X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTests%2FFace.hs;h=0949936da4817df4999d730b6e4dfb2903e273a8;hb=fc12cb4f59a423af3520aa19f8d23ce9b963e3f1;hp=2de60605b255c9d2cfe2a6bc07022de48a7c86b4;hpb=57c28cdd36989e49e59f556a5d7569768ea5ebb3;p=spline3.git

diff --git a/src/Tests/Face.hs b/src/Tests/Face.hs
index 2de6060..0949936 100644
--- a/src/Tests/Face.hs
+++ b/src/Tests/Face.hs
@@ -1,308 +1,633 @@
 module Tests.Face
 where
 
+import Control.Monad (unless)
+import Test.HUnit
 import Test.QuickCheck
 
 import Comparisons
-import Cube (Cube(grid), top)
-import Face (face0,
-             face2,
-             face5,
-             tetrahedron0,
-             tetrahedron1,
-             tetrahedron2,
-             tetrahedron3,
-             tetrahedrons)
-import Grid (Grid(h))
+import Face
+import Grid (Grid(h), make_grid)
+import Point
 import Tetrahedron
 
--- QuickCheck Tests.
-
--- | Since the grid size is necessarily positive, all tetrahedrons
---   (which comprise cubes of positive volume) must have positive volume
---   as well.
-prop_all_volumes_positive :: Cube -> Property
-prop_all_volumes_positive c =
-    (delta > 0) ==> (null nonpositive_volumes)
-    where
-      delta = h (grid c)
-      ts = tetrahedrons c
-      volumes = map volume ts
-      nonpositive_volumes = filter (<= 0) volumes
-
-
--- | Given in Sorokina and Zeilfelder, p. 78.
-prop_cijk1_identity :: Cube -> Bool
-prop_cijk1_identity cube =
-     and [ c t0' i j k 1 ~= (c t1' (i+1) j k 0) * ((b0 t0') (v3 t1')) +
-                            (c t1' i (j+1) k 0) * ((b1 t0') (v3 t1')) +
-                            (c t1' i j (k+1) 0) * ((b2 t0') (v3 t1')) +
-                            (c t1' i j k 1) * ((b3 t0') (v3 t1')) | i <- [0..2],
-                                                                    j <- [0..2],
-                                                                    k <- [0..2],
-                                                                    i + j + k == 2]
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c0120_identity1 :: Cube -> Bool
-prop_c0120_identity1 cube =
-    c t0' 0 1 2 0 ~= (c t0' 0 0 2 1 + c t1' 0 0 2 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c0210_identity1 :: Cube -> Bool
-prop_c0210_identity1 cube =
-    c t0' 0 2 1 0 ~= (c t0' 0 1 1 1 + c t1' 0 1 1 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c0300_identity1 :: Cube -> Bool
-prop_c0300_identity1 cube =
-    c t0' 0 3 0 0 ~= (c t0' 0 2 0 1 + c t1' 0 2 0 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1110_identity :: Cube -> Bool
-prop_c1110_identity cube =
-    c t0' 1 1 1 0 ~= (c t0' 1 0 1 1 + c t1' 1 0 1 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1200_identity1 :: Cube -> Bool
-prop_c1200_identity1 cube =
-    c t0' 1 2 0 0 ~= (c t0' 1 1 0 1 + c t1' 1 1 0 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c2100_identity1 :: Cube -> Bool
-prop_c2100_identity1 cube =
-    c t0' 2 1 0 0 ~= (c t0' 2 0 0 1 + c t1' 2 0 0 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron1 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c0102_identity1 :: Cube -> Bool
-prop_c0102_identity1 cube =
-    c t0' 0 1 0 2 ~= (c t0' 0 0 1 2 + c t3' 0 0 1 2) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t3 = tetrahedron3 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c0201_identity1 :: Cube -> Bool
-prop_c0201_identity1 cube =
-    c t0' 0 2 0 1 ~= (c t0' 0 1 1 1 + c t3' 0 1 1 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t3 = tetrahedron3 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c0300_identity2 :: Cube -> Bool
