g s x ⇒ g → x → x → Bool isAdjacent g a b = distance g a b ≡ 1 -- | @'adjacentTilesToward' g a b@ returns the indices of all tiles -- which are neighbours of the tile at index @a@, and which are -- closer to the tile at @b@ than @a@ is. In other words, it returns -- the possible next steps on a minimal path from @a@ to @b@. If @a@ -- or @b@ are not contained within @g@, or if there is no path from -- @a@ to @b@ (e.g., a disconnected grid), the result is undefined. adjacentTilesToward ∷ g → x → x → [x] adjacentTilesToward g a b | a ≡ b = [] | otherwise = filter f $ neighbours g a where f x = distance g x b ≡ distance g a b - 1 -- | @'minimalPaths' g a b@ returns a list of all minimal paths from -- the tile at index @a@ to the tile at index @b@ in grid @g@. A -- path is a sequence of tiles where each tile in the sequence is -- adjacent to the previous one. (Two tiles are adjacent if they -- share an edge.) If @a@ or @b@ are not contained within @g@, the -- result is undefined. -- -- Tip: The default implementation of this function calls -- @'adjacentTilesToward'@. If you want to use a custom algorithm, -- consider modifying @'adjacentTilesToward'@ instead of -- @'minimalPaths'@. minimalPaths ∷ g → x → x → [[x]] minimalPaths g a b | a ≡ b = [[a]] | distance g a b ≡ 1 = [[a,b]] | otherwise = map (a:) xs where xs = concatMap (\x → minimalPaths g x b) ys ys = adjacentTilesToward g a b sameEdge ∷ Eq t ⇒ (t, t) → (t, t) → Bool sameEdge (a,b) (c,d) = (a,b) ≡ (c,d) || (a,b) ≡ (d,c) adjacentEdges ∷ Grid g s t ⇒ t → g → [(t, t)] adjacentEdges i g = map (\j → (i,j)) $ neighbours g i -- | A regular arrangement of tiles with an edge. -- Minimal complete definition: @boundary@. class Grid g s x ⇒ BoundedGrid g s x where -- | Returns a the indices of all the tiles at the boundary of a grid, -- including corner tiles. boundary ∷ g → [x] -- | @'isBoundary' g x@' returns @True@ if the tile with index @x@ is -- on a boundary of @g@, @False@ otherwise. (Corner tiles are also -- boundary tiles.) isBoundary ∷ g → x → Bool isBoundary g x = x `elem` boundary g -- | Returns the index of the tile(s) that require the maximum number -- of moves to reach the nearest boundary tile. A grid may have more -- than one central tile (e.g., a rectangular grid with an even -- number of rows and columns will have four central tiles). centre ∷ g → [x] centre g = map fst . head . reverse . groupBy ((==) `on` snd) . sortBy (comparing snd) $ xds where xds = map (\y -> (y, minDistance g bs y)) $ indices g bs = boundary g -- | @'isCentre' g x@' returns @True@ if the tile with index @x@ is -- a centre tile of @g@, @False@ otherwise. isCentre ∷ g → x → Bool isCentre g x = x `elem` centre g -- -- Triangular tiles -- -- | For triangular tiles, it is convenient to define a third component -- z. triZ ∷ Int → Int → Int triZ x y | even y = -x - y | otherwise = -x - y + 1 triDistance ∷ Grid g s (Int, Int) ⇒ g → (Int, Int) → (Int, Int) → Int triDistance g (x1, y1) (x2, y2) = if g `contains` (x1, y1) && g `contains` (x2, y2) then maximum [abs (x2-x1), abs (y2-y1), abs(z2-z1)] else undefined where z1 = triZ x1 y1 z2 = triZ x2 y2 triNeighbours ∷ Grid g s (Int, Int) ⇒ g → (Int, Int) → [(Int, Int)] triNeighbours g (x,y) = filter (g `contains`) xs