-module Cube
+module Cube (
+ Cube(..),
+ cube_properties,
+ find_containing_tetrahedron,
+ tetrahedra,
+ tetrahedron )
where
-import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
-
-import Cardinal
-import qualified Face (Face(Face, v0, v1, v2, v3))
-import FunctionValues
-import Point
-import Tetrahedron hiding (c)
-import ThreeDimensional
-
-data Cube = Cube { h :: Double,
- i :: Int,
- j :: Int,
- k :: Int,
- fv :: FunctionValues }
+import Data.Maybe ( fromJust )
+import qualified Data.Vector as V (
+ Vector,
+ findIndex,
+ map,
+ minimum,
+ singleton,
+ snoc,
+ unsafeIndex)
+import Prelude hiding ( LT )
+import Test.Tasty ( TestTree, testGroup )
+import Test.Tasty.QuickCheck (
+ Arbitrary( arbitrary ),
+ Gen,
+ Positive( Positive ),
+ choose,
+ testProperty )
+import Cardinal (
+ Cardinal(F, B, L, R, D, T, FL, FR, FD, FT,
+ BL, BR, BD, BT, LD, LT, RD, RT, I),
+ ccwx,
+ ccwy,
+ ccwz,
+ cwx,
+ cwy,
+ cwz )
+import Comparisons ( (~=), (~~=) )
+import qualified Face ( Face(..), center )
+import FunctionValues ( FunctionValues, eval, rotate )
+import Misc ( all_equal, disjoint )
+import Point ( Point( Point ), dot )
+import Tetrahedron (
+ Tetrahedron(Tetrahedron, function_values, v0, v1, v2, v3),
+ barycenter,
+ c,
+ volume )
+
+data Cube = Cube { i :: !Int,
+ j :: !Int,
+ k :: !Int,
+ fv :: !FunctionValues,
+ tetrahedra_volume :: !Double }
deriving (Eq)
instance Arbitrary Cube where
arbitrary = do
- (Positive h') <- arbitrary :: Gen (Positive Double)
i' <- choose (coordmin, coordmax)
j' <- choose (coordmin, coordmax)
k' <- choose (coordmin, coordmax)
fv' <- arbitrary :: Gen FunctionValues
- return (Cube h' i' j' k' fv')
- where
- coordmin = -268435456 -- -(2^29 / 2)
- coordmax = 268435456 -- +(2^29 / 2)
+ (Positive tet_vol) <- arbitrary :: Gen (Positive Double)
+ return (Cube i' j' k' fv' tet_vol)
+ where
+ -- The idea here is that, when cubed in the volume formula,
+ -- these numbers don't overflow 64 bits. This number is not
+ -- magic in any other sense than that it does not cause test
+ -- failures, while 2^23 does.
+ coordmax = 4194304 :: Int -- 2^22
+ coordmin = -coordmax
instance Show Cube where
- show c =
+ show cube =
"Cube_" ++ subscript ++ "\n" ++
- " h: " ++ (show (h c)) ++ "\n" ++
- " Center: " ++ (show (center c)) ++ "\n" ++
- " xmin: " ++ (show (xmin c)) ++ "\n" ++
- " xmax: " ++ (show (xmax c)) ++ "\n" ++
- " ymin: " ++ (show (ymin c)) ++ "\n" ++
- " ymax: " ++ (show (ymax c)) ++ "\n" ++
- " zmin: " ++ (show (zmin c)) ++ "\n" ++
- " zmax: " ++ (show (zmax c)) ++ "\n" ++
- " fv: " ++ (show (Cube.fv c)) ++ "\n"
+ " Center: " ++ (show (center cube)) ++ "\n" ++
+ " xmin: " ++ (show (xmin cube)) ++ "\n" ++
+ " xmax: " ++ (show (xmax cube)) ++ "\n" ++
+ " ymin: " ++ (show (ymin cube)) ++ "\n" ++
+ " ymax: " ++ (show (ymax cube)) ++ "\n" ++
+ " zmin: " ++ (show (zmin cube)) ++ "\n" ++
+ " zmax: " ++ (show (zmax cube)) ++ "\n"
where
subscript =
- (show (i c)) ++ "," ++ (show (j c)) ++ "," ++ (show (k c))
-
-
--- | Returns an empty 'Cube'.