-prop_c0300_identity2 cube =
-    c t0' 3 0 0 0 ~= (c t0' 0 2 1 0 + c t3' 0 2 1 0) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t3 = tetrahedron3 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1101_identity :: Cube -> Bool
-prop_c1101_identity cube =
-    c t0' 1 1 0 1 ~= (c t0' 1 1 0 1 + c t3' 1 1 0 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t3 = tetrahedron3 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1200_identity2 :: Cube -> Bool
-prop_c1200_identity2 cube =
-    c t0' 1 1 1 0 ~= (c t0' 1 1 1 0 + c t3' 1 1 1 0) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t3 = tetrahedron3 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c2100_identity2 :: Cube -> Bool
-prop_c2100_identity2 cube =
-    c t0' 2 1 0 0 ~= (c t0' 2 0 1 0 + c t3' 2 0 1 0) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t3 = tetrahedron3 (face0 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c3000_identity :: Cube -> Bool
-prop_c3000_identity cube =
-    c t0' 3 0 0 0 ~= c t0' 2 1 0 0 + c t2' 2 1 0 0 - ((c t0' 2 0 1 0 + c t0' 2 0 0 1)/ 2)
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t2 = tetrahedron2 (face5 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c2010_identity :: Cube -> Bool
-prop_c2010_identity cube =
-    c t0' 2 0 1 0 ~= c t0' 1 1 1 0 + c t2' 1 1 1 0 - ((c t0' 1 0 2 0 + c t0' 1 0 1 1)/ 2)
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t2 = tetrahedron2 (face5 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c2001_identity :: Cube -> Bool
-prop_c2001_identity cube =
-    c t0' 2 0 0 1 ~= c t0' 1 1 0 1 + c t2' 1 1 0 1 - ((c t0' 1 0 0 2 + c t0' 1 0 1 1)/ 2)
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t2 = tetrahedron2 (face5 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1020_identity :: Cube -> Bool
-prop_c1020_identity cube =
-    c t0' 1 0 2 0 ~= c t0' 0 1 2 0 + c t2' 0 1 2 0 - ((c t0' 0 0 3 0 + c t0' 0 0 2 1)/ 2)
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t2 = tetrahedron2 (face5 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1002_identity :: Cube -> Bool
-prop_c1002_identity cube =
-    c t0' 1 0 0 2 ~= c t0' 0 1 0 2 + c t2' 0 1 0 2 - ((c t0' 0 0 0 3 + c t0' 0 0 1 2)/ 2)
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t2 = tetrahedron2 (face5 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
-prop_c1011_identity :: Cube -> Bool
-prop_c1011_identity cube =
-    c t0' 1 0 1 1 ~= c t0' 0 1 1 1 + c t2' 0 1 1 1 - ((c t0' 0 0 1 2 + c t0' 0 0 2 1)/ 2)
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t2 = tetrahedron2 (face5 cube)
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
-
-
--- | Given in Sorokina and Zeilfelder, p. 80.
-prop_c0120_identity2 :: Cube -> Bool
-prop_c0120_identity2 cube =
-        c t0' 0 1 2 0 ~= (c t0' 1 0 2 0 + c t1' 1 0 2 0) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron0 (face2 (top cube))
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 80.
-prop_c0102_identity2 :: Cube -> Bool
-prop_c0102_identity2 cube =
-    c t0' 0 1 0 2 ~= (c t0' 1 0 0 2 + c t1' 1 0 0 2) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron0 (face2 (top cube))
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 80.
-prop_c0111_identity :: Cube -> Bool
-prop_c0111_identity cube =
-    c t0' 0 1 1 1 ~= (c t0' 1 0 1 1 + c t1' 1 0 1 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron0 (face2 (top cube))
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 80.
