--- | The Grid module just contains the Grid type and two constructors
--- for it. We hide the main Grid constructor because we don't want
--- to allow instantiation of a grid with h <= 0.
-module Grid
+{-# LANGUAGE BangPatterns #-}
+-- | The Grid module contains the Grid type, its tests, and the 'zoom'
+-- function used to build the interpolation.
+module Grid (
+ cube_at,
+ grid_tests,
+ slow_tests,
+ zoom
+ )
where
-import Test.QuickCheck (Arbitrary(..), Gen, Positive(..))
-
-import Cube (Cube(Cube), find_containing_tetrahedra)
-import FunctionValues
-import Misc (flatten)
-import Point (Point)
-import Tetrahedron (polynomial)
+import qualified Data.Array.Repa as R
+import qualified Data.Array.Repa.Operators.Traversal as R (unsafeTraverse)
+import Test.HUnit (Assertion, assertEqual)
+import Test.Framework (Test, testGroup)
+import Test.Framework.Providers.HUnit (testCase)
+import Test.Framework.Providers.QuickCheck2 (testProperty)
+import Test.QuickCheck ((==>),
+ Arbitrary(..),
+ Gen,
+ Property,
+ choose,
+ vectorOf)
+import Assertions (assertAlmostEqual, assertTrue)
+import Comparisons ((~=))
+import Cube (Cube(Cube),
+ find_containing_tetrahedron,
+ tetrahedra,
+ tetrahedron)
+import Examples (trilinear, trilinear9x9x9, zeros)
+import FunctionValues (make_values, value_at)
+import Point (Point(..))
+import ScaleFactor (ScaleFactor)
+import Tetrahedron (
+ Tetrahedron(v0,v1,v2,v3),
+ c,
+ polynomial,
+ )
import Values (Values3D, dims, empty3d, zoom_shape)
-import qualified Data.Array.Repa as R
-- | Our problem is defined on a Grid. The grid size is given by the
--- positive number h. The function values are the values of the
--- function at the grid points, which are distance h from one
--- another in each direction (x,y,z).
-data Grid = Grid { h :: Double, -- MUST BE GREATER THAN ZERO!
- function_values :: Values3D }
- deriving (Eq, Show)
+-- positive number h, which we have defined to always be 1 for
+-- performance reasons (and simplicity). The function values are the
+-- values of the function at the grid points, which are distance h=1
+-- from one another in each direction (x,y,z).
+data Grid = Grid { function_values :: Values3D }
+ deriving (Show)
instance Arbitrary Grid where
arbitrary = do
- (Positive h') <- arbitrary :: Gen (Positive Double)
- fvs <- arbitrary :: Gen Values3D
- return (make_grid h' fvs)
-
-
--- | The constructor that we want people to use. If we're passed a
--- non-positive grid size, we throw an error.
-make_grid :: Double -> Values3D -> Grid
-make_grid grid_size values
- | grid_size <= 0 = error "grid size must be positive"
- | otherwise = Grid grid_size values
-
+ x_dim <- choose (1, 27)
+ y_dim <- choose (1, 27)
+ z_dim <- choose (1, 27)
+ elements <- vectorOf (x_dim * y_dim * z_dim) (arbitrary :: Gen Double)
+ let new_shape = (R.Z R.:. x_dim R.:. y_dim R.:. z_dim)
+ let fvs = R.fromListUnboxed new_shape elements
+ return $ Grid fvs
--- | Creates an empty grid with grid size 1.
-empty_grid :: Grid
-empty_grid = Grid 1 empty3d
--- | Returns a three-dimensional list of cubes centered on the grid
--- points of g with the appropriate 'FunctionValues'.
-cubes :: Grid -> [[[Cube]]]
-cubes g
- | xsize == 0 || ysize == 0 || zsize == 0 = [[[]]]
- | otherwise =
- [[[ Cube (h g) i j k (make_values fvs i j k) | i <- [0..xsize]]
- | j <- [0..ysize]]
- | k <- [0..zsize]]
- where
- fvs = function_values g
- (xsize, ysize, zsize) = dims fvs
+-- | Takes a grid and a position as an argument and returns the cube
+-- centered on that position. If there is no cube there, well, you
+-- shouldn't have done that. The omitted "otherwise" case actually
+-- does improve performance.
+cube_at :: Grid -> Int -> Int -> Int -> Cube
+cube_at !g !i !j !k =
+ Cube i j k fvs' tet_vol
+ where
+ fvs = function_values g
+ fvs' = make_values fvs i j k
+ tet_vol = 1/24
--- | Takes a grid and a position as an argument and returns the cube
--- centered on that position. If there is no cube there (i.e. the
--- position is outside of the grid), it will return 'Nothing'.
