-module Cube
+module Cube (
+ Cube(..),
+ cube_properties,
+ find_containing_tetrahedron,
+ tetrahedra,
+ tetrahedron
+ )
where
-import Data.List ( (\\) )
+import Data.Maybe (fromJust)
+import qualified Data.Vector as V (
+ Vector,
+ findIndex,
+ map,
+ minimum,
+ singleton,
+ snoc,
+ unsafeIndex
+ )
+import Prelude hiding (LT)
+import Test.Framework (Test, testGroup)
+import Test.Framework.Providers.QuickCheck2 (testProperty)
import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
import Cardinal
+import Comparisons ((~=), (~~=))
import qualified Face (Face(Face, v0, v1, v2, v3))
import FunctionValues
+import Misc (all_equal, disjoint)
import Point
-import Tetrahedron hiding (c)
+import Tetrahedron (
+ Tetrahedron(..),
+ c,
+ b0,
+ b1,
+ b2,
+ b3,
+ volume
+ )
import ThreeDimensional
data Cube = Cube { h :: Double,
tetrahedron :: Cube -> Int -> Tetrahedron
tetrahedron c 0 =
- Tetrahedron (Cube.fv c) v0' v1' v2' v3' vol 0
+ Tetrahedron (fv c) v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (front_face c)
vol = tetrahedra_volume c
tetrahedron c 1 =
- Tetrahedron fv' v0' v1' v2' v3' vol 1
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (front_face c)
v2' = Face.v1 (front_face c)
v3' = Face.v2 (front_face c)
- fv' = rotate ccwx (Cube.fv c)
+ fv' = rotate ccwx (fv c)
vol = tetrahedra_volume c
tetrahedron c 2 =
- Tetrahedron fv' v0' v1' v2' v3' vol 2
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (front_face c)
v2' = Face.v2 (front_face c)
v3' = Face.v3 (front_face c)
- fv' = rotate ccwx $ rotate ccwx $ Cube.fv c
+ fv' = rotate ccwx $ rotate ccwx $ fv c
vol = tetrahedra_volume c
tetrahedron c 3 =
- Tetrahedron fv' v0' v1' v2' v3' vol 3
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (front_face c)
v2' = Face.v3 (front_face c)
v3' = Face.v0 (front_face c)
- fv' = rotate cwx (Cube.fv c)
+ fv' = rotate cwx (fv c)
vol = tetrahedra_volume c
tetrahedron c 4 =
- Tetrahedron fv' v0' v1' v2' v3' vol 4
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (top_face c)
v2' = Face.v0 (top_face c)
v3' = Face.v1 (top_face c)
- fv' = rotate cwy (Cube.fv c)
+ fv' = rotate cwy (fv c)
vol = tetrahedra_volume c
tetrahedron c 5 =
- Tetrahedron fv' v0' v1' v2' v3' vol 5
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (top_face c)
v2' = Face.v1 (top_face c)
v3' = Face.v2 (top_face c)
- fv' = rotate cwy $ rotate cwz $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate cwy $ rotate cwz $ fv c
vol = tetrahedra_volume c
tetrahedron c 6 =
- Tetrahedron fv' v0' v1' v2' v3' vol 6
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (top_face c)
v3' = Face.v3 (top_face c)
fv' = rotate cwy $ rotate cwz
$ rotate cwz
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 7 =
- Tetrahedron fv' v0' v1' v2' v3' vol 7
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (top_face c)
v2' = Face.v3 (top_face c)
v3' = Face.v0 (top_face c)
- fv' = rotate cwy $ rotate ccwz $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate cwy $ rotate ccwz $ fv c
vol = tetrahedra_volume c
tetrahedron c 8 =
- Tetrahedron fv' v0' v1' v2' v3' vol 8
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (back_face c)
v2' = Face.v0 (back_face c)
v3' = Face.v1 (back_face c)
- fv' = rotate cwy $ rotate cwy $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate cwy $ rotate cwy $ fv c
vol = tetrahedra_volume c
tetrahedron c 9 =
- Tetrahedron fv' v0' v1' v2' v3' vol 9
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (back_face c)
v3' = Face.v2 (back_face c)
fv' = rotate cwy $ rotate cwy
$ rotate cwx
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 10 =
- Tetrahedron fv' v0' v1' v2' v3' vol 10
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (back_face c)
fv' = rotate cwy $ rotate cwy
$ rotate cwx
$ rotate cwx
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 11 =
