where
import Prelude hiding (LT)
-import Test.QuickCheck
import Cardinal
import Comparisons
-import Cube
+import Cube hiding (i, j, k)
import FunctionValues
-import Misc (all_equal)
-import Tests.FunctionValues ()
+import Misc (all_equal, disjoint)
import Tetrahedron (b0, b1, b2, b3, c, fv,
v0, v1, v2, v3, volume)
-instance Arbitrary Cube where
- arbitrary = do
- (Positive h') <- arbitrary :: Gen (Positive Double)
- i' <- choose (coordmin, coordmax)
- j' <- choose (coordmin, coordmax)
- k' <- choose (coordmin, coordmax)
- fv' <- arbitrary :: Gen FunctionValues
- return (Cube h' i' j' k' fv')
- where
- coordmin = -268435456 -- -(2^29 / 2)
- coordmax = 268435456 -- +(2^29 / 2)
-
-- 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 -> Bool
-prop_all_volumes_positive cube =
- null nonpositive_volumes
- where
- ts = tetrahedrons cube
- volumes = map volume ts
- nonpositive_volumes = filter (<= 0) volumes
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 c =
+ disjoint (front_left_top_tetrahedra c) (front_right_down_tetrahedra c)
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron0_volumes_exact :: Cube -> Bool
-prop_tetrahedron0_volumes_exact cube =
- volume (tetrahedron0 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 c =
+ disjoint (back_left_top_tetrahedra c) (back_right_down_tetrahedra c)
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 c =
+ disjoint (front_left_top_tetrahedra c) (back_right_top_tetrahedra c)
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron1_volumes_exact :: Cube -> Bool
-prop_tetrahedron1_volumes_exact cube =
- volume (tetrahedron1 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron2_volumes_exact :: Cube -> Bool
-prop_tetrahedron2_volumes_exact cube =
- volume (tetrahedron2 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 c =
+ disjoint (front_left_down_tetrahedra c) (back_right_down_tetrahedra c)
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron3_volumes_exact :: Cube -> Bool
-prop_tetrahedron3_volumes_exact cube =
- volume (tetrahedron3 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 c =
+ disjoint (front_left_top_tetrahedra c) (back_left_down_tetrahedra c)
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron4_volumes_exact :: Cube -> Bool
-prop_tetrahedron4_volumes_exact cube =
- volume (tetrahedron4 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 c =
+ disjoint (front_right_top_tetrahedra c) (back_right_down_tetrahedra c)
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron5_volumes_exact :: Cube -> Bool
-prop_tetrahedron5_volumes_exact cube =
- volume (tetrahedron5 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron6_volumes_exact :: Cube -> Bool
-prop_tetrahedron6_volumes_exact cube =
- volume (tetrahedron6 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron7_volumes_exact :: Cube -> Bool
-prop_tetrahedron7_volumes_exact cube =
- volume (tetrahedron7 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron8_volumes_exact :: Cube -> Bool
-prop_tetrahedron8_volumes_exact cube =
- volume (tetrahedron8 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron9_volumes_exact :: Cube -> Bool
-prop_tetrahedron9_volumes_exact cube =
- volume (tetrahedron9 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron10_volumes_exact :: Cube -> Bool
-prop_tetrahedron10_volumes_exact cube =
- volume (tetrahedron10 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron11_volumes_exact :: Cube -> Bool
-prop_tetrahedron11_volumes_exact cube =
- volume (tetrahedron11 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron12_volumes_exact :: Cube -> Bool
-prop_tetrahedron12_volumes_exact cube =
- volume (tetrahedron12 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron13_volumes_exact :: Cube -> Bool
-prop_tetrahedron13_volumes_exact cube =
- volume (tetrahedron13 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron14_volumes_exact :: Cube -> Bool
-prop_tetrahedron14_volumes_exact cube =
- volume (tetrahedron14 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron15_volumes_exact :: Cube -> Bool
-prop_tetrahedron15_volumes_exact cube =
- volume (tetrahedron15 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron16_volumes_exact :: Cube -> Bool
-prop_tetrahedron16_volumes_exact cube =
- volume (tetrahedron16 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron17_volumes_exact :: Cube -> Bool
-prop_tetrahedron17_volumes_exact cube =
- volume (tetrahedron17 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron18_volumes_exact :: Cube -> Bool
-prop_tetrahedron18_volumes_exact cube =
- volume (tetrahedron18 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron19_volumes_exact :: Cube -> Bool
-prop_tetrahedron19_volumes_exact cube =
- volume (tetrahedron19 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron20_volumes_exact :: Cube -> Bool
-prop_tetrahedron20_volumes_exact cube =
- volume (tetrahedron20 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron21_volumes_exact :: Cube -> Bool
-prop_tetrahedron21_volumes_exact cube =
- volume (tetrahedron21 cube) ~~= (1/24)*(delta^(3::Int))
- where
- delta = h cube
-
--- | In fact, since all of the tetrahedra are identical, we should
--- already know their volumes. There's 24 tetrahedra to a cube, so
--- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron22_volumes_exact :: Cube -> Bool
-prop_tetrahedron22_volumes_exact cube =
- volume (tetrahedron22 cube) ~~= (1/24)*(delta^(3::Int))
+-- | Since the grid size is necessarily positive, all tetrahedra
+-- (which comprise cubes of positive volume) must have positive volume
+-- as well.