where xs | even y = [(x-1,y+1), (x+1,y+1), (x+1,y-1)] | otherwise = [(x-1,y-1), (x-1,y+1), (x+1,y-1)] -- -- Triangular grids with triangular tiles -- -- | A triangular grid with triangular tiles. -- The grid and its indexing scheme are illustrated in the user guide, -- available at <https://github.com/mhwombat/grid/wiki>. data TriTriGrid = TriTriGrid Int [(Int, Int)] deriving Eq instance Show TriTriGrid where show (TriTriGrid s _) = "triTriGrid " ++ show s instance Grid TriTriGrid Int (Int, Int) where indices (TriTriGrid _ xs) = xs neighbours = triNeighbours distance = triDistance contains (TriTriGrid s _) (x, y) = inTriGrid (x,y) s size (TriTriGrid s _) = s inTriGrid ∷ (Int, Int) → Int → Bool inTriGrid (x, y) s = x ≥ 0 && y ≥ 0 && even (x+y) && abs z ≤ 2*s-2 where z = triZ x y instance BoundedGrid TriTriGrid Int (Int, Int) where -- corners g = if empty g -- then [] -- else nub [(0,0), (0,2*s-2), (2*s-2, 0)] -- where s = size g boundary g = west ++ east ++ south where s = size g west = [(0,k) | k ← [0,2..2*s-2]] east = [(k,2*s-2-k) | k ← [2,4..2*s-2]] south = [(k,0) | k ← [2*s-4,2*s-6..2]] centre g = case s `mod` 3 of 0 → trefoilWithTop (k-1,k+1) where k = (2*s) `div` 3 1 → [(k,k)] where k = (2*(s-1)) `div` 3 2 → [(k+1,k+1)] where k = (2*(s-2)) `div` 3 _ → error "This will never happen." where s = size g trefoilWithTop ∷ (Int, Int) → [(Int,Int)] trefoilWithTop (i,j) = [(i,j), (i+2, j-2), (i,j-2)] -- | @'triTriGrid' s@ returns a triangular grid with sides of -- length @s@, using triangular tiles. If @s@ is nonnegative, the -- resulting grid will have @s^2@ tiles. Otherwise, the resulting grid -- will be empty and the list of indices will be null. triTriGrid ∷ Int → TriTriGrid triTriGrid s = TriTriGrid s [(xx,yy) | xx ← [0..2*(s-1)], yy ← [0..2*(s-1)], (xx,yy) `inTriGrid` s] -- -- Parallelogrammatical grids with triangular tiles -- -- | A Parallelogrammatical grid with triangular tiles. -- The grid and its indexing scheme are illustrated in the user guide, -- available at <https://github.com/mhwombat/grid/wiki>. data ParaTriGrid = ParaTriGrid (Int, Int) [(Int, Int)] deriving Eq instance Show ParaTriGrid where show (ParaTriGrid (r,c) _) = "paraTriGrid " ++ show r ++ " " ++ show c instance Grid ParaTriGrid (Int, Int) (Int, Int) where indices (ParaTriGrid _ xs) = xs neighbours = triNeighbours distance = triDistance size (ParaTriGrid s _) = s instance BoundedGrid ParaTriGrid (Int, Int) (Int, Int) where boundary g = west ++ north ++ east ++ south where (r,c) = size g west = [(0,k) | k ← [0,2..2*r-2], c>0] north = [(k,2*r-1) | k ← [1,3..2*c-1], r>0] east = [(2*c-1,k) | k ← [2*r-3,2*r-5..1], c>0] south = [(k,0) | k ← [2*c-2,2*c-4..2], r>0] centre g = paraTriGridCentre . size $ g paraTriGridCentre ∷ (Int, Int) → [(Int, Int)] paraTriGridCentre (r,c) | odd r && odd c = [(c-1,r-1), (c,r)] | even r && even c && r == c = bowtie (c-1,r-1) | even r && even c && r > c = bowtie (c-1,r-3) ++ bowtie (c-1,r-1) ++ bowtie (c-1,r+1) | even r && even c && r < c = bowtie (c-3,r-1) ++ bowtie (c-1,r-1) ++ bowtie (c+1,r-1) | otherwise = [(c-1,r), (c,r-1)] bowtie :: (Int,Int) -> [(Int,Int)] bowtie (i,j) = [(i,j), (i+1,j+1)] -- | @'paraTriGrid' r c@ returns a grid in the shape of a -- parallelogram with @r@ rows and @c@ columns, using triangular -- tiles. If @r@ and @c@ are both nonnegative, the resulting grid will -- have @2*r*c@ tiles. Otherwise, the resulting grid will be empty and -- the list of indices will be null. paraTriGrid ∷ Int → Int → ParaTriGrid paraTriGrid r c = ParaTriGrid (r,c) [(x,y) | x ← [0..2*c-1], y ← [0..2*r-1], even (x+y)] -- -- Rectangular grids with square tiles -- -- | A rectangular grid with square tiles. -- The grid and its indexing scheme are illustrated in the user guide, -- available at <https://github.com/mhwombat/grid/wiki>. data RectSquareGrid = RectSquareGrid (Int, Int) [(Int, Int)] deriving Eq instance Show RectSquareGrid where show (RectSquareGrid (r,c) _) = "rectSquareGrid " ++ show r ++ " " ++ show c instance Grid RectSquareGrid (Int, Int) (Int, Int) where indices (RectSquareGrid _ xs) = xs neighbours g (x, y) = filter (g `contains`) [(x-1,y), (x,y+1), (x+1,y), (x,y-1)] distance g (x1, y1) (x2, y2) = if g `contains` (x1, y1) && g `contains` (x2, y2) then abs (x2-x1) + abs (y2-y1) else undefined size (RectSquareGrid s _) = s adjacentTilesToward g a@(x1, y1) (x2, y2) = filter (\i → g `contains` i && i ≠ a) $ nub [(x1,y1+dy),(x1+dx,y1)] where dx = signum (x2-x1) dy = signum (y2-y1) instance BoundedGrid RectSquareGrid (Int, Int) (Int, Int) where boundary g = cartesianIndices . size $ g centre g = cartesianCentre . size $ g cartesianIndices ∷ (Enum r, Enum c, Num r, Num c, Ord r, Ord c) ⇒ (r, c) → [(c, r)] cartesianIndices (r, c) = west ++ north ++ east ++ south where west = [(0,k) | k ← [0,1..r-1], c>0] north = [(k,r-1) | k ← [1,2..c-1], r>0] east = [(c-1,k) | k ← [r-2,r-3..0], c>1] south = [(k,0) | k ← [c-2,c-3..1], r>1] cartesianCentre ∷ (Int, Int) → [(Int, Int)] cartesianCentre (r,c) = [(i,j) | i ← midpoints c, j ← midpoints r] midpoints ∷ Int → [Int] midpoints k = if even k then [m-1,m] else [m] where m = floor (k'/2.0) k' = fromIntegral k ∷ Double -- | @'rectSquareGrid' r c@ produces a rectangular grid with @r@ rows -- and @c@ columns, using square tiles. If @r@ and @c@ are both -- nonnegative, the resulting grid will have @r*c@ tiles. Otherwise, -- the resulting grid will be empty and the list of indices will be -- null. rectSquareGrid ∷ Int → Int → RectSquareGrid rectSquareGrid r c = RectSquareGrid (r,c) [(x,y) | x ← [0..c-1], y ← [0..r-1]] -- -- Toroidal grids with square tiles. -- -- | A toroidal grid with square tiles. -- The grid and its indexing scheme are illustrated in the user guide, -- available at <https://github.com/mhwombat/grid/wiki>. data TorSquareGrid = TorSquareGrid (Int, Int) [(Int, Int)] deriving Eq instance Show TorSquareGrid where show (TorSquareGrid (r,c) _) = "torSquareGrid " ++ show r ++ " " ++ show c instance Grid TorSquareGrid (Int, Int) (Int, Int) where indices (TorSquareGrid _ xs) = xs neighbours (TorSquareGrid (r,c) _) (x,y) = nub $ filter (\(xx,yy) → xx ≠ x || yy ≠ y) [((x-1) `mod` c,y), (x,(y+1) `mod` r), ((x+1) `mod` c,y), (x,(y-1) `mod` r)] distance g@(TorSquareGrid (r,c) _) (x1, y1) (x2, y2) = if g `contains` (x1, y1) && g `contains` (x2, y2) then min adx (abs (c-adx)) + min ady (abs (r-ady)) else undefined where adx = abs (x2 - x1) ady = abs (y2 - y1) size (TorSquareGrid s _) = s -- | @'torSquareGrid' r c@ returns a toroidal grid