--- | 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.
+{-# 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,
- make_grid,
slow_tests,
zoom
)
where
-import Data.Array (Array, array, (!))
import qualified Data.Array.Repa as R
-import Test.HUnit
+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, Positive(..), choose)
-
-import Assertions
-import Comparisons
+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
-import FunctionValues
-import Point (Point)
-import ScaleFactor
-import Tetrahedron (Tetrahedron, c, polynomial, v0, v1, v2, v3)
-import ThreeDimensional
+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)
-type CubeGrid = Array (Int,Int,Int) Cube
-
-
-- | 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,
- cube_grid :: CubeGrid }
- 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 (cubes grid_size values)
-
-
--- | Returns a three-dimensional array of cubes centered on the grid
--- points (h*i, h*j, h*k) with the appropriate 'FunctionValues'.
-cubes :: Double -> Values3D -> CubeGrid
-cubes delta fvs
- = array (lbounds, ubounds)
- [ ((i,j,k), cube_ijk)
- | i <- [0..xmax],
- j <- [0..ymax],
- k <- [0..zmax],
- let tet_vol = (1/24)*(delta^(3::Int)),
- let cube_ijk =
- Cube delta i j k (make_values fvs i j k) tet_vol]
- where
- xmax = xsize - 1
- ymax = ysize - 1
- zmax = zsize - 1
- lbounds = (0, 0, 0)
- ubounds = (xmax, ymax, zmax)
- (xsize, ysize, zsize) = dims fvs
+ 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
+
-- | 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 throw an error.
+-- 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
- | i < 0 = error "i < 0 in cube_at"
- | i >= xsize = error "i >= xsize in cube_at"
- | j < 0 = error "j < 0 in cube_at"
- | j >= ysize = error "j >= ysize in cube_at"
- | k < 0 = error "k < 0 in cube_at"
- | k >= zsize = error "k >= zsize in cube_at"
- | otherwise = (cube_grid g) ! (i,j,k)
- where
- fvs = function_values g
- (xsize, ysize, zsize) = dims fvs
-
--- 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.
+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
+
+
+-- 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/2' 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
-- exists.
| coord < offset = 0
| coord == offset && (xsize > 1 && ysize > 1 && zsize > 1) = 1
- | otherwise = (ceiling ( (coord + offset) / cube_width )) - 1
+ | otherwise = (ceiling (coord + offset)) - 1
where
(xsize, ysize, zsize) = dims (function_values g)
- cube_width = (h g)
- offset = cube_width / 2
+ 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 =
+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 -> ScaleFactor -> a -> (R.DIM3 -> Double)
-zoom_lookup g scale_factor _ =
- zoom_result g scale_factor
+zoom_lookup :: Values3D -> ScaleFactor -> a -> (R.DIM3 -> Double)
+zoom_lookup v3d scale_factor _ =
+ zoom_result v3d scale_factor
-{-# INLINE zoom_result #-}
-zoom_result :: Grid -> ScaleFactor -> R.DIM3 -> Double
-zoom_result g (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
+zoom_result :: Values3D -> ScaleFactor -> R.DIM3 -> Double
+zoom_result v3d (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
f p
where
- offset = (h g)/2
+ 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 = (m', n', o') :: Point
+ p = Point m' n' o'
cube = find_containing_cube g p
- -- Figure out i,j,k without importing those functions.
t = find_containing_tetrahedron cube p
f = polynomial t
-
-zoom :: Grid -> ScaleFactor -> Values3D
-zoom g scale_factor
- | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
- | otherwise =
- R.force $ R.unsafeTraverse arr transExtent (zoom_lookup g scale_factor)
- where
- arr = function_values g
- (xsize, ysize, zsize) = dims arr
- transExtent = zoom_shape scale_factor
+--
+-- 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.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
testCase "v3 is correct" test_trilinear_f0_t0_v3]
]
where
- g = make_grid 1 trilinear
+ g = Grid trilinear
cube = cube_at g 1 1 1
t = tetrahedron cube 0
test_trilinear_f0_t0_v0 :: Assertion
test_trilinear_f0_t0_v0 =
- assertEqual "v0 is correct" (v0 t) (1, 1, 1)
+ 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) (0.5, 1, 1)
+ 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) (0.5, 0.5, 1.5)
+ assertEqual "v2 is correct" (v2 t) (Point 0.5 0.5 1.5)
test_trilinear_f0_t0_v3 :: Assertion
test_trilinear_f0_t0_v3 =
- assertClose "v3 is correct" (v3 t) (0.5, 1.5, 1.5)
+ 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 (i', j', k') ~= value_at trilinear i j k
+ 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 = make_grid 1 trilinear
- c0 = cube_at g 1 1 1
+ 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 (i', j', k') ~= value_at zeros i j k
+ 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]
+ let k' = fromIntegral k,
+ c0 <- cs,
+ t0 <- tetrahedra c0,
+ let p = polynomial t0 ]
where
- g = make_grid 1 zeros
- c0 = cube_at g 1 1 1
- t0 = tetrahedron c0 0
- p = polynomial t0
+ 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 (i', j', k') ~= value_at trilinear9x9x9 i j k
+ and [p (Point i' j' k') ~= value_at trilinear9x9x9 i j k
| i <- [0..8],
j <- [0..8],
k <- [0..8],
let j' = (fromIntegral j) * 0.5,
let k' = (fromIntegral k) * 0.5]
where
- g = make_grid 1 trilinear
+ g = Grid trilinear
c0 = cube_at g 1 1 1
--- | The point 'p' in this test lies on the boundary of tetrahedra 12 and 15.
--- However, the 'contains_point' test fails due to some numerical innacuracy.
--- This bug should have been fixed by setting a positive tolerance level.
---
--- Example from before the fix:
---
--- b1 (tetrahedron c 20) (0, 17.5, 0.5)
--- -0.0
---
-test_tetrahedra_collision_sensitivity :: Assertion
-test_tetrahedra_collision_sensitivity =
- assertTrue "tetrahedron collision tests isn't too sensitive" $
- contains_point t20 p
- where
- g = make_grid 1 naturals_1d
- cube = cube_at g 0 18 0
- p = (0, 17.5, 0.5) :: Point
- t20 = tetrahedron cube 20
-
prop_cube_indices_never_go_out_of_bounds :: Grid -> Gen Bool
prop_cube_indices_never_go_out_of_bounds g =
do
- let delta = Grid.h g
- let coordmin = negate (delta/2)
+ let coordmin = negate (1/2)
let (xsize, ysize, zsize) = dims $ function_values g
- let xmax = delta*(fromIntegral xsize) - (delta/2)
- let ymax = delta*(fromIntegral ysize) - (delta/2)
- let zmax = delta*(fromIntegral zsize) - (delta/2)
+ 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)
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,
- testCase "tetrahedra collision test isn't too sensitive"
- test_tetrahedra_collision_sensitivity,
- testCase "trilinear reproduced" test_trilinear_reproduced,
- testCase "zeros reproduced" test_zeros_reproduced ]
+ 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" [
- testProperty "cube indices within bounds"
- prop_cube_indices_never_go_out_of_bounds,
- testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced ]
+ testCase "trilinear reproduced" test_trilinear_reproduced,
+ testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced,
+ testCase "zeros reproduced" test_zeros_reproduced ]
projects / spline3.git / blobdiff