- Tetrahedron fv' v0' v1' v2' v3' vol 11
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (back_face c)
v3' = Face.v0 (back_face c)
fv' = rotate cwy $ rotate cwy
$ rotate ccwx
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 12 =
- Tetrahedron fv' v0' v1' v2' v3' vol 12
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (down_face c)
v2' = Face.v0 (down_face c)
v3' = Face.v1 (down_face c)
- fv' = rotate ccwy (Tetrahedron.fv (tetrahedron c 0))
+ fv' = rotate ccwy $ fv c
vol = tetrahedra_volume c
tetrahedron c 13 =
- Tetrahedron fv' v0' v1' v2' v3' vol 13
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (down_face c)
v2' = Face.v1 (down_face c)
v3' = Face.v2 (down_face c)
- fv' = rotate ccwy $ rotate ccwz $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate ccwy $ rotate ccwz $ fv c
vol = tetrahedra_volume c
tetrahedron c 14 =
- Tetrahedron fv' v0' v1' v2' v3' vol 14
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (down_face c)
v3' = Face.v3 (down_face c)
fv' = rotate ccwy $ rotate ccwz
$ rotate ccwz
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 15 =
- Tetrahedron fv' v0' v1' v2' v3' vol 15
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (down_face c)
v2' = Face.v3 (down_face c)
v3' = Face.v0 (down_face c)
- fv' = rotate ccwy $ rotate cwz $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate ccwy $ rotate cwz $ fv c
vol = tetrahedra_volume c
tetrahedron c 16 =
- Tetrahedron fv' v0' v1' v2' v3' vol 16
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (right_face c)
v2' = Face.v0 (right_face c)
v3' = Face.v1 (right_face c)
- fv' = rotate ccwz (Tetrahedron.fv (tetrahedron c 0))
+ fv' = rotate ccwz $ fv c
vol = tetrahedra_volume c
tetrahedron c 17 =
- Tetrahedron fv' v0' v1' v2' v3' vol 17
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (right_face c)
v2' = Face.v1 (right_face c)
v3' = Face.v2 (right_face c)
- fv' = rotate ccwz $ rotate cwy $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate ccwz $ rotate cwy $ fv c
vol = tetrahedra_volume c
tetrahedron c 18 =
- Tetrahedron fv' v0' v1' v2' v3' vol 18
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (right_face c)
v3' = Face.v3 (right_face c)
fv' = rotate ccwz $ rotate cwy
$ rotate cwy
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 19 =
- Tetrahedron fv' v0' v1' v2' v3' vol 19
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (right_face c)
v2' = Face.v3 (right_face c)
v3' = Face.v0 (right_face c)
fv' = rotate ccwz $ rotate ccwy
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 20 =
- Tetrahedron fv' v0' v1' v2' v3' vol 20
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (left_face c)
v2' = Face.v0 (left_face c)
v3' = Face.v1 (left_face c)
- fv' = rotate cwz (Tetrahedron.fv (tetrahedron c 0))
+ fv' = rotate cwz $ fv c
vol = tetrahedra_volume c
tetrahedron c 21 =
- Tetrahedron fv' v0' v1' v2' v3' vol 21
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (left_face c)
v2' = Face.v1 (left_face c)
v3' = Face.v2 (left_face c)
- fv' = rotate cwz $ rotate ccwy $ Tetrahedron.fv (tetrahedron c 0)
+ fv' = rotate cwz $ rotate ccwy $ fv c
vol = tetrahedra_volume c
tetrahedron c 22 =
- Tetrahedron fv' v0' v1' v2' v3' vol 22
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (left_face c)
v3' = Face.v3 (left_face c)
fv' = rotate cwz $ rotate ccwy
$ rotate ccwy
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
tetrahedron c 23 =
- Tetrahedron fv' v0' v1' v2' v3' vol 23
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
v0' = center c
v1' = center (left_face c)
v2' = Face.v3 (left_face c)
v3' = Face.v0 (left_face c)
fv' = rotate cwz $ rotate cwy
- $ Tetrahedron.fv (tetrahedron c 0)
+ $ fv c
vol = tetrahedra_volume c
-- 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 c =
- [ tetrahedron c n | n <- [0..23] ]
-
--- | All completely contained in the front half of the cube.
