module Cube
where
-import Grid
+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
-class Gridded a where
- back :: a -> Cube
- down :: a -> Cube
- front :: a -> Cube
- left :: a -> Cube
- right :: a -> Cube
- top :: a -> Cube
-
-
-data Cube = Cube { grid :: Grid,
+data Cube = Cube { h :: Double,
i :: Int,
j :: Int,
k :: Int,
- datum :: Double }
+ fv :: FunctionValues }
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)
+
+
instance Show Cube where
show c =
- "Cube_" ++ (show (i c)) ++ "," ++ (show (j c)) ++ "," ++ (show (k c)) ++
- " (Grid: " ++ (show (grid c)) ++ ")" ++
- " (Center: " ++ (show (center c)) ++ ")" ++
- " (xmin: " ++ (show (xmin c)) ++ ")" ++
- " (xmax: " ++ (show (xmax c)) ++ ")" ++
- " (ymin: " ++ (show (ymin c)) ++ ")" ++
- " (ymax: " ++ (show (ymax c)) ++ ")" ++
- " (zmin: " ++ (show (zmin c)) ++ ")" ++
- " (zmax: " ++ (show (zmax c)) ++ ")" ++
- " (datum: " ++ (show (datum c)) ++ ")\n\n"
-
+ "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"
+ where
+ subscript =
+ (show (i c)) ++ "," ++ (show (j c)) ++ "," ++ (show (k c))
+
+
+-- | Returns an empty 'Cube'.
empty_cube :: Cube
-empty_cube = Cube empty_grid 0 0 0 0
-
--- TODO: The paper considers 'i' to be the front/back direction,
--- whereas I have it in the left/right direction.
-instance Gridded Cube where
- back c = cube_at (grid c) ((i c) + 1) (j c) (k c)
- down c = cube_at (grid c) (i c) (j c) ((k c) - 1)
- front c = cube_at (grid c) ((i c) - 1) (j c) (k c)
- left c = cube_at (grid c) (i c) ((j c) - 1) (k c)
- right c = cube_at (grid c) (i c) ((j c) + 1) (k c)
- top c = cube_at (grid c) (i c) (j c) ((k c) + 1)
+empty_cube = Cube 0 0 0 0 empty_values
+
-- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
xmin c = (2*i' - 1)*delta / 2
where
i' = fromIntegral (i c) :: Double
- delta = h (grid c)
+ delta = h c
-- | The right-side boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
xmax c = (2*i' + 1)*delta / 2
where
i' = fromIntegral (i c) :: Double
- delta = h (grid c)
+ delta = h c
-- | The front boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
ymin c = (2*j' - 1)*delta / 2
where
j' = fromIntegral (j c) :: Double
- delta = h (grid c)
+ delta = h c
-- | The back boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
ymax c = (2*j' + 1)*delta / 2
where
j' = fromIntegral (j c) :: Double
- delta = h (grid c)
+ delta = h c
-- | The bottom boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
zmin c = (2*k' - 1)*delta / 2
where
k' = fromIntegral (k c) :: Double
- delta = h (grid c)
+ delta = h c
-- | The top boundary of the cube. See Sorokina and Zeilfelder,
-- p. 76.
zmax c = (2*k' + 1)*delta / 2
where
k' = fromIntegral (k c) :: Double
- delta = h (grid c)
+ 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.
+ -- (ih, jh, kh). See Sorokina and Zeilfelder, p. 76.
center c = (x, y, z)
where
- delta = h (grid c)
+ delta = h c
i' = fromIntegral (i c) :: Double
j' = fromIntegral (j c) :: Double
k' = fromIntegral (k c) :: Double
y = delta * j'
z = delta * k'
- contains_point c p
- | (x_coord p) < (xmin c) = False
- | (x_coord p) > (xmax c) = False
- | (y_coord p) < (ymin c) = False
- | (y_coord p) > (ymax c) = False
- | (z_coord p) < (zmin c) = False
- | (z_coord p) > (zmax c) = False
+ -- | 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
-instance Num Cube where
- (Cube g1 i1 j1 k1 d1) + (Cube _ i2 j2 k2 d2) =
- Cube g1 (i1 + i2) (j1 + j2) (k1 + k2) (d1 + d2)
- (Cube g1 i1 j1 k1 d1) - (Cube _ i2 j2 k2 d2) =
- Cube g1 (i1 - i2) (j1 - j2) (k1 - k2) (d1 - d2)
+-- 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'
+ 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)
+
+
+
+-- | 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'
+ 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)
+
+
+-- 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'
+ 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)
+
+
+
+-- | 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'
+ 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)
+
+-- | 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'
+ 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)
+
+
+-- | 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)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate cwx
+ $ rotate cwx
+ $ Tetrahedron.fv (tetrahedron0 c)
+
+
+tetrahedron11 :: Cube -> Tetrahedron
+tetrahedron11 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ where
+ v0' = center c
+ v1' = center (back_face c)
+ v2' = Face.v3 (back_face c)
+ v3' = Face.v0 (back_face c)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate ccwx
+ $ Tetrahedron.fv (tetrahedron0 c)
+
+
+tetrahedron12 :: Cube -> Tetrahedron
