X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=3c202a7f08b23ab188a2e62166a553bf01a96d8e;hb=993490fd9d940f5e8dea4f934c07c1a5a6c1f8ff;hp=882233f0b656295ccda8ad8f4902fe31cd90cd82;hpb=f96b07dc2004d60bc87a7ff849959ad76fa7bc45;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index 882233f..3c202a7 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -1,52 +1,69 @@
 module Cube
 where
 
-import Grid
+import Data.Maybe (fromJust)
+import qualified Data.Vector as V (
+  Vector,
+  findIndex,
+  map,
+  minimum,
+  singleton,
+  snoc,
+  unsafeIndex
+  )
+import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
+
+import Cardinal
+import qualified Face (Face(Face, v0, v1, v2, v3))
+import FunctionValues
 import Point
+import Tetrahedron (Tetrahedron(Tetrahedron))
 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,
+                   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
+      (Positive tet_vol) <- arbitrary :: Gen (Positive Double)
+      return (Cube h' i' j' k' fv' tet_vol)
+        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 0
+
 
 -- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
@@ -54,7 +71,7 @@ xmin :: Cube -> Double
 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.
@@ -62,7 +79,7 @@ xmax :: Cube -> Double
 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.
@@ -70,7 +87,7 @@ ymin :: Cube -> Double
 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.
@@ -78,7 +95,7 @@ ymax :: Cube -> Double
 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.
@@ -86,7 +103,7 @@ zmin :: Cube -> Double
 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.
@@ -94,93 +111,515 @@ zmax :: Cube -> Double
 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.
     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
-             x = (delta * i')
-             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
+             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
 
 
-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.
 
-    (Cube g1 i1 j1 k1 d1) * (Cube _ i2 j2 k2 d2) =
-        Cube g1 (i1 * i2) (j1 * j2) (k1 * k2) (d1 * d2)
+-- | 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)
 
-    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)
 
-    fromInteger x = empty_cube { datum = (fromIntegral x) }
+-- | 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)
 
-instance Fractional Cube where
-    (Cube g1 i1 j1 k1 d1) / (Cube _ _ _ _ d2) =
-        Cube g1 i1 j1 k1 (d1 / d2)
 
-    recip (Cube g1 i1 j1 k1 d1) =
-        Cube g1 i1 j1 k1 (recip d1)
+-- 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)
 
-    fromRational q = empty_cube { datum = fromRational q }
 
-reverse_cube :: Grid -> Int -> Int -> Int -> Double -> Cube
-reverse_cube g k' j' i' datum' = Cube g i' j' k' datum'
 
+-- | 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)
 
-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.
+-- | 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)
 
--- | 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..]
 
--- | 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..]
+tetrahedron :: Cube -> Int -> Tetrahedron
 
--- | 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)
+tetrahedron c 0 =
+    Tetrahedron (Cube.fv c) v0' v1' v2' v3' vol
+    where
+      v0' = center c
+      v1' = center (front_face c)
+      v2' = Face.v0 (front_face c)
+      v3' = Face.v1 (front_face c)
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      vol = tetrahedra_volume c
+
+tetrahedron c 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
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 6 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    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
+                       $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 9 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    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
+                       $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 10 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    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
+                       $ fv c
+
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      fv' = rotate cwy $ rotate cwy
+                       $ rotate ccwx
+                       $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      fv' = rotate ccwy $ rotate ccwz
+                        $ rotate ccwz
+                        $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      fv' = rotate ccwz $ rotate cwy
+                        $ rotate cwy
+                        $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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
+                        $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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 $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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)
+      fv' = rotate cwz $ rotate ccwy
+                       $ rotate ccwy
+                       $ fv c
+      vol = tetrahedra_volume c
+
+tetrahedron c 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
+                       $ 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] ]
+
+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) =
+  distance_from_top <= distance_from_bottom
+  where
+    distance_from_top = abs $ (zmax c) - z
+    distance_from_bottom = abs $ (zmin c) - z
+
+in_front_half :: Cube -> Point -> Bool
+in_front_half c (x,_,_) =
+    distance_from_front <= distance_from_back
+  where
+    distance_from_front = abs $ (xmin c) - x
+    distance_from_back = abs $ (xmax c) - x
+
+
+in_left_half :: Cube -> Point -> Bool
+in_left_half c (_,y,_) =
+    distance_from_left <= distance_from_right
+  where
+    distance_from_left = abs $ (ymin c) - y
+    distance_from_right = abs $ (ymax c) - y
+
+
+-- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
+--   contain the given 'Point'. This should be faster than checking
+--   every tetrahedron individually, since we determine which half
+--   (hemisphere?) of the cube the point lies in three times: once in
+--   each dimension. This allows us to eliminate non-candidates
+--   quickly.
+--
+--   This can throw an exception, but the use of 'head' might
+--   save us some unnecessary computations.
+--
+find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
+find_containing_tetrahedron c p =
+  candidates `V.unsafeIndex` (fromJust lucky_idx)
+  where
+    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
+          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