+prop_all_volumes_positive :: Cube -> Bool
+prop_all_volumes_positive cube =
+ null nonpositive_volumes
where
- delta = h cube
+ ts = tetrahedra cube
+ volumes = map volume ts
+ nonpositive_volumes = filter (<= 0) volumes
-- | In fact, since all of the tetrahedra are identical, we should
-- already know their volumes. There's 24 tetrahedra to a cube, so
-- we'd expect the volume of each one to be (1/24)*h^3.
-prop_tetrahedron23_volumes_exact :: Cube -> Bool
-prop_tetrahedron23_volumes_exact cube =
- volume (tetrahedron23 cube) ~~= (1/24)*(delta^(3::Int))
+prop_all_volumes_exact :: Cube -> Bool
+prop_all_volumes_exact cube =
+ and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
where
delta = h cube
prop_v0_all_equal :: Cube -> Bool
prop_v0_all_equal cube = (v0 t0) == (v0 t1)
where
- t0 = head (tetrahedrons cube) -- Doesn't matter which two we choose.
- t1 = head $ tail (tetrahedrons cube)
-
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron0_volumes_positive :: Cube -> Bool
-prop_tetrahedron0_volumes_positive cube =
- volume (tetrahedron0 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron1_volumes_positive :: Cube -> Bool
-prop_tetrahedron1_volumes_positive cube =
- volume (tetrahedron1 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron2_volumes_positive :: Cube -> Bool
-prop_tetrahedron2_volumes_positive cube =
- volume (tetrahedron2 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron3_volumes_positive :: Cube -> Bool
-prop_tetrahedron3_volumes_positive cube =
- volume (tetrahedron3 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron4_volumes_positive :: Cube -> Bool
-prop_tetrahedron4_volumes_positive cube =
- volume (tetrahedron4 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron5_volumes_positive :: Cube -> Bool
-prop_tetrahedron5_volumes_positive cube =
- volume (tetrahedron5 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron6_volumes_positive :: Cube -> Bool
-prop_tetrahedron6_volumes_positive cube =
- volume (tetrahedron6 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron7_volumes_positive :: Cube -> Bool
-prop_tetrahedron7_volumes_positive cube =
- volume (tetrahedron7 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron8_volumes_positive :: Cube -> Bool
-prop_tetrahedron8_volumes_positive cube =
- volume (tetrahedron8 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron9_volumes_positive :: Cube -> Bool
-prop_tetrahedron9_volumes_positive cube =
- volume (tetrahedron9 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron10_volumes_positive :: Cube -> Bool
-prop_tetrahedron10_volumes_positive cube =
- volume (tetrahedron10 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron11_volumes_positive :: Cube -> Bool
-prop_tetrahedron11_volumes_positive cube =
- volume (tetrahedron11 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron12_volumes_positive :: Cube -> Bool
-prop_tetrahedron12_volumes_positive cube =
- volume (tetrahedron12 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron13_volumes_positive :: Cube -> Bool
-prop_tetrahedron13_volumes_positive cube =
- volume (tetrahedron13 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron14_volumes_positive :: Cube -> Bool
-prop_tetrahedron14_volumes_positive cube =
- volume (tetrahedron14 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron15_volumes_positive :: Cube -> Bool
-prop_tetrahedron15_volumes_positive cube =
- volume (tetrahedron15 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron16_volumes_positive :: Cube -> Bool
-prop_tetrahedron16_volumes_positive cube =
