src/Grid.hs

   1 -- | The Grid module just contains the Grid type and two constructors
   2 --   for it. We hide the main Grid constructor because we don't want
   3 --   to allow instantiation of a grid with h <= 0.
   4 module Grid
   5 where
   6
   7 import Test.QuickCheck (Arbitrary(..), Gen, Positive(..))
   8
   9 import Cube (Cube(Cube), find_containing_tetrahedra)
  10 import FunctionValues
  11 import Misc (flatten)
  12 import Point (Point)
  13 import Tetrahedron (polynomial)
  14 import Values (Values3D, dims, empty3d, zoom_shape)
  15
  16 import qualified Data.Array.Repa as R
  17
  18 -- | Our problem is defined on a Grid. The grid size is given by the
  19 --   positive number h. The function values are the values of the
  20 --   function at the grid points, which are distance h from one
  21 --   another in each direction (x,y,z).
  22 data Grid = Grid { h :: Double, -- MUST BE GREATER THAN ZERO!
  23                    function_values :: Values3D }
  24           deriving (Eq, Show)
  25
  26
  27 instance Arbitrary Grid where
  28     arbitrary = do
  29       (Positive h') <- arbitrary :: Gen (Positive Double)
  30       fvs <- arbitrary :: Gen Values3D
  31       return (make_grid h' fvs)
  32
  33
  34 -- | The constructor that we want people to use. If we're passed a
  35 --   non-positive grid size, we throw an error.
  36 make_grid :: Double -> Values3D -> Grid
  37 make_grid grid_size values
  38     | grid_size <= 0 = error "grid size must be positive"
  39     | otherwise = Grid grid_size values
  40
  41
  42 -- | Creates an empty grid with grid size 1.
  43 empty_grid :: Grid
  44 empty_grid = Grid 1 empty3d
  45
  46
  47 -- | Returns a three-dimensional list of cubes centered on the grid
  48 --   points of g with the appropriate 'FunctionValues'.
  49 cubes :: Grid -> [[[Cube]]]
  50 cubes g
  51     | xsize == 0 || ysize == 0 || zsize == 0 = [[[]]]
  52     | otherwise =
  53         [[[ Cube (h g) i j k (make_values fvs i j k) | i <- [0..xsize]]
  54               | j <- [0..ysize]]
  55               | k <- [0..zsize]]
  56     where
  57       fvs = function_values g
  58       (xsize, ysize, zsize) = dims fvs
  59
  60
  61 -- | Takes a grid and a position as an argument and returns the cube
  62 --   centered on that position. If there is no cube there (i.e. the
  63 --   position is outside of the grid), it will return 'Nothing'.
  64 cube_at :: Grid -> Int -> Int -> Int -> Maybe Cube
  65 cube_at g i j k
  66     | i < 0 = Nothing
  67     | j < 0 = Nothing
  68     | k < 0 = Nothing
  69     | i >= length (cubes g) = Nothing
  70     | j >= length ((cubes g) !! i) = Nothing
  71     | k >= length (((cubes g) !! i) !! j) = Nothing
  72     | otherwise = Just $ (((cubes g) !! i) !! j) !! k
  73
  74
  75
  76 --   The first cube along any axis covers (-h/2, h/2). The second
  77 --   covers (h/2, 3h/2).  The third, (3h/2, 5h/2), and so on.
  78 --
  79 --   We translate the (x,y,z) coordinates forward by 'h' so that the
  80 --   first covers (0, h), the second covers (h, 2h), etc. This makes
  81 --   it easy to figure out which cube contains the given point.
  82 calculate_containing_cube_coordinate :: Grid -> Double -> Int
  83 calculate_containing_cube_coordinate g coord
  84     -- Don't use a cube on the boundary if we can help it.
  85     | coord == delta && (xsize > 0 && ysize > 0 && zsize > 0) = 1
  86     | otherwise = (ceiling ( (coord + delta) / cube_width )) - 1
  87     where
  88       (xsize, ysize, zsize) = dims (function_values g)
  89       delta = (h g)
  90       cube_width = 2 * delta
  91
  92
  93 -- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
  94 --   Since our grid is rectangular, we can figure this out without having
  95 --   to check every cube.
  96 find_containing_cube :: Grid -> Point -> Cube
  97 find_containing_cube g p =
  98     case cube_at g i j k of
  99       Just c -> c
 100       Nothing -> error "No cube contains the given point."
 101     where
 102       (x, y, z) = p
 103       i = calculate_containing_cube_coordinate g x
 104       j = calculate_containing_cube_coordinate g y
 105       k = calculate_containing_cube_coordinate g z
 106
 107
 108 zoom :: Grid -> Int -> Values3D
 109 zoom g scale_factor
 110     | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
 111     | otherwise =
 112         R.traverse arr transExtent (\_ -> newlookup)
 113           where
 114             fvs = function_values g
 115             (xsize, ysize, zsize) = dims fvs
 116             arr = fvs
 117             transExtent = zoom_shape scale_factor
 118             newlookup :: R.DIM3 -> Double
 119             newlookup (R.Z R.:. i R.:. j R.:. k) =
 120                 f p
 121                   where
 122                     i' = fromIntegral i
 123                     j' = fromIntegral j
 124                     k' = fromIntegral k
 125                     p = (i', j', k') :: Point
 126                     c = find_containing_cube g p
 127                     t = head (find_containing_tetrahedra c p)
 128                     f = polynomial t