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 qualified Data.Array.Repa as R
   8 import Test.QuickCheck (Arbitrary(..), Gen, Positive(..))
   9
  10 import Cube (Cube(Cube), find_containing_tetrahedra)
  11 import FunctionValues
  12 import Point (Point)
  13 import ScaleFactor
  14 import Tetrahedron (polynomial)
  15 import Values (Values3D, dims, empty3d, zoom_shape)
  16
  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 throw an error.
  64 cube_at :: Grid -> Int -> Int -> Int -> Cube
  65 cube_at g i j k
  66     | i < 0 = error "i < 0 in cube_at"
  67     | j < 0 = error "j < 0 in cube_at"
  68     | k < 0 = error "k < 0 in cube_at"
  69     | otherwise =
  70         let zsize = length (cubes g) in
  71           if k >= zsize then
  72             error "k >= xsize in cube_at"
  73           else
  74             let ysize = length ((cubes g) !! k) in
  75               if j >= ysize then
  76                 error "j >= ysize in cube_at"
  77               else
  78                 let xsize = length (((cubes g) !! k) !! j) in
  79                   if i >= xsize then
  80                     error "i >= xsize in cube_at"
  81                   else
  82                     (((cubes g) !! k) !! j) !! i
  83
  84
  85 --   The first cube along any axis covers (-h/2, h/2). The second
  86 --   covers (h/2, 3h/2).  The third, (3h/2, 5h/2), and so on.
  87 --
  88 --   We translate the (x,y,z) coordinates forward by 'h/2' so that the
  89 --   first covers (0, h), the second covers (h, 2h), etc. This makes
  90 --   it easy to figure out which cube contains the given point.
  91 calculate_containing_cube_coordinate :: Grid -> Double -> Int
  92 calculate_containing_cube_coordinate g coord
  93     -- Don't use a cube on the boundary if we can help it. This
  94     -- returns cube #1 if we would have returned cube #0 and cube #1
  95     -- exists.
  96     | coord == offset && (xsize > 0 && ysize > 0 && zsize > 0) = 1
  97     | otherwise = (ceiling ( (coord + offset) / cube_width )) - 1
  98     where
  99       (xsize, ysize, zsize) = dims (function_values g)
 100       cube_width = (h g)
 101       offset = cube_width / 2
 102
 103
 104 -- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
 105 --   Since our grid is rectangular, we can figure this out without having
 106 --   to check every cube.
 107 find_containing_cube :: Grid -> Point -> Cube
 108 find_containing_cube g p =
 109     cube_at g i j k
 110     where
 111       (x, y, z) = p
 112       i = calculate_containing_cube_coordinate g x
 113       j = calculate_containing_cube_coordinate g y
 114       k = calculate_containing_cube_coordinate g z
 115
 116
 117 {-# INLINE zoom_lookup #-}
 118 zoom_lookup :: Grid -> ScaleFactor -> a -> (R.DIM3 -> Double)
 119 zoom_lookup g scale_factor _ = zoom_result g scale_factor
 120
 121
 122 {-# INLINE zoom_result #-}
 123 zoom_result :: Grid -> ScaleFactor -> R.DIM3 -> Double
 124 zoom_result g (sfx, sfy, sfz) (R.Z R.:. i R.:. j R.:. k) =
 125   f p
 126   where
 127     i' = (fromIntegral i) / (fromIntegral sfx)
 128     j' = (fromIntegral j) / (fromIntegral sfy)
 129     k' = (fromIntegral k) / (fromIntegral sfz)
 130     p  = (i', j', k') :: Point
 131     c = find_containing_cube g p
 132     t = head (find_containing_tetrahedra c p)
 133     f = polynomial t
 134
 135
 136 zoom :: Grid -> ScaleFactor -> Values3D
 137 zoom g scale_factor
 138     | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
 139     | otherwise =
 140         R.force $ R.traverse arr transExtent (zoom_lookup g scale_factor)
 141           where
 142             arr = function_values g
 143             (xsize, ysize, zsize) = dims arr
 144             transExtent = zoom_shape scale_factor