-front_half_tetrahedra :: Cube -> [Tetrahedron]
-front_half_tetrahedra c =
- [ tetrahedron c n | n <- [0,1,2,3,6,12,19,21] ]
-
--- | All tetrahedra completely contained in the top half of the cube.
-top_half_tetrahedra :: Cube -> [Tetrahedron]
-top_half_tetrahedra c =
- [ tetrahedron c n | n <- [4,5,6,7,0,10,16,20] ]
-
--- | All tetrahedra completely contained in the back half of the cube.
-back_half_tetrahedra :: Cube -> [Tetrahedron]
-back_half_tetrahedra c =
- [ tetrahedron c n | n <- [8,9,10,11,4,14,17,23] ]
-
--- | All tetrahedra completely contained in the down half of the cube.
-down_half_tetrahedra :: Cube -> [Tetrahedron]
-down_half_tetrahedra c =
- [ tetrahedron c n | n <- [12,13,14,15,2,8,18,22] ]
-
--- | All tetrahedra completely contained in the right half of the cube.
-right_half_tetrahedra :: Cube -> [Tetrahedron]
-right_half_tetrahedra c =
- [ tetrahedron c n | n <- [16,17,18,19,1,5,9,13] ]
-
--- | All tetrahedra completely contained in the left half of the cube.
-left_half_tetrahedra :: Cube -> [Tetrahedron]
-left_half_tetrahedra c =
- [ tetrahedron c n | n <- [20,21,22,23,3,7,11,15] ]
+tetrahedra c = [ tetrahedron c n | n <- [0..23] ]
+
+front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_top_tetrahedra c =
+ V.singleton (tetrahedron c 0) `V.snoc`
+ (tetrahedron c 3) `V.snoc`
+ (tetrahedron c 6) `V.snoc`
+ (tetrahedron c 7) `V.snoc`
+ (tetrahedron c 20) `V.snoc`
+ (tetrahedron c 21)
+
+front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_down_tetrahedra c =
+ V.singleton (tetrahedron c 0) `V.snoc`
+ (tetrahedron c 2) `V.snoc`
+ (tetrahedron c 3) `V.snoc`
+ (tetrahedron c 12) `V.snoc`
+ (tetrahedron c 15) `V.snoc`
+ (tetrahedron c 21)
+
+front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_top_tetrahedra c =
+ V.singleton (tetrahedron c 0) `V.snoc`
+ (tetrahedron c 1) `V.snoc`
+ (tetrahedron c 5) `V.snoc`
+ (tetrahedron c 6) `V.snoc`
+ (tetrahedron c 16) `V.snoc`
+ (tetrahedron c 19)
+
+front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_down_tetrahedra c =
+ V.singleton (tetrahedron c 1) `V.snoc`
+ (tetrahedron c 2) `V.snoc`
+ (tetrahedron c 12) `V.snoc`
+ (tetrahedron c 13) `V.snoc`
+ (tetrahedron c 18) `V.snoc`
+ (tetrahedron c 19)
+
+back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_top_tetrahedra c =
+ V.singleton (tetrahedron c 0) `V.snoc`
+ (tetrahedron c 3) `V.snoc`
+ (tetrahedron c 6) `V.snoc`
+ (tetrahedron c 7) `V.snoc`
+ (tetrahedron c 20) `V.snoc`
+ (tetrahedron c 21)
+
+back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_down_tetrahedra c =
+ V.singleton (tetrahedron c 8) `V.snoc`
+ (tetrahedron c 11) `V.snoc`
+ (tetrahedron c 14) `V.snoc`
+ (tetrahedron c 15) `V.snoc`
+ (tetrahedron c 22) `V.snoc`
+ (tetrahedron c 23)
+
+back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_top_tetrahedra c =
+ V.singleton (tetrahedron c 4) `V.snoc`
+ (tetrahedron c 5) `V.snoc`
+ (tetrahedron c 9) `V.snoc`
+ (tetrahedron c 10) `V.snoc`
+ (tetrahedron c 16) `V.snoc`
+ (tetrahedron c 17)
+
+back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_down_tetrahedra c =
+ V.singleton (tetrahedron c 8) `V.snoc`
+ (tetrahedron c 9) `V.snoc`
+ (tetrahedron c 13) `V.snoc`
+ (tetrahedron c 14) `V.snoc`
+ (tetrahedron c 17) `V.snoc`
+ (tetrahedron c 18)
in_top_half :: Cube -> Point -> Bool
in_top_half c (_,_,z) =
--
find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