- volume (tetrahedron16 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron17_volumes_positive :: Cube -> Bool
-prop_tetrahedron17_volumes_positive cube =
- volume (tetrahedron17 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron18_volumes_positive :: Cube -> Bool
-prop_tetrahedron18_volumes_positive cube =
- volume (tetrahedron18 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron19_volumes_positive :: Cube -> Bool
-prop_tetrahedron19_volumes_positive cube =
- volume (tetrahedron19 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron20_volumes_positive :: Cube -> Bool
-prop_tetrahedron20_volumes_positive cube =
- volume (tetrahedron20 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron21_volumes_positive :: Cube -> Bool
-prop_tetrahedron21_volumes_positive cube =
- volume (tetrahedron21 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron22_volumes_positive :: Cube -> Bool
-prop_tetrahedron22_volumes_positive cube =
- volume (tetrahedron22 cube) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
--- but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron23_volumes_positive :: Cube -> Bool
-prop_tetrahedron23_volumes_positive cube =
- volume (tetrahedron23 cube) > 0
-
-
--- | Given in Sorokina and Zeilfelder, p. 79, (2.6). It appears that
--- the assumptions in sections (2.6) and (2.7) have been
--- switched. From the description, one would expect 'tetrahedron0'
--- and 'tetrahedron3' to share face \<v0,v1,v2\>; however, we have
--- to use 'tetrahedron0' and 'tetahedron1' for all of the tests in
--- section (2.6). Also note that the third and fourth indices of
--- c-t1 have been switched. This is because we store the triangles
--- oriented such that their volume is positive. If T and T-tilde
--- share \<v0,v1,v2\> and v3,v3-tilde point in opposite directions,
--- one of them has to have negative volume!
+ t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
+ t1 = head $ tail (tetrahedra cube)
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
+-- third and fourth indices of c-t1 have been switched. This is
+-- because we store the triangles oriented such that their volume is
+-- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
+-- in opposite directions, one of them has to have negative volume!
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 1 2) / 2
+ c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
-- 'prop_c0120_identity1' with tetrahedrons 1 and 2.
prop_c0120_identity2 :: Cube -> Bool
prop_c0120_identity2 cube =
- c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t2 0 0 1 2) / 2
+ c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
where
- t1 = tetrahedron1 cube
- t2 = tetrahedron2 cube
-
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
+
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
--- 'prop_c0120_identity1' with tetrahedrons 2 and 3.
+-- 'prop_c0120_identity1' with tetrahedrons 1 and 2.
prop_c0120_identity3 :: Cube -> Bool
prop_c0120_identity3 cube =
- c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t3 0 0 1 2) / 2
+ c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
+ where
+ t1 = tetrahedron cube 1
+ t2 = tetrahedron cube 2
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- 'prop_c0120_identity1' with tetrahedrons 2 and 3.
+prop_c0120_identity4 :: Cube -> Bool
+prop_c0120_identity4 cube =
+ c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
where
- t2 = tetrahedron2 cube
- t3 = tetrahedron3 cube
+ t2 = tetrahedron cube 2
+ t3 = tetrahedron cube 3
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
-- 'prop_c0120_identity1' with tetrahedrons 4 and 5.