-prop_c0210_identity2 :: Cube -> Bool
-prop_c0210_identity2 cube =
-    c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t1 1 1 1 0) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron0 (face2 (top cube))
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 80.
-prop_c0201_identity2 :: Cube -> Bool
-prop_c0201_identity2 cube =
-    c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t1 1 1 0 1) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron0 (face2 (top cube))
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
-
-
--- | Given in Sorokina and Zeilfelder, p. 80.
-prop_c0300_identity3 :: Cube -> Bool
-prop_c0300_identity3 cube =
-    c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t1 1 2 0 0) / 2
-      where
-        t0 = tetrahedron0 (face0 cube)
-        t1 = tetrahedron0 (face2 (top cube))
-        t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
-        t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+-- HUnit tests.
+
+-- | An HUnit assertion that wraps the almost_equals function. Stolen
+--   from the definition of assertEqual in Test/HUnit/Base.hs.
+assertAlmostEqual :: String -> Double -> Double -> Assertion
+assertAlmostEqual preface expected actual =
+  unless (actual ~= expected) (assertFailure msg)
+ where msg = (if null preface then "" else preface ++ "\n") ++
+             "expected: " ++ show expected ++ "\n but got: " ++ show actual
+
+
+-- | An HUnit assertion that wraps the is_close function. Stolen
+--   from the definition of assertEqual in Test/HUnit/Base.hs.
+assertClose :: String -> Point -> Point -> Assertion
+assertClose preface expected actual =
+  unless (actual `is_close` expected) (assertFailure msg)
+ where msg = (if null preface then "" else preface ++ "\n") ++
+             "expected: " ++ show expected ++ "\n but got: " ++ show actual
+
+
+-- | Values of the function f(x,y,z) = 1 + x + xy + xyz taken at nine
+--   points (hi, hj, jk) with h = 1. From example one in the paper.
+--   Used in the next bunch of tests.
+trilinear :: [[[Double]]]
+trilinear = [ [ [ 1, 2, 3 ],
+                [ 1, 3, 5 ],
+                [ 1, 4, 7 ] ],
+              [ [ 1, 2, 3 ],
+                [ 1, 4, 7 ],
+                [ 1, 6, 11 ] ],
+              [ [ 1, 2, 3 ],
+                [ 1, 5, 9 ],
+                [ 1, 8, 15 ]]]
+
+-- | Check the value of c0030 for any tetrahedron belonging to the
+--   cube centered on (1,1,1) with a grid constructed from the
+--   trilinear values. See example one in the paper.
+-- test_trilinear_c0030 :: Test
+-- test_trilinear_c0030 =
+--     TestCase $ assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0003 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0003 :: Test
+-- test_trilinear_c0003 =
+--     TestCase $ assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0021 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0021 :: Test
+-- test_trilinear_c0021 =
+--     TestCase $ assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0012 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0012 :: Test
+-- test_trilinear_c0012 =
+--     TestCase $ assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0120 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0120 :: Test
+-- test_trilinear_c0120 =
+--     TestCase $ assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0102 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0102 :: Test
+-- test_trilinear_c0102 =
+--     TestCase $ assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0111 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0111 :: Test
+-- test_trilinear_c0111 =
+--     TestCase $ assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0210 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0210 :: Test
+-- test_trilinear_c0210 =
+--     TestCase $ assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0201 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0201 :: Test
+-- test_trilinear_c0201 =
+--     TestCase $ assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c0300 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c0300 :: Test
