volume (tetrahedron23 cube) > 0
--- | Given in Sorokina and Zeilfelder, p. 79. 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, (2.6). 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!
prop_c0120_identity1 :: Cube -> Bool
prop_c0120_identity1 cube =
- c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
+ c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t1 0 0 1 2) / 2
where
t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity2 with tetrahedrons 3 and 2.
+-- | 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 t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
+ c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t2 0 0 1 2) / 2
where
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
t2 = tetrahedron2 cube
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 2 and 1.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- 'prop_c0120_identity1' with tetrahedrons 2 and 3.
prop_c0120_identity3 :: Cube -> Bool
prop_c0120_identity3 cube =
- c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
+ c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t3 0 0 1 2) / 2
where
t2 = tetrahedron2 cube
- t1 = tetrahedron1 cube
-
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 4 and 7.
-prop_c0120_identity4 :: Cube -> Bool
-prop_c0120_identity4 cube =
- c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t7 0 0 1 2) / 2
- where
- t4 = tetrahedron4 cube
- t7 = tetrahedron7 cube
-
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 7 and 6.
-prop_c0120_identity5 :: Cube -> Bool
-prop_c0120_identity5 cube =
- c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
- where
- t7 = tetrahedron7 cube
- t6 = tetrahedron6 cube
-
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 6 and 5.
-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
- t6 = tetrahedron6 cube
- t5 = tetrahedron5 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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, (2.6). 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
+
+-- -- | 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
+
+
+-- -- | 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
+
+
+-- | 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 t3 0 1 1 1) / 2
+ c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
where
t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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, (2.6). 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 t3 0 2 1 0) / 2
+ c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t1 0 2 1 0) / 2
where
t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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, (2.6). 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 t3 1 0 1 1) / 2
+ c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
where
t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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, (2.6). 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 t3 1 1 1 0) / 2
+ c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t1 1 1 1 0) / 2
where
t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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, (2.6). 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 t3 2 0 1 0) / 2
+ c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t1 2 0 1 0) / 2
where
t0 = tetrahedron0 cube
- t3 = tetrahedron3 cube
+ t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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!
prop_c0102_identity1 :: Cube -> Bool
prop_c0102_identity1 cube =
- c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
+ c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t3 0 0 2 1) / 2
where
t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 0 1 1 1) / 2
+ c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
where
t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 0 2 0 1) / 2
+ c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t3 0 2 0 1) / 2
where
t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 1 0 1 1) / 2
+ c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
where
t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 1 1 0 1) / 2
+ c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t3 1 1 0 1) / 2
where
t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 2 0 0 1) / 2
+ c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t3 2 0 0 1) / 2
where
t0 = tetrahedron0 cube
- t1 = tetrahedron1 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
+-- fourth indices of c-t6 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! We also switch the third and fourth vertices of t6, but
+-- as of now, why this works is a mystery.
prop_c3000_identity :: Cube -> Bool
prop_c3000_identity cube =
- 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)
+ 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) { v2 = (v3 t6), v3 = (v2 t6) }
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+-- 'prop_c3000_identity'.
prop_c2010_identity :: Cube -> Bool
prop_c2010_identity cube =
- c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 1 0 - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
+ 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+-- 'prop_c3000_identity'.
prop_c2001_identity :: Cube -> Bool
prop_c2001_identity cube =
- c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 0 1 - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
+ 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | Given in Sorokina and Zeilfelder, p. 79.
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+-- 'prop_c3000_identity'.
prop_c1020_identity :: Cube -> Bool
prop_c1020_identity cube =
- c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 2 0 - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
+ 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+-- 'prop_c3000_identity'.
prop_c1002_identity :: Cube -> Bool
prop_c1002_identity cube =
- c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 0 2 - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
+ 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+-- 'prop_c3000_identity'.
prop_c1011_identity :: Cube -> Bool
prop_c1011_identity cube =
- 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)
+ 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
-- This test checks the actual value based on the FunctionValues of the cube.
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 == (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)
where
t0 = tetrahedron0 cube
t6 = tetrahedron6 cube
projects / spline3.git / commitdiff