find_containing_tetrahedron c p =
- head containing_tetrahedra
+ candidates `V.unsafeIndex` (fromJust lucky_idx)
where
- candidates = tetrahedra c
- non_candidates_x =
- if (in_front_half c p) then
- back_half_tetrahedra c
+ front_half = in_front_half c p
+ top_half = in_top_half c p
+ left_half = in_left_half c p
+
+ candidates =
+ if front_half then
+
+ if left_half then
+ if top_half then
+ front_left_top_tetrahedra c
+ else
+ front_left_down_tetrahedra c
else
- front_half_tetrahedra c
+ if top_half then
+ front_right_top_tetrahedra c
+ else
+ front_right_down_tetrahedra c
+
+ else -- bottom half
+
+ if left_half then
+ if top_half then
+ back_left_top_tetrahedra c
+ else
+ back_left_down_tetrahedra c
+ else
+ if top_half then
+ back_right_top_tetrahedra c
+ else
+ back_right_down_tetrahedra c
+
+ -- 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.
- candidates_x = candidates \\ non_candidates_x
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 c =
+ disjoint (front_left_top_tetrahedra c) (front_right_down_tetrahedra c)
- non_candidates_y =
- if (in_left_half c p) then
- right_half_tetrahedra c
- else
- left_half_tetrahedra c
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 c =
+ disjoint (back_left_top_tetrahedra c) (back_right_down_tetrahedra c)
- candidates_xy = candidates_x \\ non_candidates_y
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 c =
+ disjoint (front_left_top_tetrahedra c) (back_right_top_tetrahedra c)
- non_candidates_z =
- if (in_top_half c p) then
- down_half_tetrahedra c
- else
- top_half_tetrahedra c
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 c =
+ disjoint (front_left_down_tetrahedra c) (back_right_down_tetrahedra c)
- candidates_xyz = candidates_xy \\ non_candidates_z
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 c =
+ disjoint (front_left_top_tetrahedra c) (back_left_down_tetrahedra c)
- contains_our_point = flip contains_point p
- containing_tetrahedra = filter contains_our_point candidates_xyz
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 c =
+ disjoint (front_right_top_tetrahedra c) (back_right_down_tetrahedra c)
+
+-- | 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-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
+ 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
+
+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 = 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 = tetrahedron cube 7
+ t20 = tetrahedron cube 20
+
+
+
+
+
+p78_25_properties :: Test.Framework.Test
+p78_25_properties =
+ testGroup "p. 78, Section (2.5) Properties" [
+ testProperty "c_ijk1 identity" prop_cijk1_identity ]
+
+p79_26_properties :: Test.Framework.Test
+p79_26_properties =
+ testGroup "p. 79, Section (2.6) Properties" [
+ testProperty "c0120 identity1" prop_c0120_identity1,
+ testProperty "c0120 identity2" prop_c0120_identity2,
+ testProperty "c0120 identity3" prop_c0120_identity3,
+ testProperty "c0120 identity4" prop_c0120_identity4,
+ testProperty "c0120 identity5" prop_c0120_identity5,
+ testProperty "c0120 identity6" prop_c0120_identity6,
+ testProperty "c0120 identity7" prop_c0120_identity7,
+ testProperty "c0210 identity1" prop_c0210_identity1,
+ testProperty "c0300 identity1" prop_c0300_identity1,
+ testProperty "c1110 identity" prop_c1110_identity,
+ testProperty "c1200 identity1" prop_c1200_identity1,