--- prop_c0120_identity4 :: Cube -> Bool
--- prop_c0120_identity4 cube =
--- sum [trace ("c_t4_0120: " ++ (show tmp1)) tmp1,
--- trace ("c_t5_0012: " ++ (show tmp2)) tmp2,
--- trace ("c_t5_0102: " ++ (show tmp3)) tmp3,
--- trace ("c_t5_1002: " ++ (show tmp4)) tmp4,
--- trace ("c_t5_0120: " ++ (show tmp5)) tmp5,
--- trace ("c_t5_1020: " ++ (show tmp6)) tmp6,
--- trace ("c_t5_1200: " ++ (show tmp7)) tmp7,
--- trace ("c_t5_0021: " ++ (show tmp8)) tmp8,
--- trace ("c_t5_0201: " ++ (show tmp9)) tmp9,
--- trace ("c_t5_2001: " ++ (show tmp10)) tmp10,
--- trace ("c_t5_0210: " ++ (show tmp11)) tmp11,
--- trace ("c_t5_2010: " ++ (show tmp12)) tmp12,
--- trace ("c_t5_2100: " ++ (show tmp13)) tmp13] == 10
--- -- c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t5 0 0 1 2) / 2
--- where
--- t4 = tetrahedron4 cube
--- t5 = tetrahedron5 cube
--- tmp1 = c t4 0 1 2 0
--- tmp2 = (c t4 0 0 2 1 + c t5 0 0 1 2) / 2
--- tmp3 = (c t4 0 0 2 1 + c t5 0 1 0 2) / 2
--- tmp4 = (c t4 0 0 2 1 + c t5 1 0 0 2) / 2
--- tmp5 = (c t4 0 0 2 1 + c t5 0 1 2 0) / 2
--- tmp6 = (c t4 0 0 2 1 + c t5 1 0 2 0) / 2
--- tmp7 = (c t4 0 0 2 1 + c t5 1 2 0 0) / 2
--- tmp8 = (c t4 0 0 2 1 + c t5 0 0 2 1) / 2
--- tmp9 = (c t4 0 0 2 1 + c t5 0 2 0 1) / 2
--- tmp10 = (c t4 0 0 2 1 + c t5 2 0 0 1) / 2
--- tmp11 = (c t4 0 0 2 1 + c t5 0 2 1 0) / 2
--- tmp12 = (c t4 0 0 2 1 + c t5 2 0 1 0) / 2
--- tmp13 = (c t4 0 0 2 1 + c t5 2 1 0 0) / 2
+prop_c0120_identity5 :: Cube -> Bool
+prop_c0120_identity5 cube =
+ c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
+ where
+ t4 = tetrahedron cube 4
+ t5 = tetrahedron cube 5
-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
-- -- 'prop_c0120_identity1' with tetrahedrons 5 and 6.
--- prop_c0120_identity5 :: Cube -> Bool
--- prop_c0120_identity5 cube =
--- c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t6 0 0 1 2) / 2
--- where
--- t5 = tetrahedron5 cube
--- t6 = tetrahedron6 cube
+prop_c0120_identity6 :: Cube -> Bool
+prop_c0120_identity6 cube =
+ c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
+ where
+ t5 = tetrahedron cube 5
+ t6 = tetrahedron cube 6
-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
-- -- 'prop_c0120_identity1' with tetrahedrons 6 and 7.
--- prop_c0120_identity6 :: Cube -> Bool
--- prop_c0120_identity6 cube =
--- c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t7 0 0 1 2) / 2
--- where
--- t6 = tetrahedron6 cube
--- t7 = tetrahedron7 cube
+prop_c0120_identity7 :: Cube -> Bool
+prop_c0120_identity7 cube =
+ c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
+ where
+ t6 = tetrahedron cube 6
+ t7 = tetrahedron cube 7
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
-- 'prop_c0120_identity1'.
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
+ c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
-- 'prop_c0120_identity1'.
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 1 0) / 2
+ c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
-- 'prop_c0120_identity1'.
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
+ c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
-- 'prop_c0120_identity1'.
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 1 0) / 2
+ c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
-- 'prop_c0120_identity1'.
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 1 0) / 2
+ c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
--- | Given in Sorokina and Zeilfelder, p. 79, (2.7). It appears that
--- the assumptions in sections (2.6) and (2.7) have been
--- switched. From the description, one would expect 'tetrahedron0'
--- and 'tetrahedron1' to share face \<v0,v1,v3\>; however, we have
--- to use 'tetrahedron0' and 'tetahedron3' for all of the tests in
--- section (2.7). Also note that the third and fourth indices of
--- c-t3 have been switched. This is because we store the triangles
--- oriented such that their volume is positive. If T and T-tilde
--- share \<v0,v1,v2\> and v3,v3-tilde point in opposite directions,
--- one of them has to have negative volume!