+-- test_trilinear_c0300 =
+--     TestCase $ assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c1020 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c1020 :: Test
+-- test_trilinear_c1020 =
+--     TestCase $ assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c1002 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c1002 :: Test
+-- test_trilinear_c1002 =
+--     TestCase $ assertAlmostEqual "c1002 is correct" (c t 1 0 0 2) (23/6)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c1011 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c1011 :: Test
+-- test_trilinear_c1011 =
+--     TestCase $ assertAlmostEqual "c1011 is correct" (c t 1 0 1 1) (13/4)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c1110 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c1110 :: Test
+-- test_trilinear_c1110 =
+--     TestCase $ assertAlmostEqual "c1110 is correct" (c t 1 1 1 0) (23/8)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c1101 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c1101 :: Test
+-- test_trilinear_c1101 =
+--     TestCase $ assertAlmostEqual "c1101 is correct" (c t 1 1 0 1) (27/8)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c1200 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c1200 :: Test
+-- test_trilinear_c1200 =
+--     TestCase $ assertAlmostEqual "c1200 is correct" (c t 1 2 0 0) 3
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c2010 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c2010 :: Test
+-- test_trilinear_c2010 =
+--     TestCase $ assertAlmostEqual "c2010 is correct" (c t 2 0 1 0) (10/3)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c2001 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c2001 :: Test
+-- test_trilinear_c2001 =
+--     TestCase $ assertAlmostEqual "c2001 is correct" (c t 2 0 0 1) 4
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c2100 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- test_trilinear_c2100 :: Test
+-- test_trilinear_c2100 =
+--     TestCase $ assertAlmostEqual "c2100 is correct" (c t 2 1 0 0) (7/2)
+--       where
+--         g = make_grid 1 trilinear
+--         cube = cube_at g 1 1 1
+--         t = head (tetrahedrons cube) -- Any one will do.
+
+
+-- -- | Check the value of c3000 for any tetrahedron belonging to the
+-- --   cube centered on (1,1,1) with a grid constructed from the
+-- --   trilinear values. See example one in the paper.
+-- -- test_trilinear_c3000 :: Test
+-- -- test_trilinear_c3000 =
+-- --     TestCase $ assertAlmostEqual "c3000 is correct" (c t 3 0 0 0) 4
+-- --       where
+-- --         g = make_grid 1 trilinear
+-- --         cube = cube_at g 1 1 1
+-- --         t = head (tetrahedrons cube) -- Any one will do.
+
+
+
+-- -- test_trilinear_f0_t0_v0 :: Test
+-- -- test_trilinear_f0_t0_v0 =
+-- --     TestCase $ assertClose "v0 is correct" (v0 t) (0.5, 1.5, 1.5)
+-- --       where
+-- --         g = make_grid 1 trilinear
+-- --         cube = cube_at g 1 1 1
+-- --         t = tetrahedron0 (face0 cube) -- Any one will do.
+
+
+-- -- test_trilinear_f0_t0_v1 :: Test
+-- -- test_trilinear_f0_t0_v1 =
+-- --     TestCase $ assertClose "v1 is correct" (v1 t) (1.5, 1.5, 1.5)
+-- --       where
+-- --         g = make_grid 1 trilinear
+-- --         cube = cube_at g 1 1 1
+-- --         t = tetrahedron0 (face0 cube) -- Any one will do.
+
+
+-- -- test_trilinear_f0_t0_v2 :: Test
+-- -- test_trilinear_f0_t0_v2 =
+-- --     TestCase $ assertClose "v2 is correct" (v2 t) (1, 1, 1.5)
+-- --       where
+-- --         g = make_grid 1 trilinear
+-- --         cube = cube_at g 1 1 1
+-- --         t = tetrahedron0 (face0 cube) -- Any one will do.
+
+
+
+-- -- test_trilinear_f0_t0_v3 :: Test
+-- -- test_trilinear_f0_t0_v3 =
+-- --     TestCase $ assertClose "v3 is correct" (v3 t) (1, 1, 1)
+-- --       where
+-- --         g = make_grid 1 trilinear
+-- --         cube = cube_at g 1 1 1
+-- --         t = tetrahedron0 (face0 cube) -- Any one will do.
