Math.Geometry.Grid

A regular arrangement of tiles. Grids have a variety of uses, including games and self-organising maps. The userguide is available at https://github.com/mhwombat/grid/wiki.

NOTE: Version 3.0 changed the order of parameters for many functions. This makes it easier for the user to write mapping and folding operations.

The Grid class

class Eq x => Grid g s x | g -> s, g -> x where

A regular arrangement of tiles. Minimal complete definition: indices, distance and size.

Methods

indices :: g -> [x]

Returns the indices of all tiles in a grid.

distance :: g -> x -> x -> Int

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.

minDistance :: 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.

size :: g -> s

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.

neighbours :: g -> x -> [x]

neighbours g x returns the indices of the tiles in the grid g which are adjacent to the tile with index x.

numNeighbours :: g -> x -> Int

numNeighbours g x returns the number of tiles in the grid g which are adjacent to the tile with index x.

contains :: g -> x -> Bool

g `'contains'` x returns True if the index x is contained within the grid g, otherwise it returns false.

viewpoint :: g -> x -> [(x, Int)]

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.

tileCount :: g -> Int

Returns the number of tiles in a grid. Compare with size.

empty :: g -> Bool

Returns True if the number of tiles in a grid is zero, False otherwise.

nonEmpty :: g -> Bool

Returns False if the number of tiles in a grid is zero, True otherwise.

edges :: g -> [(x, x)]

A list of all edges in a grid, where the edges are represented by a pair of indices of adjacent tiles.

isAdjacent :: Grid g s x => g -> x -> x -> Bool

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.

adjacentTilesToward :: g -> x -> x -> [x]

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.

minimalPaths :: g -> x -> x -> [[x]]

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.

Instances

Grid HexHexGrid Int (Int, Int)

Grid TriTriGrid Int (Int, Int)

Grid ParaHexGrid (Int, Int) (Int, Int)

Grid TorSquareGrid (Int, Int) (Int, Int)

Grid RectSquareGrid (Int, Int) (Int, Int)

Grid ParaTriGrid (Int, Int) (Int, Int)

(Eq k, Grid g s k) => Grid (GridMap g k v) s k

class Grid g s x => BoundedGrid g s x where

A regular arrangement of tiles with an edge. Minimal complete definition: boundary.

Methods

boundary :: g -> [x]

Returns a the indices of all the tiles at the boundary of a grid, including corner tiles.

isBoundary :: g -> x -> Bool

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.)

centre :: g -> [x]

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).

isCentre :: g -> x -> Bool

isCentre g x' returns True if the tile with index x is a centre tile of g, False otherwise.

Instances

BoundedGrid HexHexGrid Int (Int, Int)

BoundedGrid TriTriGrid Int (Int, Int)

BoundedGrid ParaHexGrid (Int, Int) (Int, Int)

BoundedGrid RectSquareGrid (Int, Int) (Int, Int)

BoundedGrid ParaTriGrid (Int, Int) (Int, Int)

Grids with triangular tiles

data TriTriGrid

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.

Instances

Eq TriTriGrid

Show TriTriGrid

BoundedGrid TriTriGrid Int (Int, Int)

Grid TriTriGrid Int (Int, Int)

triTriGrid :: Int -> TriTriGrid

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.

data ParaTriGrid

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.

Instances

Eq ParaTriGrid

Show ParaTriGrid

BoundedGrid ParaTriGrid (Int, Int) (Int, Int)

Grid ParaTriGrid (Int, Int) (Int, Int)

paraTriGrid :: Int -> Int -> ParaTriGrid

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.

Grids with square tiles

data RectSquareGrid

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.

Instances

Eq RectSquareGrid

Show RectSquareGrid

BoundedGrid RectSquareGrid (Int, Int) (Int, Int)

Grid RectSquareGrid (Int, Int) (Int, Int)

rectSquareGrid :: Int -> Int -> RectSquareGrid

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.

data TorSquareGrid

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.

Instances

Eq TorSquareGrid

Show TorSquareGrid

Grid TorSquareGrid (Int, Int) (Int, Int)

torSquareGrid :: Int -> Int -> TorSquareGrid

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.

Grids with hexagonal tiles

data HexHexGrid

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.

Instances

Eq HexHexGrid

Show HexHexGrid

BoundedGrid HexHexGrid Int (Int, Int)

Grid HexHexGrid Int (Int, Int)

hexHexGrid :: Int -> HexHexGrid

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.

data ParaHexGrid

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.

Instances

Eq ParaHexGrid

Show ParaHexGrid

BoundedGrid ParaHexGrid (Int, Int) (Int, Int)

Grid ParaHexGrid (Int, Int) (Int, Int)

paraHexGrid :: Int -> Int -> ParaHexGrid

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.

Example

Create a grid.

ghci> let g = hexHexGrid 3
ghci> indices g
[(-2,0),(-2,1),(-2,2),(-1,-1),(-1,0),(-1,1),(-1,2),(0,-2),(0,-1),(0,0),(0,1),(0,2),(1,-2),(1,-1),(1,0),(1,1),(2,-2),(2,-1),(2,0)]

Find out the minimum number of moves to go from one tile in a grid to another tile, moving between adjacent tiles at each step.

ghci> distance g (0,-2) (0,2)
4

Find out the minimum number of moves to go from one tile in a grid to any other tile, moving between adjacent tiles at each step.

ghci> viewpoint g (1,-2)
[((-2,0),3),((-2,1),3),((-2,2),4),((-1,-1),2),((-1,0),2),((-1,1),3),((-1,2),4),((0,-2),1),((0,-1),1),((0,0),2),((0,1),3),((0,2),4),((1,-2),0),((1,-1),1),((1,0),2),((1,1),3),((2,-2),1),((2,-1),2),((2,0),3)]

Find out which tiles are adjacent to a particular tile.

ghci> neighbours g (-1,1)
[(-2,1),(-2,2),(-1,2),(0,1),(0,0),(-1,0)]

Find out if a tile is within the grid boundary.

ghci> g `contains` (0,0)
True
ghci> g `contains` (0,12)
False

Find out the physical dimensions of the grid.

ghci> size g
3

Find out the number of tiles in the grid.

ghci> tileCount g
19

Check if a grid is empty (contains no tiles).

ghci> empty g
False
ghci> nonEmpty g
True

Find all of the minimal paths between two points.

ghci> let g = hexHexGrid 3 ghci> minimalPaths g (0,0) (2,-1) [[(0,0),(1,0),(2,-1)],[(0,0),(1,-1),(2,-1)]]