+-- Face stuff.
+
+-- | The top (in the direction of z) face of the cube.
+top_face :: Cube -> Face.Face
+top_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2)*(h cube)
+ v0' = (center cube) + (delta, -delta, delta)
+ v1' = (center cube) + (delta, delta, delta)
+ v2' = (center cube) + (-delta, delta, delta)
+ v3' = (center cube) + (-delta, -delta, delta)
+
+
+
+-- | The back (in the direction of x) face of the cube.
+back_face :: Cube -> Face.Face
+back_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2)*(h cube)
+ v0' = (center cube) + (delta, -delta, -delta)
+ v1' = (center cube) + (delta, delta, -delta)
+ v2' = (center cube) + (delta, delta, delta)
+ v3' = (center cube) + (delta, -delta, delta)
+
+
+-- The bottom face (in the direction of -z) of the cube.
+down_face :: Cube -> Face.Face
+down_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2)*(h cube)
+ v0' = (center cube) + (-delta, -delta, -delta)
+ v1' = (center cube) + (-delta, delta, -delta)
+ v2' = (center cube) + (delta, delta, -delta)
+ v3' = (center cube) + (delta, -delta, -delta)
+
+
+
+-- | The front (in the direction of -x) face of the cube.
+front_face :: Cube -> Face.Face
+front_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2)*(h cube)
+ v0' = (center cube) + (-delta, -delta, delta)
+ v1' = (center cube) + (-delta, delta, delta)
+ v2' = (center cube) + (-delta, delta, -delta)
+ v3' = (center cube) + (-delta, -delta, -delta)
+
+-- | The left (in the direction of -y) face of the cube.
+left_face :: Cube -> Face.Face
+left_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2)*(h cube)
+ v0' = (center cube) + (delta, -delta, delta)
+ v1' = (center cube) + (-delta, -delta, delta)
+ v2' = (center cube) + (-delta, -delta, -delta)
+ v3' = (center cube) + (delta, -delta, -delta)
+
+
+-- | The right (in the direction of y) face of the cube.
+right_face :: Cube -> Face.Face
+right_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2)*(h cube)
+ v0' = (center cube) + (-delta, delta, delta)
+ v1' = (center cube) + (delta, delta, delta)
+ v2' = (center cube) + (delta, delta, -delta)
+ v3' = (center cube) + (-delta, delta, -delta)
+
+
+tetrahedron :: Cube -> Int -> Tetrahedron
+
+tetrahedron cube 0 =
+ Tetrahedron (fv cube) v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v0 (front_face cube)
+ v3' = Face.v1 (front_face cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 1 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v1 (front_face cube)
+ v3' = Face.v2 (front_face cube)
+ fv' = rotate ccwx (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 2 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v2 (front_face cube)
+ v3' = Face.v3 (front_face cube)
+ fv' = rotate ccwx $ rotate ccwx $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 3 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v3 (front_face cube)
+ v3' = Face.v0 (front_face cube)
+ fv' = rotate cwx (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 4 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v0 (top_face cube)
+ v3' = Face.v1 (top_face cube)
+ fv' = rotate cwy (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 5 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v1 (top_face cube)
+ v3' = Face.v2 (top_face cube)
+ fv' = rotate cwy $ rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 6 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v2 (top_face cube)
+ v3' = Face.v3 (top_face cube)
+ fv' = rotate cwy $ rotate cwz
+ $ rotate cwz
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 7 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v3 (top_face cube)
+ v3' = Face.v0 (top_face cube)
+ fv' = rotate cwy $ rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 8 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v0 (back_face cube)
+ v3' = Face.v1 (back_face cube)
+ fv' = rotate cwy $ rotate cwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 9 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v1 (back_face cube)
+ v3' = Face.v2 (back_face cube)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate cwx
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 10 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v2 (back_face cube)
+ v3' = Face.v3 (back_face cube)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate cwx
+ $ rotate cwx
+ $ fv cube
+
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 11 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v3 (back_face cube)
+ v3' = Face.v0 (back_face cube)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate ccwx
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 12 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v0 (down_face cube)
+ v3' = Face.v1 (down_face cube)
+ fv' = rotate ccwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 13 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v1 (down_face cube)
+ v3' = Face.v2 (down_face cube)
+ fv' = rotate ccwy $ rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 14 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v2 (down_face cube)
+ v3' = Face.v3 (down_face cube)
+ fv' = rotate ccwy $ rotate ccwz
+ $ rotate ccwz
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 15 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v3 (down_face cube)
+ v3' = Face.v0 (down_face cube)
+ fv' = rotate ccwy $ rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 16 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (right_face cube)
+ v2' = Face.v0 (right_face cube)
+ v3' = Face.v1 (right_face cube)
+ fv' = rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 17 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (right_face cube)
+ v2' = Face.v1 (right_face cube)
+ v3' = Face.v2 (right_face cube)
+ 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
+ v1' = center (right_face cube)
+ v2' = Face.v2 (right_face cube)
+ v3' = Face.v3 (right_face cube)
+ 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
+ v1' = center (right_face cube)
+ v2' = Face.v3 (right_face cube)
+ v3' = Face.v0 (right_face cube)
+ 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
+ v1' = center (left_face cube)
+ v2' = Face.v0 (left_face cube)
+ v3' = Face.v1 (left_face cube)
+ fv' = rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 21 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (left_face cube)
+ v2' = Face.v1 (left_face cube)
+ v3' = Face.v2 (left_face cube)
+ 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
+ v1' = center (left_face cube)
+ v2' = Face.v2 (left_face cube)
+ v3' = Face.v3 (left_face cube)
+ 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
+ v1' = center (left_face cube)
+ v2' = Face.v3 (left_face cube)
+ v3' = Face.v0 (left_face cube)
+ fv' = rotate cwz $ rotate cwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+-- Feels dirty, but whatever.
+tetrahedron _ _ = error "asked for a nonexistent tetrahedron"
+
+
+-- 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 (_,_,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 (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 (_,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.
+--
+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 =
+ 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 'distance' here to save a
+ -- sqrt(). So, "distances" below really means "distances squared."
+ distances = V.map ((dot p) . center) candidates
+ shortest_distance = V.minimum distances
+ lucky_idx = V.findIndex
+ (\t -> (center 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 =
+ null nonpositive_volumes
+ where
+ 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_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
+
+-- | 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
+
+
+
+-- | 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 = tetrahedron cube 0
+ t1 = tetrahedron cube 1
+
+
+-- | 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
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+
+ -- What gets computed for c2100 of t6.
+ expr1 = eval (function_values t6) $
+ (3/8)*I +
+ (1/12)*(T + R + L + D) +
+ (1/64)*(FT + FR + FL + FD) +
+ (7/48)*F +
+ (1/48)*B +
+ (1/96)*(RT + LD + LT + RD) +
+ (1/192)*(BT + BR + BL + BD)
+
+ -- What should be computed for c2100 of t6.
+ expr2 = eval (function_values t0) $
+ (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)
+
+
+-- | 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 == expected
+ where
+ t0 = tetrahedron cube 0
+ t6 = tetrahedron cube 6
+ fvs = function_values t0
+ 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = 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 = tetrahedron cube 5
+ t16 = tetrahedron cube 16