+
+
+
+-- face_tests :: [Test]
+face_tests = []
+-- face_tests = [test_trilinear_c0030,
+--               test_trilinear_c0003,
+--               test_trilinear_c0021,
+--               test_trilinear_c0012,
+--               test_trilinear_c0120,
+--               test_trilinear_c0102,
+--               test_trilinear_c0111,
+--               test_trilinear_c0210,
+--               test_trilinear_c0201,
+--               test_trilinear_c0300,
+--               test_trilinear_c1020,
+--               test_trilinear_c1002,
+--               test_trilinear_c1011,
+--               test_trilinear_c1110,
+--               test_trilinear_c1101,
+--               test_trilinear_c1200,
+--               test_trilinear_c2010,
+--               test_trilinear_c2001,
+--               test_trilinear_c2100,
+--               test_trilinear_c3000,
+--               test_trilinear_f0_t0_v0,
+--               test_trilinear_f0_t0_v1,
+--               test_trilinear_f0_t0_v2,
+--               test_trilinear_f0_t0_v3]
+
+
+-- -- QuickCheck Tests.
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 78.
+-- prop_cijk1_identity :: Cube -> Bool
+-- prop_cijk1_identity cube =
+--      and [ c t0' i j k 1 ~= (c t1' (i+1) j k 0) * ((b0 t0') (v3 t1')) +
+--                             (c t1' i (j+1) k 0) * ((b1 t0') (v3 t1')) +
+--                             (c t1' i j (k+1) 0) * ((b2 t0') (v3 t1')) +
+--                             (c t1' i j k 1) * ((b3 t0') (v3 t1')) | i <- [0..2],
+--                                                                     j <- [0..2],
+--                                                                     k <- [0..2],
+--                                                                     i + j + k == 2]
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c0120_identity1 :: Cube -> Bool
+-- prop_c0120_identity1 cube =
+--     c t0' 0 1 2 0 ~= (c t0' 0 0 2 1 + c t1' 0 0 2 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c0210_identity1 :: Cube -> Bool
+-- prop_c0210_identity1 cube =
+--     c t0' 0 2 1 0 ~= (c t0' 0 1 1 1 + c t1' 0 1 1 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c0300_identity1 :: Cube -> Bool
+-- prop_c0300_identity1 cube =
+--     c t0' 0 3 0 0 ~= (c t0' 0 2 0 1 + c t1' 0 2 0 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1110_identity :: Cube -> Bool
+-- prop_c1110_identity cube =
+--     c t0' 1 1 1 0 ~= (c t0' 1 0 1 1 + c t1' 1 0 1 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1200_identity1 :: Cube -> Bool
+-- prop_c1200_identity1 cube =
+--     c t0' 1 2 0 0 ~= (c t0' 1 1 0 1 + c t1' 1 1 0 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c2100_identity1 :: Cube -> Bool
+-- prop_c2100_identity1 cube =
+--     c t0' 2 1 0 0 ~= (c t0' 2 0 0 1 + c t1' 2 0 0 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron1 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c0102_identity1 :: Cube -> Bool
+-- prop_c0102_identity1 cube =
+--     c t0' 0 1 0 2 ~= (c t0' 0 0 1 2 + c t3' 0 0 1 2) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t3 = tetrahedron3 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c0201_identity1 :: Cube -> Bool
+-- prop_c0201_identity1 cube =
+--     c t0' 0 2 0 1 ~= (c t0' 0 1 1 1 + c t3' 0 1 1 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t3 = tetrahedron3 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c0300_identity2 :: Cube -> Bool
+-- prop_c0300_identity2 cube =
+--     c t0' 3 0 0 0 ~= (c t0' 0 2 1 0 + c t3' 0 2 1 0) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t3 = tetrahedron3 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1101_identity :: Cube -> Bool
+-- prop_c1101_identity cube =
+--     c t0' 1 1 0 1 ~= (c t0' 1 1 0 1 + c t3' 1 1 0 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t3 = tetrahedron3 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1200_identity2 :: Cube -> Bool
+-- prop_c1200_identity2 cube =