-cube_at :: Grid -> Int -> Int -> Int -> Maybe Cube
-cube_at g i j k
- | i < 0 = Nothing
- | j < 0 = Nothing
- | k < 0 = Nothing
- | i >= length (cubes g) = Nothing
- | j >= length ((cubes g) !! i) = Nothing
- | k >= length (((cubes g) !! i) !! j) = Nothing
- | otherwise = Just $ (((cubes g) !! i) !! j) !! k
-
-
-
--- The first cube along any axis covers (-h/2, h/2). The second
--- covers (h/2, 3h/2). The third, (3h/2, 5h/2), and so on.
+-- The first cube along any axis covers (-1/2, 1/2). The second
+-- covers (1/2, 3/2). The third, (3/2, 5/2), and so on.
--
--- We translate the (x,y,z) coordinates forward by 'h' so that the
--- first covers (0, h), the second covers (h, 2h), etc. This makes
+-- We translate the (x,y,z) coordinates forward by 1/2 so that the
+-- first covers (0, 1), the second covers (1, 2), etc. This makes
-- it easy to figure out which cube contains the given point.
calculate_containing_cube_coordinate :: Grid -> Double -> Int
calculate_containing_cube_coordinate g coord
- -- Don't use a cube on the boundary if we can help it.
- | coord == delta && (xsize > 0 && ysize > 0 && zsize > 0) = 1
- | otherwise = (ceiling ( (coord + delta) / cube_width )) - 1
+ -- Don't use a cube on the boundary if we can help it. This
+ -- returns cube #1 if we would have returned cube #0 and cube #1
+ -- exists.
+ | coord < offset = 0
+ | coord == offset && (xsize > 1 && ysize > 1 && zsize > 1) = 1
+ | otherwise = (ceiling (coord + offset)) - 1
where
(xsize, ysize, zsize) = dims (function_values g)
- delta = (h g)
- cube_width = 2 * delta
+ offset = 1/2
-- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
-- Since our grid is rectangular, we can figure this out without having
-- to check every cube.
find_containing_cube :: Grid -> Point -> Cube
-find_containing_cube g p =
- case cube_at g i j k of
- Just c -> c
- Nothing -> error "No cube contains the given point."
+find_containing_cube g (Point x y z) =
+ cube_at g i j k
where
- (x, y, z) = p
i = calculate_containing_cube_coordinate g x
j = calculate_containing_cube_coordinate g y
k = calculate_containing_cube_coordinate g z
-{-# INLINE zoom_lookup #-}
-zoom_lookup :: Grid -> a -> (R.DIM3 -> Double)
-zoom_lookup g _ = zoom_result g
+zoom_lookup :: Values3D -> ScaleFactor -> a -> (R.DIM3 -> Double)
+zoom_lookup v3d scale_factor _ =
+ zoom_result v3d scale_factor
-{-# INLINE zoom_result #-}
-zoom_result :: Grid -> R.DIM3 -> Double
-zoom_result g (R.Z R.:. i R.:. j R.:. k) =
+zoom_result :: Values3D -> ScaleFactor -> R.DIM3 -> Double
+zoom_result v3d (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
f p
where
- i' = fromIntegral i
- j' = fromIntegral j
- k' = fromIntegral k
- p = (i', j', k') :: Point
- c = find_containing_cube g p
- t = head (find_containing_tetrahedra c p)
+ g = Grid v3d
+ offset = 1/2
+ m' = (fromIntegral m) / (fromIntegral sfx) - offset
+ n' = (fromIntegral n) / (fromIntegral sfy) - offset
+ o' = (fromIntegral o) / (fromIntegral sfz) - offset
+ p = Point m' n' o'
+ cube = find_containing_cube g p
+ t = find_containing_tetrahedron cube p
f = polynomial t
-zoom :: Grid -> Int -> Values3D
-zoom g scale_factor
- | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
+--
+-- Instead of IO, we could get away with a generic monad 'm'
+-- here. However, /we/ only call this function from within IO.
