+tetrahedron cube 16 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ rf = right_face cube
+ v1' = Face.center rf
+ v2' = Face.v0 rf
+ v3' = Face.v1 rf
+ fv' = rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 17 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ rf = right_face cube
+ v1' = Face.center rf
+ v2' = Face.v1 rf
+ v3' = Face.v2 rf
+ fv' = rotate ccwz $ rotate cwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 18 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ rf = right_face cube
+ v1' = Face.center rf
+ v2' = Face.v2 rf
+ v3' = Face.v3 rf
+ fv' = rotate ccwz $ rotate cwy
+ $ rotate cwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 19 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ rf = right_face cube
+ v1' = Face.center rf
+ v2' = Face.v3 rf
+ v3' = Face.v0 rf
+ fv' = rotate ccwz $ rotate ccwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 20 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ lf = left_face cube
+ v1' = Face.center lf
+ v2' = Face.v0 lf
+ v3' = Face.v1 lf
+ fv' = rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 21 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ lf = left_face cube
+ v1' = Face.center lf
+ v2' = Face.v1 lf
+ v3' = Face.v2 lf
+ fv' = rotate cwz $ rotate ccwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 22 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ lf = left_face cube
+ v1' = Face.center lf
+ v2' = Face.v2 lf
+ v3' = Face.v3 lf
+ fv' = rotate cwz $ rotate ccwy
+ $ rotate ccwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 23 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ lf = left_face cube
+ v1' = Face.center lf
+ v2' = Face.v3 lf
+ v3' = Face.v0 lf
+ fv' = rotate cwz $ rotate cwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+
+-- Only used in tests, so we don't need the added speed
+-- of Data.Vector.
+tetrahedra :: Cube -> [Tetrahedron]
+tetrahedra cube = [ tetrahedron cube n | n <- [0..23] ]
+
+front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 3) `V.snoc`
+ (tetrahedron cube 6) `V.snoc`
+ (tetrahedron cube 7) `V.snoc`
+ (tetrahedron cube 20) `V.snoc`
+ (tetrahedron cube 21)
+
+front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 2) `V.snoc`
+ (tetrahedron cube 3) `V.snoc`
+ (tetrahedron cube 12) `V.snoc`
+ (tetrahedron cube 15) `V.snoc`
+ (tetrahedron cube 21)
+
+front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 1) `V.snoc`
+ (tetrahedron cube 5) `V.snoc`
+ (tetrahedron cube 6) `V.snoc`
+ (tetrahedron cube 16) `V.snoc`
+ (tetrahedron cube 19)
+
+front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 1) `V.snoc`
+ (tetrahedron cube 2) `V.snoc`
+ (tetrahedron cube 12) `V.snoc`
+ (tetrahedron cube 13) `V.snoc`
+ (tetrahedron cube 18) `V.snoc`
+ (tetrahedron cube 19)
+
+back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 3) `V.snoc`
+ (tetrahedron cube 6) `V.snoc`
+ (tetrahedron cube 7) `V.snoc`
+ (tetrahedron cube 20) `V.snoc`
+ (tetrahedron cube 21)
+
+back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 8) `V.snoc`
+ (tetrahedron cube 11) `V.snoc`
+ (tetrahedron cube 14) `V.snoc`
+ (tetrahedron cube 15) `V.snoc`
+ (tetrahedron cube 22) `V.snoc`
+ (tetrahedron cube 23)
+
+back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 4) `V.snoc`
+ (tetrahedron cube 5) `V.snoc`
+ (tetrahedron cube 9) `V.snoc`
+ (tetrahedron cube 10) `V.snoc`
+ (tetrahedron cube 16) `V.snoc`
+ (tetrahedron cube 17)
+
+back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 8) `V.snoc`
+ (tetrahedron cube 9) `V.snoc`
+ (tetrahedron cube 13) `V.snoc`
+ (tetrahedron cube 14) `V.snoc`
+ (tetrahedron cube 17) `V.snoc`
+ (tetrahedron cube 18)
+
+in_top_half :: Cube -> Point -> Bool
+in_top_half cube (Point _ _ z) =
+ distance_from_top <= distance_from_bottom
+ where
+ distance_from_top = abs $ (zmax cube) - z
+ distance_from_bottom = abs $ (zmin cube) - z
+
+in_front_half :: Cube -> Point -> Bool
+in_front_half cube (Point x _ _) =
+ distance_from_front <= distance_from_back
+ where
+ distance_from_front = abs $ (xmin cube) - x
+ distance_from_back = abs $ (xmax cube) - x
+
+
+in_left_half :: Cube -> Point -> Bool
+in_left_half cube (Point _ y _) =
+ distance_from_left <= distance_from_right
+ where
+ distance_from_left = abs $ (ymin cube) - y
+ distance_from_right = abs $ (ymax cube) - y
+
+
+-- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
+-- contain the given 'Point'. This should be faster than checking
+-- every tetrahedron individually, since we determine which half
+-- (hemisphere?) of the cube the point lies in three times: once in
+-- each dimension. This allows us to eliminate non-candidates
+-- quickly.