-empty_cube :: Cube
-empty_cube = Cube 0 0 0 0 empty_values
+ (show (i cube)) ++ "," ++ (show (j cube)) ++ "," ++ (show (k cube))
-- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
xmin :: Cube -> Double
-xmin c = (2*i' - 1)*delta / 2
+xmin cube = (i' - 1/2)
where
- i' = fromIntegral (i c) :: Double
- delta = h c
+ i' = fromIntegral (i cube) :: Double
-- | The right-side boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
xmax :: Cube -> Double
-xmax c = (2*i' + 1)*delta / 2
+xmax cube = (i' + 1/2)
where
- i' = fromIntegral (i c) :: Double
- delta = h c
+ i' = fromIntegral (i cube) :: Double
-- | The front boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
ymin :: Cube -> Double
-ymin c = (2*j' - 1)*delta / 2
+ymin cube = (j' - 1/2)
where
- j' = fromIntegral (j c) :: Double
- delta = h c
+ j' = fromIntegral (j cube) :: Double
-- | The back boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
ymax :: Cube -> Double
-ymax c = (2*j' + 1)*delta / 2
+ymax cube = (j' + 1/2)
where
- j' = fromIntegral (j c) :: Double
- delta = h c
+ j' = fromIntegral (j cube) :: Double
-- | The bottom boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
zmin :: Cube -> Double
-zmin c = (2*k' - 1)*delta / 2
+zmin cube = (k' - 1/2)
where
- k' = fromIntegral (k c) :: Double
- delta = h c
+ k' = fromIntegral (k cube) :: Double
-- | The top boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
zmax :: Cube -> Double
-zmax c = (2*k' + 1)*delta / 2
- where
- k' = fromIntegral (k c) :: Double
- delta = h c
-
-instance ThreeDimensional Cube where
- -- | The center of Cube_ijk coincides with v_ijk at
- -- (ih, jh, kh). See Sorokina and Zeilfelder, p. 76.
- center c = (x, y, z)
- where
- delta = h c
- i' = fromIntegral (i c) :: Double
- j' = fromIntegral (j c) :: Double
- k' = fromIntegral (k c) :: Double
- x = delta * i'
- y = delta * j'
- z = delta * k'
-
- -- | It's easy to tell if a point is within a cube; just make sure
- -- that it falls on the proper side of each of the cube's faces.
- contains_point c (x, y, z)
- | x < (xmin c) = False
- | x > (xmax c) = False
- | y < (ymin c) = False
- | y > (ymax c) = False
- | z < (zmin c) = False
- | z > (zmax c) = False
- | otherwise = True
+zmax cube = (k' + 1/2)
+ where
+ k' = fromIntegral (k cube) :: Double
+-- | The center of Cube_ijk coincides with v_ijk at
+-- (i, j, k). See Sorokina and Zeilfelder, p. 76.
+center :: Cube -> Point
+center cube =
+ Point x y z
+ where
+ x = fromIntegral (i cube) :: Double
+ y = fromIntegral (j cube) :: Double
+ z = fromIntegral (k cube) :: Double
+
-- Face stuff.
-- | The top (in the direction of z) face of the cube.
top_face :: Cube -> Face.Face
-top_face c = Face.Face v0' v1' v2' v3'
+top_face cube = Face.Face v0' v1' v2' v3'
where
- delta = (1/2)*(h c)
- v0' = (center c) + (delta, -delta, delta)
- v1' = (center c) + (delta, delta, delta)
- v2' = (center c) + (-delta, delta, delta)
- v3' = (center c) + (-delta, -delta, delta)
+ delta = (1/2) :: Double
+ cc = center cube
+ v0' = cc + ( Point delta (-delta) delta )
+ v1' = cc + ( Point delta delta delta )
+ v2' = cc + ( Point (-delta) delta delta )
+ v3' = cc + ( Point (-delta) (-delta) delta )
-- | The back (in the direction of x) face of the cube.
back_face :: Cube -> Face.Face
-back_face c = Face.Face v0' v1' v2' v3'
+back_face cube = Face.Face v0' v1' v2' v3'
where
- delta = (1/2)*(h c)
- v0' = (center c) + (delta, -delta, -delta)
- v1' = (center c) + (delta, delta, -delta)
- v2' = (center c) + (delta, delta, delta)
- v3' = (center c) + (delta, -delta, delta)
+ delta = (1/2) :: Double
+ cc = center cube
+ v0' = cc + ( Point delta (-delta) (-delta) )
+ v1' = cc + ( Point delta delta (-delta) )
+ v2' = cc + ( Point delta delta delta )
+ v3' = cc + ( Point delta (-delta) delta )
-- The bottom face (in the direction of -z) of the cube.