+--     c t0' 1 1 1 0 ~= (c t0' 1 1 1 0 + c t3' 1 1 1 0) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t3 = tetrahedron3 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c2100_identity2 :: Cube -> Bool
+-- prop_c2100_identity2 cube =
+--     c t0' 2 1 0 0 ~= (c t0' 2 0 1 0 + c t3' 2 0 1 0) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t3 = tetrahedron3 (face0 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c3000_identity :: Cube -> Bool
+-- prop_c3000_identity cube =
+--     c t0' 3 0 0 0 ~= c t0' 2 1 0 0 + c t2' 2 1 0 0 - ((c t0' 2 0 1 0 + c t0' 2 0 0 1)/ 2)
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t2 = tetrahedron2 (face5 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c2010_identity :: Cube -> Bool
+-- prop_c2010_identity cube =
+--     c t0' 2 0 1 0 ~= c t0' 1 1 1 0 + c t2' 1 1 1 0 - ((c t0' 1 0 2 0 + c t0' 1 0 1 1)/ 2)
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t2 = tetrahedron2 (face5 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c2001_identity :: Cube -> Bool
+-- prop_c2001_identity cube =
+--     c t0' 2 0 0 1 ~= c t0' 1 1 0 1 + c t2' 1 1 0 1 - ((c t0' 1 0 0 2 + c t0' 1 0 1 1)/ 2)
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t2 = tetrahedron2 (face5 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1020_identity :: Cube -> Bool
+-- prop_c1020_identity cube =
+--     c t0' 1 0 2 0 ~= c t0' 0 1 2 0 + c t2' 0 1 2 0 - ((c t0' 0 0 3 0 + c t0' 0 0 2 1)/ 2)
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t2 = tetrahedron2 (face5 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1002_identity :: Cube -> Bool
+-- prop_c1002_identity cube =
+--     c t0' 1 0 0 2 ~= c t0' 0 1 0 2 + c t2' 0 1 0 2 - ((c t0' 0 0 0 3 + c t0' 0 0 1 2)/ 2)
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t2 = tetrahedron2 (face5 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79.
+-- prop_c1011_identity :: Cube -> Bool
+-- prop_c1011_identity cube =
+--     c t0' 1 0 1 1 ~= c t0' 0 1 1 1 + c t2' 0 1 1 1 - ((c t0' 0 0 1 2 + c t0' 0 0 2 1)/ 2)
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t2 = tetrahedron2 (face5 cube)
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 80.
+-- prop_c0120_identity2 :: Cube -> Bool
+-- prop_c0120_identity2 cube =
+--         c t0' 0 1 2 0 ~= (c t0' 1 0 2 0 + c t1' 1 0 2 0) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron0 (face2 (top cube))
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 80.
+-- prop_c0102_identity2 :: Cube -> Bool
+-- prop_c0102_identity2 cube =
+--     c t0' 0 1 0 2 ~= (c t0' 1 0 0 2 + c t1' 1 0 0 2) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron0 (face2 (top cube))
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 80.
+-- prop_c0111_identity :: Cube -> Bool
+-- prop_c0111_identity cube =
+--     c t0' 0 1 1 1 ~= (c t0' 1 0 1 1 + c t1' 1 0 1 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron0 (face2 (top cube))
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 80.
+-- prop_c0210_identity2 :: Cube -> Bool
+-- prop_c0210_identity2 cube =
+--     c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t1 1 1 1 0) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron0 (face2 (top cube))
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 80.
+-- prop_c0201_identity2 :: Cube -> Bool
+-- prop_c0201_identity2 cube =
+--     c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t1 1 1 0 1) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron0 (face2 (top cube))
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 80.
+-- prop_c0300_identity3 :: Cube -> Bool
+-- prop_c0300_identity3 cube =
+--     c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t1 1 2 0 0) / 2
+--       where
+--         t0 = tetrahedron0 (face0 cube)
+--         t1 = tetrahedron0 (face2 (top cube))
+--         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
+--         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)