+--
+zoom :: Values3D -> ScaleFactor -> IO Values3D
+zoom v3d scale_factor
+ | xsize == 0 || ysize == 0 || zsize == 0 = return empty3d
| otherwise =
- R.force $ R.traverse arr transExtent (zoom_lookup g)
- where
- arr = function_values g
- (xsize, ysize, zsize) = dims arr
- transExtent = zoom_shape scale_factor
+ R.computeUnboxedP $ R.unsafeTraverse v3d transExtent f
+ where
+ (xsize, ysize, zsize) = dims v3d
+ transExtent = zoom_shape scale_factor
+ f = zoom_lookup v3d scale_factor
+
+
+-- | Check all coefficients of tetrahedron0 belonging to the cube
+-- centered on (1,1,1) with a grid constructed from the trilinear
+-- values. See example one in the paper.
+--
+-- We also verify that the four vertices on face0 of the cube are
+-- in the correct location.
+--
+trilinear_c0_t0_tests :: Test.Framework.Test
+trilinear_c0_t0_tests =
+ testGroup "trilinear c0 t0"
+ [testGroup "coefficients"
+ [testCase "c0030 is correct" test_trilinear_c0030,
+ testCase "c0003 is correct" test_trilinear_c0003,
+ testCase "c0021 is correct" test_trilinear_c0021,
+ testCase "c0012 is correct" test_trilinear_c0012,
+ testCase "c0120 is correct" test_trilinear_c0120,
+ testCase "c0102 is correct" test_trilinear_c0102,
+ testCase "c0111 is correct" test_trilinear_c0111,
+ testCase "c0210 is correct" test_trilinear_c0210,
+ testCase "c0201 is correct" test_trilinear_c0201,
+ testCase "c0300 is correct" test_trilinear_c0300,
+ testCase "c1020 is correct" test_trilinear_c1020,
+ testCase "c1002 is correct" test_trilinear_c1002,
+ testCase "c1011 is correct" test_trilinear_c1011,
+ testCase "c1110 is correct" test_trilinear_c1110,
+ testCase "c1101 is correct" test_trilinear_c1101,
+ testCase "c1200 is correct" test_trilinear_c1200,
+ testCase "c2010 is correct" test_trilinear_c2010,
+ testCase "c2001 is correct" test_trilinear_c2001,
+ testCase "c2100 is correct" test_trilinear_c2100,
+ testCase "c3000 is correct" test_trilinear_c3000],
+
+ testGroup "face0 vertices"
+ [testCase "v0 is correct" test_trilinear_f0_t0_v0,
+ testCase "v1 is correct" test_trilinear_f0_t0_v1,
+ testCase "v2 is correct" test_trilinear_f0_t0_v2,
+ testCase "v3 is correct" test_trilinear_f0_t0_v3]
+ ]
+ where
+ g = Grid trilinear
+ cube = cube_at g 1 1 1
+ t = tetrahedron cube 0
+
+ test_trilinear_c0030 :: Assertion
+ test_trilinear_c0030 =
+ assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
+
+ test_trilinear_c0003 :: Assertion
+ test_trilinear_c0003 =
+ assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
+
+ test_trilinear_c0021 :: Assertion
+ test_trilinear_c0021 =
+ assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
+
+ test_trilinear_c0012 :: Assertion
+ test_trilinear_c0012 =
+ assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
+
+ test_trilinear_c0120 :: Assertion
+ test_trilinear_c0120 =
+ assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
+
+ test_trilinear_c0102 :: Assertion
+ test_trilinear_c0102 =
+ assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
+
+ test_trilinear_c0111 :: Assertion
+ test_trilinear_c0111 =
+ assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
+
+ test_trilinear_c0210 :: Assertion
+ test_trilinear_c0210 =
+ assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
+
+ test_trilinear_c0201 :: Assertion
+ test_trilinear_c0201 =
+ assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
+
+ test_trilinear_c0300 :: Assertion
+ test_trilinear_c0300 =
+ assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
+
+ test_trilinear_c1020 :: Assertion
+ test_trilinear_c1020 =
+ assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
+
+ test_trilinear_c1002 :: Assertion
+ test_trilinear_c1002 =
+ assertAlmostEqual "c1002 is correct" (c t 1 0 0 2) (23/6)
+
+ test_trilinear_c1011 :: Assertion
+ test_trilinear_c1011 =
+ assertAlmostEqual "c1011 is correct" (c t 1 0 1 1) (13/4)