with @r@ -- rows and @c@ columns, using square tiles. If @r@ and @c@ are -- both nonnegative, the resulting grid will have @r*c@ tiles. Otherwise, -- the resulting grid will be empty and the list of indices will be null. torSquareGrid ∷ Int → Int → TorSquareGrid torSquareGrid r c = TorSquareGrid (r,c) [(x, y) | x ← [0..c-1], y ← [0..r-1]] -- -- Hexagonal tiles -- hexDistance ∷ Grid g s (Int, Int) ⇒ g → (Int, Int) → (Int, Int) → Int hexDistance g (x1, y1) (x2, y2) = if g `contains` (x1, y1) && g `contains` (x2, y2) then maximum [abs (x2-x1), abs (y2-y1), abs(z2-z1)] else undefined where z1 = -x1 - y1 z2 = -x2 - y2 -- -- Hexagonal grids with hexagonal tiles -- -- | A hexagonal grid with hexagonal tiles -- The grid and its indexing scheme are illustrated in the user guide, -- available at <https://github.com/mhwombat/grid/wiki>. data HexHexGrid = HexHexGrid Int [(Int, Int)] deriving Eq instance Show HexHexGrid where show (HexHexGrid s _) = "hexHexGrid " ++ show s instance Grid HexHexGrid Int (Int, Int) where indices (HexHexGrid _ xs) = xs neighbours g (x,y) = filter (g `contains`) [(x-1,y), (x-1,y+1), (x,y+1), (x+1,y), (x+1,y-1), (x,y-1)] distance = hexDistance size (HexHexGrid s _) = s instance BoundedGrid HexHexGrid Int (Int, Int) where boundary g = north ++ northeast ++ southeast ++ south ++ southwest ++ northwest where s = size g north = [(k,s-1) | k ← [-s+1,-s+2..0]] northeast = [(k,s-1-k) | k ← [1,2..s-1]] southeast = [(s-1,k) | k ← [-1,-2..(-s)+1]] south = [(k,(-s)+1) | k ← [s-2,s-3..0]] southwest = [(k,(-s)+1-k) | k ← [-1,-2..(-s)+1]] northwest = [(-s+1,k) | k ← [1,2..s-2]] centre _ = [(0,0)] -- | @'hexHexGrid' s@ returns a grid of hexagonal shape, with -- sides of length @s@, using hexagonal tiles. If @s@ is nonnegative, the -- resulting grid will have @3*s*(s-1) + 1@ tiles. Otherwise, the resulting -- grid will be empty and the list of indices will be null. hexHexGrid ∷ Int → HexHexGrid hexHexGrid r = HexHexGrid r [(x, y) | x ← [-r+1..r-1], y ← f x] where f x = if x < 0 then [1-r-x .. r-1] else [1-r .. r-1-x] -- -- Parallelogrammatical grids with hexagonal tiles -- -- | A parallelogramatical grid with hexagonal tiles -- The grid and its indexing scheme are illustrated in the user guide, -- available at <https://github.com/mhwombat/grid/wiki>. data ParaHexGrid = ParaHexGrid (Int, Int) [(Int, Int)] deriving Eq instance Show ParaHexGrid where show (ParaHexGrid (r,c) _) = "paraHexGrid " ++ show r ++ " " ++ show c instance Grid ParaHexGrid (Int, Int) (Int, Int) where indices (ParaHexGrid _ xs) = xs neighbours g (x,y) = filter (g `contains`) [(x-1,y), (x-1,y+1), (x,y+1), (x+1,y), (x+1,y-1), (x,y-1)] distance = hexDistance size (ParaHexGrid s _) = s instance BoundedGrid ParaHexGrid (Int, Int) (Int, Int) where boundary g = cartesianIndices . size $ g centre g = cartesianCentre . size $ g -- | @'paraHexGrid' r c@ returns a grid in the shape of a -- parallelogram with @r@ rows and @c@ columns, using hexagonal tiles. If -- @r@ and @c@ are both nonnegative, the resulting grid will have @r*c@ tiles. -- Otherwise, the resulting grid will be empty and the list of indices will -- be null. paraHexGrid ∷ Int → Int → ParaHexGrid paraHexGrid r c = ParaHexGrid (r,c) [(x, y) | x ← [0..c-1], y ← [0..r-1]]