+ testProperty "c2100 identity1" prop_c2100_identity1]
+
+p79_27_properties :: Test.Framework.Test
+p79_27_properties =
+ testGroup "p. 79, Section (2.7) Properties" [
+ testProperty "c0102 identity1" prop_c0102_identity1,
+ testProperty "c0201 identity1" prop_c0201_identity1,
+ testProperty "c0300 identity2" prop_c0300_identity2,
+ testProperty "c1101 identity" prop_c1101_identity,
+ testProperty "c1200 identity2" prop_c1200_identity2,
+ testProperty "c2100 identity2" prop_c2100_identity2 ]
+
+
+p79_28_properties :: Test.Framework.Test
+p79_28_properties =
+ testGroup "p. 79, Section (2.8) Properties" [
+ testProperty "c3000 identity" prop_c3000_identity,
+ testProperty "c2010 identity" prop_c2010_identity,
+ testProperty "c2001 identity" prop_c2001_identity,
+ testProperty "c1020 identity" prop_c1020_identity,
+ testProperty "c1002 identity" prop_c1002_identity,
+ testProperty "c1011 identity" prop_c1011_identity ]
+
+
+edge_incidence_tests :: Test.Framework.Test
+edge_incidence_tests =
+ testGroup "Edge Incidence Tests" [
+ testProperty "t0 shares edge with t6" prop_t0_shares_edge_with_t6,
+ testProperty "t0 shares edge with t1" prop_t0_shares_edge_with_t1,
+ testProperty "t0 shares edge with t3" prop_t0_shares_edge_with_t3,
+ testProperty "t1 shares edge with t2" prop_t1_shares_edge_with_t2,
+ testProperty "t1 shares edge with t19" prop_t1_shares_edge_with_t19,
+ testProperty "t2 shares edge with t3" prop_t2_shares_edge_with_t3,
+ testProperty "t2 shares edge with t12" prop_t2_shares_edge_with_t12,
+ testProperty "t3 shares edge with t21" prop_t3_shares_edge_with_t21,
+ testProperty "t4 shares edge with t5" prop_t4_shares_edge_with_t5,
+ testProperty "t4 shares edge with t7" prop_t4_shares_edge_with_t7,
+ testProperty "t4 shares edge with t10" prop_t4_shares_edge_with_t10,
+ testProperty "t5 shares edge with t6" prop_t5_shares_edge_with_t6,
+ testProperty "t5 shares edge with t16" prop_t5_shares_edge_with_t16,
+ testProperty "t6 shares edge with t7" prop_t6_shares_edge_with_t7,
+ testProperty "t7 shares edge with t20" prop_t7_shares_edge_with_t20 ]
+
+cube_properties :: Test.Framework.Test
+cube_properties =
+ testGroup "Cube Properties" [
+ p78_25_properties,
+ p79_26_properties,
+ p79_27_properties,
+ p79_28_properties,
+ edge_incidence_tests,
+ testProperty "opposite octant tetrahedra are disjoint (1)"
+ prop_opposite_octant_tetrahedra_disjoint1,
+ testProperty "opposite octant tetrahedra are disjoint (2)"
+ prop_opposite_octant_tetrahedra_disjoint2,
+ testProperty "opposite octant tetrahedra are disjoint (3)"
+ prop_opposite_octant_tetrahedra_disjoint3,
+ testProperty "opposite octant tetrahedra are disjoint (4)"
+ prop_opposite_octant_tetrahedra_disjoint4,
+ testProperty "opposite octant tetrahedra are disjoint (5)"
+ prop_opposite_octant_tetrahedra_disjoint5,
+ testProperty "opposite octant tetrahedra are disjoint (6)"
+ prop_opposite_octant_tetrahedra_disjoint6,
+ testProperty "all volumes positive" prop_all_volumes_positive,
+ testProperty "all volumes exact" prop_all_volumes_exact,
+ testProperty "v0 all equal" prop_v0_all_equal,
+ testProperty "interior values all identical"
+ prop_interior_values_all_identical,
+ testProperty "c-tilde_2100 rotation correct"
+ prop_c_tilde_2100_rotation_correct,
+ testProperty "c-tilde_2100 correct"
+ prop_c_tilde_2100_correct ]
projects / spline3.git / blobdiff