+tetrahedron12 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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))
+
- (Cube g1 i1 j1 k1 d1) * (Cube _ i2 j2 k2 d2) =
- Cube g1 (i1 * i2) (j1 * j2) (k1 * k2) (d1 * d2)
+tetrahedron13 :: Cube -> Tetrahedron
+tetrahedron13 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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)
- abs (Cube g1 i1 j1 k1 d1) =
- Cube g1 (abs i1) (abs j1) (abs k1) (abs d1)
- signum (Cube g1 i1 j1 k1 d1) =
- Cube g1 (signum i1) (signum j1) (signum k1) (signum d1)
+tetrahedron14 :: Cube -> Tetrahedron
+tetrahedron14 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ where
+ v0' = center c
+ v1' = center (down_face c)
+ v2' = Face.v2 (down_face c)
+ v3' = Face.v3 (down_face c)
+ fv' = rotate ccwy $ rotate ccwz
+ $ rotate ccwz
+ $ Tetrahedron.fv (tetrahedron0 c)
+
+
+tetrahedron15 :: Cube -> Tetrahedron
+tetrahedron15 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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)
- fromInteger x = empty_cube { datum = (fromIntegral x) }
-instance Fractional Cube where
- (Cube g1 i1 j1 k1 d1) / (Cube _ _ _ _ d2) =
- Cube g1 i1 j1 k1 (d1 / d2)
+tetrahedron16 :: Cube -> Tetrahedron
+tetrahedron16 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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))
- recip (Cube g1 i1 j1 k1 d1) =
- Cube g1 i1 j1 k1 (recip d1)
- fromRational q = empty_cube { datum = fromRational q }
+tetrahedron17 :: Cube -> Tetrahedron
+tetrahedron17 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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)
--- | Constructs a cube, switching the i and k axes.
-reverse_cube :: Grid -> Int -> Int -> Int -> Double -> Cube
-reverse_cube g k' j' i' = Cube g i' j' k'
+tetrahedron18 :: Cube -> Tetrahedron
+tetrahedron18 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ where
+ v0' = center c
+ v1' = center (right_face c)
+ v2' = Face.v2 (right_face c)
+ v3' = Face.v3 (right_face c)
+ fv' = rotate ccwz $ rotate cwy
+ $ rotate cwy
+ $ Tetrahedron.fv (tetrahedron0 c)
+
+
+tetrahedron19 :: Cube -> Tetrahedron
+tetrahedron19 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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 (tetrahedron0 c)
-cube_at :: Grid -> Int -> Int -> Int -> Cube
-cube_at g i' j' k'
- | i' >= length (function_values g) = Cube g i' j' k' 0
- | i' < 0 = Cube g i' j' k' 0
- | j' >= length ((function_values g) !! i') = Cube g i' j' k' 0
- | j' < 0 = Cube g i' j' k' 0
- | k' >= length (((function_values g) !! i') !! j') = Cube g i' j' k' 0
- | k' < 0 = Cube g i' j' k' 0
- | otherwise =
- Cube g i' j' k' ((((function_values g) !! i') !! j') !! k')
--- These next three functions basically form a 'for' loop, looping
--- through the xs, ys, and zs in that order.
+tetrahedron20 :: Cube -> Tetrahedron
+tetrahedron20 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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))
+
--- | The cubes_from_values function will return a function that takes
--- a list of values and returns a list of cubes. It could just as
--- well be written to take the values as a parameter; the omission
--- of the last parameter is known as an eta reduce.
-cubes_from_values :: Grid -> Int -> Int -> ([Double] -> [Cube])
-cubes_from_values g i' j' =
- zipWith (reverse_cube g i' j') [0..]
+tetrahedron21 :: Cube -> Tetrahedron
+tetrahedron21 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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)
--- | The cubes_from_planes function will return a function that takes
--- a list of planes and returns a list of cubes. It could just as
--- well be written to take the planes as a parameter; the omission
--- of the last parameter is known as an eta reduce.
-cubes_from_planes :: Grid -> Int -> ([[Double]] -> [[Cube]])
-cubes_from_planes g i' =
- zipWith (cubes_from_values g i') [0..]
--- | Takes a grid as an argument, and returns a three-dimensional list
--- of cubes centered on its grid points.
-cubes :: Grid -> [[[Cube]]]
-cubes g = zipWith (cubes_from_planes g) [0..] (function_values g)
+tetrahedron22 :: Cube -> Tetrahedron
+tetrahedron22 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ where
+ v0' = center c
+ v1' = center (left_face c)
+ v2' = Face.v2 (left_face c)
+ v3' = Face.v3 (left_face c)
+ fv' = rotate cwz $ rotate ccwy
+ $ rotate ccwy
+ $ Tetrahedron.fv (tetrahedron0 c)
+
+
+tetrahedron23 :: Cube -> Tetrahedron
+tetrahedron23 c =
+ Tetrahedron fv' v0' v1' v2' v3'
+ 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 (tetrahedron0 c)
+
+
+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