down_face :: Cube -> Face.Face
-down_face c = Face.Face v0' v1' v2' v3'
+down_face cube = Face.Face v0' v1' v2' v3'
where
- delta = (1/2)*(h c)
- v0' = (center c) + (-delta, -delta, -delta)
- v1' = (center c) + (-delta, delta, -delta)
- v2' = (center c) + (delta, delta, -delta)
- v3' = (center c) + (delta, -delta, -delta)
+ delta = (1/2) :: Double
+ cc = center cube
+ v0' = cc + ( Point (-delta) (-delta) (-delta) )
+ v1' = cc + ( Point (-delta) delta (-delta) )
+ v2' = cc + ( Point delta delta (-delta) )
+ v3' = cc + ( Point delta (-delta) (-delta) )
-- | The front (in the direction of -x) face of the cube.
front_face :: Cube -> Face.Face
-front_face c = Face.Face v0' v1' v2' v3'
+front_face cube = Face.Face v0' v1' v2' v3'
where
- delta = (1/2)*(h c)
- v0' = (center c) + (-delta, -delta, delta)
- v1' = (center c) + (-delta, delta, delta)
- v2' = (center c) + (-delta, delta, -delta)
- v3' = (center c) + (-delta, -delta, -delta)
+ delta = (1/2) :: Double
+ cc = center cube
+ v0' = cc + ( Point (-delta) (-delta) delta )
+ v1' = cc + ( Point (-delta) delta delta )
+ v2' = cc + ( Point (-delta) delta (-delta) )
+ v3' = cc + ( Point (-delta) (-delta) (-delta) )
-- | The left (in the direction of -y) face of the cube.
left_face :: Cube -> Face.Face
-left_face c = Face.Face v0' v1' v2' v3'
+left_face cube = Face.Face v0' v1' v2' v3'
where
- delta = (1/2)*(h c)
- v0' = (center c) + (delta, -delta, delta)
- v1' = (center c) + (-delta, -delta, delta)
- v2' = (center c) + (-delta, -delta, -delta)
- v3' = (center c) + (delta, -delta, -delta)
+ delta = (1/2) :: Double
+ cc = center cube
+ v0' = cc + ( Point delta (-delta) delta )
+ v1' = cc + ( Point (-delta) (-delta) delta )
+ v2' = cc + ( Point (-delta) (-delta) (-delta) )
+ v3' = cc + ( Point delta (-delta) (-delta) )
-- | The right (in the direction of y) face of the cube.
right_face :: Cube -> Face.Face
-right_face c = Face.Face v0' v1' v2' v3'
- where
- delta = (1/2)*(h c)
- v0' = (center c) + (-delta, delta, delta)
- v1' = (center c) + (delta, delta, delta)
- v2' = (center c) + (delta, delta, -delta)
- v3' = (center c) + (-delta, delta, -delta)
-
-
-tetrahedron0 :: Cube -> Tetrahedron
-tetrahedron0 c =
- Tetrahedron (Cube.fv c) v0' v1' v2' v3'
- where
- v0' = center c
- v1' = center (front_face c)
- v2' = Face.v0 (front_face c)
- v3' = Face.v1 (front_face c)
-
-tetrahedron1 :: Cube -> Tetrahedron
-tetrahedron1 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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)
-
-tetrahedron2 :: Cube -> Tetrahedron
-tetrahedron2 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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
-
-tetrahedron3 :: Cube -> Tetrahedron
-tetrahedron3 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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)
-
-tetrahedron4 :: Cube -> Tetrahedron
-tetrahedron4 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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)
-
-tetrahedron5 :: Cube -> Tetrahedron
-tetrahedron5 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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 (tetrahedron0 c)
-
-tetrahedron6 :: Cube -> Tetrahedron
-tetrahedron6 c =
- Tetrahedron fv' v0' v1' v2' v3'
- where
- v0' = center c
- v1' = center (top_face c)
- v2' = Face.v2 (top_face c)
- v3' = Face.v3 (top_face c)
- fv' = rotate cwy $ rotate cwz $ rotate cwz $ Tetrahedron.fv (tetrahedron0 c)
-
-tetrahedron7 :: Cube -> Tetrahedron
-tetrahedron7 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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 (tetrahedron0 c)
-
-tetrahedron8 :: Cube -> Tetrahedron
-tetrahedron8 c =
- Tetrahedron fv' v0' v1' v2' v3'
- 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 (tetrahedron0 c)
-
-tetrahedron9 :: Cube -> Tetrahedron
-tetrahedron9 c =
- Tetrahedron fv' v0' v1' v2' v3'
- where
- v0' = center c
- v1' = center (back_face c)
- v2' = Face.v1 (back_face c)
- v3' = Face.v2 (back_face c)
- fv' = rotate cwy $ rotate cwy $ rotate cwx $ Tetrahedron.fv (tetrahedron0 c)
-
-tetrahedron10 :: Cube -> Tetrahedron
-tetrahedron10 c =
- Tetrahedron fv' v0' v1' v2' v3'
- where
- v0' = center c