+
+ test_trilinear_c1110 :: Assertion
+ test_trilinear_c1110 =
+ assertAlmostEqual "c1110 is correct" (c t 1 1 1 0) (23/8)
+
+ test_trilinear_c1101 :: Assertion
+ test_trilinear_c1101 =
+ assertAlmostEqual "c1101 is correct" (c t 1 1 0 1) (27/8)
+
+ test_trilinear_c1200 :: Assertion
+ test_trilinear_c1200 =
+ assertAlmostEqual "c1200 is correct" (c t 1 2 0 0) 3
+
+ test_trilinear_c2010 :: Assertion
+ test_trilinear_c2010 =
+ assertAlmostEqual "c2010 is correct" (c t 2 0 1 0) (10/3)
+
+ test_trilinear_c2001 :: Assertion
+ test_trilinear_c2001 =
+ assertAlmostEqual "c2001 is correct" (c t 2 0 0 1) 4
+
+ test_trilinear_c2100 :: Assertion
+ test_trilinear_c2100 =
+ assertAlmostEqual "c2100 is correct" (c t 2 1 0 0) (7/2)
+
+ test_trilinear_c3000 :: Assertion
+ test_trilinear_c3000 =
+ assertAlmostEqual "c3000 is correct" (c t 3 0 0 0) 4
+
+ test_trilinear_f0_t0_v0 :: Assertion
+ test_trilinear_f0_t0_v0 =
+ assertEqual "v0 is correct" (v0 t) (Point 1 1 1)
+
+ test_trilinear_f0_t0_v1 :: Assertion
+ test_trilinear_f0_t0_v1 =
+ assertEqual "v1 is correct" (v1 t) (Point 0.5 1 1)
+
+ test_trilinear_f0_t0_v2 :: Assertion
+ test_trilinear_f0_t0_v2 =
+ assertEqual "v2 is correct" (v2 t) (Point 0.5 0.5 1.5)
+
+ test_trilinear_f0_t0_v3 :: Assertion
+ test_trilinear_f0_t0_v3 =
+ assertEqual "v3 is correct" (v3 t) (Point 0.5 1.5 1.5)
+
+
+test_trilinear_reproduced :: Assertion
+test_trilinear_reproduced =
+ assertTrue "trilinears are reproduced correctly" $
+ and [p (Point i' j' k') ~= value_at trilinear i j k
+ | i <- [0..2],
+ j <- [0..2],
+ k <- [0..2],
+ c0 <- cs,
+ t <- tetrahedra c0,
+ let p = polynomial t,
+ let i' = fromIntegral i,
+ let j' = fromIntegral j,
+ let k' = fromIntegral k]
+ where
+ g = Grid trilinear
+ cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
+
+
+test_zeros_reproduced :: Assertion
+test_zeros_reproduced =
+ assertTrue "the zero function is reproduced correctly" $
+ and [p (Point i' j' k') ~= value_at zeros i j k
+ | i <- [0..2],
+ j <- [0..2],
+ k <- [0..2],
+ let i' = fromIntegral i,
+ let j' = fromIntegral j,
+ let k' = fromIntegral k,
+ c0 <- cs,
+ t0 <- tetrahedra c0,
+ let p = polynomial t0 ]
+ where
+ g = Grid zeros
+ cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
+
+
+-- | Make sure we can reproduce a 9x9x9 trilinear from the 3x3x3 one.
+test_trilinear9x9x9_reproduced :: Assertion
+test_trilinear9x9x9_reproduced =
+ assertTrue "trilinear 9x9x9 is reproduced correctly" $
+ and [p (Point i' j' k') ~= value_at trilinear9x9x9 i j k
+ | i <- [0..8],
+ j <- [0..8],
+ k <- [0..8],
+ t <- tetrahedra c0,
+ let p = polynomial t,
+ let i' = (fromIntegral i) * 0.5,
+ let j' = (fromIntegral j) * 0.5,
+ let k' = (fromIntegral k) * 0.5]
+ where
+ g = Grid trilinear
+ c0 = cube_at g 1 1 1
+
+
+
+prop_cube_indices_never_go_out_of_bounds :: Grid -> Gen Bool
+prop_cube_indices_never_go_out_of_bounds g =
+ do
+ let coordmin = negate (1/2)
+
+ let (xsize, ysize, zsize) = dims $ function_values g
+ let xmax = (fromIntegral xsize) - (1/2)
+ let ymax = (fromIntegral ysize) - (1/2)
+ let zmax = (fromIntegral zsize) - (1/2)
+
+ x <- choose (coordmin, xmax)
+ y <- choose (coordmin, ymax)
+ z <- choose (coordmin, zmax)
+
+ let idx_x = calculate_containing_cube_coordinate g x
+ let idx_y = calculate_containing_cube_coordinate g y
+ let idx_z = calculate_containing_cube_coordinate g z
+
+ return $
+ idx_x >= 0 &&
+ idx_x <= xsize - 1 &&
+ idx_y >= 0 &&
+ idx_y <= ysize - 1 &&
+ idx_z >= 0 &&
+ idx_z <= zsize - 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). Note that the
+-- third and fourth indices of c-t10 have been switched. This is
+-- because we store the triangles oriented such that their volume is
+-- positive. If T and T-tilde share \<v1,v2,v3\> and v0,v0-tilde point
+-- in opposite directions, one of them has to have negative volume!