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
+-- third and fourth indices of c-t3 have been switched. This is
+-- because we store the triangles oriented such that their volume is
+-- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
+-- point in opposite directions, one of them has to have negative
+-- volume!
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 2 1) / 2
+ c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
-- 'prop_c0102_identity1'.
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
+ c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
-- 'prop_c0102_identity1'.
prop_c0300_identity2 :: Cube -> Bool
prop_c0300_identity2 cube =
- c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t3 0 2 0 1) / 2
+ c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
-- 'prop_c0102_identity1'.
prop_c1101_identity :: Cube -> Bool
prop_c1101_identity cube =
- c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
+ c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
-- 'prop_c0102_identity1'.
prop_c1200_identity2 :: Cube -> Bool
prop_c1200_identity2 cube =
- c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t3 1 1 0 1) / 2
+ c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
-- 'prop_c0102_identity1'.
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 0 1) / 2
+ c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
- ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
- ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
- ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
- ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
- ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
k <- [0..2],
i + j + k == 2]
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
--- | The function values at the interior should be the same for all tetrahedra.
+-- | The function values at the interior should be the same for all
+-- tetrahedra.
prop_interior_values_all_identical :: Cube -> Bool
prop_interior_values_all_identical cube =
- all_equal [i0, i1, i2, i3, i4, i5, i6, i7, i8,
- i9, i10, i11, i12, i13, i14, i15, i16,
- i17, i18, i19, i20, i21, i22, i23]
- where
- i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) $ I
- i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) $ I
- i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) $ I
- i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) $ I
- i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) $ I
- i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) $ I
- i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) $ I
- i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) $ I
- i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) $ I
- i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) $ I
- i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) $ I
- i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) $ I
- i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) $ I
- i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) $ I
- i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) $ I
- i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) $ I
- i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) $ I
- i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) $ I
- i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) $ I
- i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) $ I
- i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) $ I
- i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) $ I
- i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) $ I
- i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) $ I
+ all_equal [ eval (Tetrahedron.fv tet) I | tet <- tetrahedra cube ]
-- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
prop_c_tilde_2100_rotation_correct cube =
expr1 == expr2
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
-- What gets computed for c2100 of t6.
expr1 = eval (Tetrahedron.fv t6) $
(1/192)*(FD + RD + LD + BD)
--- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
--- This test checks the actual value based on the FunctionValues of the cube.
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and
+-- Zeilfelder, p. 87. This test checks the actual value based on
+-- the FunctionValues of the cube.
+--
+-- If 'prop_c_tilde_2100_rotation_correct' passes, then this test is
+-- even meaningful!
prop_c_tilde_2100_correct :: Cube -> Bool
prop_c_tilde_2100_correct cube =
- c t6 2 1 0 0 == (3/8)*int
- + (1/12)*(f + r + l + b)
- + (1/64)*(ft + rt + lt + bt)
- + (7/48)*t + (1/48)*d + (1/96)*(fr + fl + br + bl)
- + (1/192)*(fd + rd + ld + bd)
+ c t6 2 1 0 0 == expected
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
fvs = Tetrahedron.fv t0
- int = interior fvs
- f = front fvs
- r = right fvs
- l = left fvs
- b = back fvs
- ft = front_top fvs
- rt = right_top fvs
- lt = left_top fvs
- bt = back_top fvs
- t = top fvs
- d = down fvs
- fr = front_right fvs
- fl = front_left fvs
- br = back_right fvs
- bl = back_left fvs
- fd = front_down fvs
- rd = right_down fvs
- ld = left_down fvs
- bd = back_down fvs
+ expected = eval fvs $
+ (3/8)*I +
+ (1/12)*(F + R + L + B) +
+ (1/64)*(FT + RT + LT + BT) +
+ (7/48)*T +
+ (1/48)*D +
+ (1/96)*(FR + FL + BR + BL) +
+ (1/192)*(FD + RD + LD + BD)
+
-- Tests to check that the correct edges are incidental.