+--
+-- This can throw an exception, but the use of 'head' might
+-- save us some unnecessary computations.
+--
+{-# INLINE find_containing_tetrahedron #-}
+find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
+find_containing_tetrahedron cube p =
+ candidates `V.unsafeIndex` (fromJust lucky_idx)
+ where
+ front_half = in_front_half cube p
+ top_half = in_top_half cube p
+ left_half = in_left_half cube p
+
+ candidates :: V.Vector Tetrahedron
+ candidates =
+ if front_half then
+
+ if left_half then
+ if top_half then
+ front_left_top_tetrahedra cube
+ else
+ front_left_down_tetrahedra cube
+ else
+ if top_half then
+ front_right_top_tetrahedra cube
+ else
+ front_right_down_tetrahedra cube
+
+ else -- bottom half
+
+ if left_half then
+ if top_half then
+ back_left_top_tetrahedra cube
+ else
+ back_left_down_tetrahedra cube
+ else
+ if top_half then
+ back_right_top_tetrahedra cube
+ else
+ back_right_down_tetrahedra cube
+
+ -- Use the dot product instead of Euclidean distance here to save
+ -- a sqrt(). So, "distances" below really means "distances
+ -- squared."
+ distances :: V.Vector Double
+ distances = V.map ((dot p) . barycenter) candidates
+
+ shortest_distance :: Double
+ shortest_distance = V.minimum distances
+
+ -- Compute the index of the tetrahedron with the center closest to
+ -- p. This is a bad algorithm, but don't change it! If you make it
+ -- smarter by finding the index of shortest_distance in distances
+ -- (this should give the same answer and avoids recomputing the
+ -- dot product), the program gets slower. Seriously!
+ lucky_idx :: Maybe Int
+ lucky_idx = V.findIndex
+ (\t -> (barycenter t) `dot` p == shortest_distance)
+ candidates
+
+
+
+
+
+
+-- Tests
+
+-- Quickcheck tests.
+
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 cube =
+ disjoint (front_left_top_tetrahedra cube) (front_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 cube =
+ disjoint (back_left_top_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 cube =
+ disjoint (front_left_top_tetrahedra cube) (back_right_top_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 cube =
+ disjoint (front_left_down_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 cube =
+ disjoint (front_left_top_tetrahedra cube) (back_left_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 cube =
+ disjoint (front_right_top_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+
+-- | 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 =
+ all (>= 0) volumes
+ where
+ ts = tetrahedra cube
+ volumes = map volume ts
+
+
+-- | 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.
+prop_all_volumes_exact :: Cube -> Bool
+prop_all_volumes_exact cube =
+ and [volume t ~~= 1/24 | t <- tetrahedra cube]
+
+-- | All tetrahedron should have their v0 located at the center of the
+-- cube.
+prop_v0_all_equal :: Cube -> Bool
+prop_v0_all_equal cube = (v0 t0) == (v0 t1)
+ where
+ 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-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_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
+ where
+ 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 t0 0 0 1 2) / 2
+ where
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- '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 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 = 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_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_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_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 t3 0 1 1 1) / 2
+ where
+ 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 t3 0 2 1 0) / 2
+ where
+ 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 t3 1 0 1 1) / 2
+ where
+ 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 t3 1 1 1 0) / 2
+ where
+ 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 t3 2 0 1 0) / 2
+ where
+ t0 = tetrahedron cube 0
+ t3 = tetrahedron cube 3
+
+
+
+-- | 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 t1 0 0 2 1) / 2
+ where
+ 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 t1 0 1 1 1) / 2
+ where
+ 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 t1 0 2 0 1) / 2
+ where
+ 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 t1 1 0 1 1) / 2
+ where
+ 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 t1 1 1 0 1) / 2
+ where
+ 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 t1 2 0 0 1) / 2
+ where
+ t0 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
+
+
+-- | 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!
+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)
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+
+-- | 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 0 1
+ - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+
+-- | 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 1 0
+ - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+
+-- | 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 0 2
+ - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+
+-- | 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 2 0
+ - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+
+-- | 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)
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+
+-- | 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 [ eval (function_values tet) I | tet <- tetrahedra cube ]
+
+
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
+-- This test checks the rotation works as expected.
+prop_c_tilde_2100_rotation_correct :: Cube -> Bool
+prop_c_tilde_2100_rotation_correct cube =
+ expr1 ~= expr2