+prop_c0120_identity :: Grid -> Property
+prop_c0120_identity g =
+ xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
+ c t0 0 1 2 0 ~= (c t0 1 0 2 0 + c t10 1 0 0 2) / 2
+ where
+ fvs = function_values g
+ (xsize, ysize, zsize) = dims fvs
+ cube0 = cube_at g 1 1 1
+ cube1 = cube_at g 0 1 1
+ t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+ t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+-- 'prop_c0120_identity'.
+prop_c0111_identity :: Grid -> Property
+prop_c0111_identity g =
+ xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
+ c t0 0 1 1 1 ~= (c t0 1 0 1 1 + c t10 1 0 1 1) / 2
+ where
+ fvs = function_values g
+ (xsize, ysize, zsize) = dims fvs
+ cube0 = cube_at g 1 1 1
+ cube1 = cube_at g 0 1 1
+ t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+ t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+-- 'prop_c0120_identity'.
+prop_c0201_identity :: Grid -> Property
+prop_c0201_identity g =
+ xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
+ c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t10 1 1 1 0) / 2
+ where
+ fvs = function_values g
+ (xsize, ysize, zsize) = dims fvs
+ cube0 = cube_at g 1 1 1
+ cube1 = cube_at g 0 1 1
+ t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+ t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+-- 'prop_c0120_identity'.
+prop_c0102_identity :: Grid -> Property
+prop_c0102_identity g =
+ xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
+ c t0 0 1 0 2 ~= (c t0 1 0 0 2 + c t10 1 0 2 0) / 2
+ where
+ fvs = function_values g
+ (xsize, ysize, zsize) = dims fvs
+ cube0 = cube_at g 1 1 1
+ cube1 = cube_at g 0 1 1
+ t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+ t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+-- 'prop_c0120_identity'.
+prop_c0210_identity :: Grid -> Property
+prop_c0210_identity g =
+ xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
+ c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t10 1 1 0 1) / 2
+ where
+ fvs = function_values g
+ (xsize, ysize, zsize) = dims fvs
+ cube0 = cube_at g 1 1 1
+ cube1 = cube_at g 0 1 1
+ t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+ t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+-- 'prop_c0120_identity'.
+prop_c0300_identity :: Grid -> Property
+prop_c0300_identity g =
+ xsize >= 3 && ysize >= 3 && zsize >= 3 ==>
+ c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t10 1 2 0 0) / 2
+ where
+ fvs = function_values g
+ (xsize, ysize, zsize) = dims fvs
+ cube0 = cube_at g 1 1 1
+ cube1 = cube_at g 0 1 1
+ t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+ t10 = tetrahedron cube1 10
+
+
+-- | All of the properties from Section (2.9), p. 80. These require a
+-- grid since they refer to two adjacent cubes.
+p80_29_properties :: Test.Framework.Test
+p80_29_properties =
+ testGroup "p. 80, Section (2.9) Properties" [
+ testProperty "c0120 identity" prop_c0120_identity,
+ testProperty "c0111 identity" prop_c0111_identity,
+ testProperty "c0201 identity" prop_c0201_identity,
+ testProperty "c0102 identity" prop_c0102_identity,
+ testProperty "c0210 identity" prop_c0210_identity,
+ testProperty "c0300 identity" prop_c0300_identity ]
+
+
+grid_tests :: Test.Framework.Test
+grid_tests =
+ testGroup "Grid Tests" [
+ trilinear_c0_t0_tests,
+ p80_29_properties,
+ testProperty "cube indices within bounds"
+ prop_cube_indices_never_go_out_of_bounds ]
+
+
+-- Do the slow tests last so we can stop paying attention.
+slow_tests :: Test.Framework.Test
+slow_tests =
+ testGroup "Slow Tests" [
+ testCase "trilinear reproduced" test_trilinear_reproduced,
+ testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced,
+ testCase "zeros reproduced" test_zeros_reproduced ]
projects / spline3.git / blobdiff