prop_t0_shares_edge_with_t1 :: Cube -> Bool
prop_t0_shares_edge_with_t1 cube =
(v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
where
- t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
prop_t0_shares_edge_with_t3 :: Cube -> Bool
prop_t0_shares_edge_with_t3 cube =
(v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
where
- t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
prop_t0_shares_edge_with_t6 :: Cube -> Bool
prop_t0_shares_edge_with_t6 cube =
(v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
where
- t0 = tetrahedron0 cube
- t6 = tetrahedron6 cube
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
prop_t1_shares_edge_with_t2 :: Cube -> Bool
prop_t1_shares_edge_with_t2 cube =
(v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
where
- t1 = tetrahedron1 cube
- t2 = tetrahedron2 cube
+ t1 = tetrahedron cube 1
+ t2 = tetrahedron cube 2
prop_t1_shares_edge_with_t19 :: Cube -> Bool
prop_t1_shares_edge_with_t19 cube =
(v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
where
- t1 = tetrahedron1 cube
- t19 = tetrahedron19 cube
+ t1 = tetrahedron cube 1
+ t19 = tetrahedron cube 19
prop_t2_shares_edge_with_t3 :: Cube -> Bool
prop_t2_shares_edge_with_t3 cube =
(v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
where
- t1 = tetrahedron1 cube
- t2 = tetrahedron2 cube
+ t1 = tetrahedron cube 1
+ t2 = tetrahedron cube 2
prop_t2_shares_edge_with_t12 :: Cube -> Bool
prop_t2_shares_edge_with_t12 cube =
(v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
where
- t2 = tetrahedron2 cube
- t12 = tetrahedron12 cube
+ t2 = tetrahedron cube 2
+ t12 = tetrahedron cube 12
prop_t3_shares_edge_with_t21 :: Cube -> Bool
prop_t3_shares_edge_with_t21 cube =
(v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
where
- t3 = tetrahedron3 cube
- t21 = tetrahedron21 cube
+ t3 = tetrahedron cube 3
+ t21 = tetrahedron cube 21
prop_t4_shares_edge_with_t5 :: Cube -> Bool
prop_t4_shares_edge_with_t5 cube =
(v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
where
- t4 = tetrahedron4 cube
- t5 = tetrahedron5 cube
+ t4 = tetrahedron cube 4
+ t5 = tetrahedron cube 5
prop_t4_shares_edge_with_t7 :: Cube -> Bool
prop_t4_shares_edge_with_t7 cube =
(v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
where
- t4 = tetrahedron4 cube
- t7 = tetrahedron7 cube
+ t4 = tetrahedron cube 4
+ t7 = tetrahedron cube 7
prop_t4_shares_edge_with_t10 :: Cube -> Bool
prop_t4_shares_edge_with_t10 cube =
(v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
where
- t4 = tetrahedron4 cube
- t10 = tetrahedron10 cube
+ t4 = tetrahedron cube 4
+ t10 = tetrahedron cube 10
prop_t5_shares_edge_with_t6 :: Cube -> Bool
prop_t5_shares_edge_with_t6 cube =
(v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
where
- t5 = tetrahedron5 cube
- t6 = tetrahedron6 cube
+ t5 = tetrahedron cube 5
+ t6 = tetrahedron cube 6
prop_t5_shares_edge_with_t16 :: Cube -> Bool
prop_t5_shares_edge_with_t16 cube =
(v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
where
- t5 = tetrahedron5 cube
- t16 = tetrahedron16 cube
+ t5 = tetrahedron cube 5
+ t16 = tetrahedron cube 16
prop_t6_shares_edge_with_t7 :: Cube -> Bool
prop_t6_shares_edge_with_t7 cube =
(v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
where
- t6 = tetrahedron6 cube
- t7 = tetrahedron7 cube
+ t6 = tetrahedron cube 6
+ t7 = tetrahedron cube 7
prop_t7_shares_edge_with_t20 :: Cube -> Bool
prop_t7_shares_edge_with_t20 cube =
(v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
where
- t7 = tetrahedron7 cube
- t20 = tetrahedron20 cube
+ t7 = tetrahedron cube 7
+ t20 = tetrahedron cube 20
projects / spline3.git / blobdiff