- v1' = center (back_face c)
- v2' = Face.v2 (back_face c)
- v3' = Face.v3 (back_face c)
+right_face cube = Face.Face v0' v1' v2' v3'
+ where
+ delta = (1/2) :: Double
+ cc = center cube
+ v0' = cc + ( Point (-delta) delta delta)
+ v1' = cc + ( Point delta delta delta )
+ v2' = cc + ( Point delta delta (-delta) )
+ v3' = cc + ( Point (-delta) delta (-delta) )
+
+
+tetrahedron :: Cube -> Int -> Tetrahedron
+
+tetrahedron cube 0 =
+ Tetrahedron (fv cube) v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ ff = front_face cube
+ v1' = Face.center ff
+ v2' = Face.v0 ff
+ v3' = Face.v1 ff
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 1 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ ff = front_face cube
+ v1' = Face.center ff
+ v2' = Face.v1 ff
+ v3' = Face.v2 ff
+ fv' = rotate ccwx (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 2 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ ff = front_face cube
+ v1' = Face.center ff
+ v2' = Face.v2 ff
+ v3' = Face.v3 ff
+ 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
+ ff = front_face cube
+ v1' = Face.center ff
+ v2' = Face.v3 ff
+ v3' = Face.v0 ff
+ fv' = rotate cwx (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 4 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ tf = top_face cube
+ v1' = Face.center tf
+ v2' = Face.v0 tf
+ v3' = Face.v1 tf
+ fv' = rotate cwy (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 5 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ tf = top_face cube
+ v1' = Face.center tf
+ v2' = Face.v1 tf
+ v3' = Face.v2 tf
+ 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
+ tf = top_face cube
+ v1' = Face.center tf
+ v2' = Face.v2 tf
+ v3' = Face.v3 tf
+ 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
+ tf = top_face cube
+ v1' = Face.center tf
+ v2' = Face.v3 tf
+ v3' = Face.v0 tf
+ 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
+ bf = back_face cube
+ v1' = Face.center bf
+ v2' = Face.v0 bf
+ v3' = Face.v1 bf
+ 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
+ bf = back_face cube
+ v1' = Face.center bf
+ v2' = Face.v1 bf
+ v3' = Face.v2 bf
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
+ bf = back_face cube
+ v1' = Face.center bf
+ v2' = Face.v2 bf
+ v3' = Face.v3 bf
+ fv' = rotate cwy $ rotate cwy
$ rotate cwx
- $ Tetrahedron.fv (tetrahedron0 c)
+ $ rotate cwx
+ $ fv cube
+ vol = tetrahedra_volume cube
-tetrahedron11 :: Cube -> Tetrahedron
-tetrahedron11 c =
- Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron cube 11 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
- v0' = center c
- v1' = center (back_face c)
- v2' = Face.v3 (back_face c)
- v3' = Face.v0 (back_face c)
+ v0' = center cube
+ bf = back_face cube
+ v1' = Face.center bf
+ v2' = Face.v3 bf
+ v3' = Face.v0 bf
fv' = rotate cwy $ rotate cwy
$ rotate ccwx
- $ Tetrahedron.fv (tetrahedron0 c)
-
+ $ fv cube
+ vol = tetrahedra_volume cube
-tetrahedron12 :: Cube -> Tetrahedron
-tetrahedron12 c =
- Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron cube 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 (tetrahedron0 c))
-
-
-tetrahedron13 :: Cube -> Tetrahedron
-tetrahedron13 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ v0' = center cube
+ df = down_face cube
+ v1' = Face.center df
+ v2' = Face.v0 df
+ v3' = Face.v1 df
+ fv' = rotate ccwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 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 (tetrahedron0 c)
-
-
-tetrahedron14 :: Cube -> Tetrahedron
-tetrahedron14 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ v0' = center cube
+ df = down_face cube
+ v1' = Face.center df
+ v2' = Face.v1 df
+ v3' = Face.v2 df
+ fv' = rotate ccwy $ rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 14 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
where
- v0' = center c
- v1' = center (down_face c)
- v2' = Face.v2 (down_face c)
- v3' = Face.v3 (down_face c)
+ v0' = center cube
+ df = down_face cube
+ v1' = Face.center df
+ v2' = Face.v2 df
+ v3' = Face.v3 df
fv' = rotate ccwy $ rotate ccwz
$ rotate ccwz
- $ Tetrahedron.fv (tetrahedron0 c)
-
+ $ fv cube
+ vol = tetrahedra_volume cube
-tetrahedron15 :: Cube -> Tetrahedron
-tetrahedron15 c =
- Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron cube 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 (tetrahedron0 c)
-
-
-tetrahedron16 :: Cube -> Tetrahedron
-tetrahedron16 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ v0' = center cube
+ df = down_face cube
+ v1' = Face.center df
+ v2' = Face.v3 df
+ v3' = Face.v0 df
+ fv' = rotate ccwy $ rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 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 (tetrahedron0 c))
-
-
-tetrahedron17 :: Cube -> Tetrahedron
-tetrahedron17 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ 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 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 (tetrahedron0 c)
-
-
-tetrahedron18 :: Cube -> Tetrahedron
-tetrahedron18 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ 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 c
- v1' = center (right_face c)
- v2' = Face.v2 (right_face c)
- v3' = Face.v3 (right_face c)
+ 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
- $ Tetrahedron.fv (tetrahedron0 c)
+ $ fv cube
+ vol = tetrahedra_volume cube
-
-tetrahedron19 :: Cube -> Tetrahedron
-tetrahedron19 c =
- Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron cube 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)
+ v0' = center cube
+ rf = right_face cube
+ v1' = Face.center rf
+ v2' = Face.v3 rf
+ v3' = Face.v0 rf
fv' = rotate ccwz $ rotate ccwy
- $ Tetrahedron.fv (tetrahedron0 c)
-
+ $ fv cube
+ vol = tetrahedra_volume cube
-tetrahedron20 :: Cube -> Tetrahedron
-tetrahedron20 c =
- Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron cube 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 (tetrahedron0 c))
-
-
-tetrahedron21 :: Cube -> Tetrahedron
-tetrahedron21 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ 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 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 (tetrahedron0 c)
-
-
-tetrahedron22 :: Cube -> Tetrahedron
-tetrahedron22 c =
- Tetrahedron fv' v0' v1' v2' v3'
+ 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 c
- v1' = center (left_face c)
- v2' = Face.v2 (left_face c)
- v3' = Face.v3 (left_face c)
+ 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
- $ Tetrahedron.fv (tetrahedron0 c)
+ $ fv cube
+ vol = tetrahedra_volume cube
-
-tetrahedron23 :: Cube -> Tetrahedron
-tetrahedron23 c =
- Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron cube 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)
+ v0' = center cube
+ lf = left_face cube
+ v1' = Face.center lf
+ v2' = Face.v3 lf
+ v3' = Face.v0 lf
fv' = rotate cwz $ rotate cwy
- $ Tetrahedron.fv (tetrahedron0 c)
+ $ 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 c =
- [tetrahedron0 c,
- tetrahedron1 c,
- tetrahedron2 c,
- tetrahedron3 c,
- tetrahedron4 c,
- tetrahedron5 c,
- tetrahedron6 c,
- tetrahedron7 c,
- tetrahedron8 c,
- tetrahedron9 c,
- tetrahedron10 c,
- tetrahedron11 c,
- tetrahedron12 c,
- tetrahedron13 c,
- tetrahedron14 c,
- tetrahedron15 c,
- tetrahedron16 c,
- tetrahedron17 c,
- tetrahedron18 c,
- tetrahedron19 c,
- tetrahedron20 c,
- tetrahedron21 c,
- tetrahedron22 c,
- tetrahedron23 c]
-
-
--- | Takes a 'Cube', and returns all Tetrahedra belonging to it that
--- contain the given 'Point'.
-find_containing_tetrahedra :: Cube -> Point -> [Tetrahedron]
-find_containing_tetrahedra c p =
- filter contains_our_point all_tetrahedra
- where
- contains_our_point = flip contains_point p
- all_tetrahedra = tetrahedra c
+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
+ | front_half =
+ 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
+
+ | otherwise = -- back 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
+
+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
+ 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
+
+
+p79_26_properties :: TestTree
+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 :: TestTree
+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 :: TestTree
+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 :: TestTree
+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 :: TestTree
+cube_properties =
+ testGroup "Cube 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