X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=b0b153e782e3fd6ab98b4856d7f0cf589ee60c40;hb=4d695b8d0b05a02d562bdee3d2a1b98ce6a5e747;hp=d873a379ddfca39eba80bf7cb3022b258addfacc;hpb=408f4da9058366cd8047592aa96f97bcc348d329;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index d873a37..b0b153e 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -1,34 +1,68 @@
 module Cube
 where
 
+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 Face (Face(Face, v0, v1, v2, v3))
+import qualified Face (Face(Face, v0, v1, v2, v3))
 import FunctionValues
 import Point
-import Tetrahedron (Tetrahedron(Tetrahedron), fv)
+import Tetrahedron hiding (c, fv)
 import ThreeDimensional
 
 data Cube = Cube { h :: Double,
                    i :: Int,
                    j :: Int,
                    k :: Int,
-                   fv :: FunctionValues }
+                   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)) ++
-        " (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)) ++ ")"
-
+        "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 0 0 0 0 empty_values
+empty_cube = Cube 0 0 0 0 empty_values 0
 
 
 -- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
@@ -92,104 +126,59 @@ instance ThreeDimensional Cube where
              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)
-
---     (Cube g1 i1 j1 k1 d1) * (Cube _ i2 j2 k2 d2) =
---         Cube g1 (i1 * i2) (j1 * j2) (k1 * k2) (d1 * d2)
-
---     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) }
-
--- 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)
-
---     fromRational q = empty_cube { datum = fromRational q }
-
-
-
--- | Return the cube corresponding to the grid point i,j,k. The list
---   of cubes is stored as [z][y][x] but we'll be requesting it by
---   [x][y][z] so we flip the indices in the last line.
--- 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 =
---             (((cubes g) !! k') !! j') !! i'
-
-
-
-
-
 
 -- Face stuff.
 
 -- | The top (in the direction of z) face of the cube.
-top_face :: Cube -> Face
-top_face c = Face v0' v1' v2' v3'
+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)
+      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
-back_face c = Face v0' v1' v2' v3'
+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)
+      v0' = (center c) + (delta, -delta, -delta)
       v1' = (center c) + (delta, delta, -delta)
-      v2' = (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
-down_face c = Face v0' v1' v2' v3'
+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)
+      v0' = (center c) + (-delta, -delta, -delta)
       v1' = (center c) + (-delta, delta, -delta)
-      v2' = (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
-front_face c = Face v0' v1' v2' v3'
+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)
@@ -197,134 +186,440 @@ front_face c = Face v0' v1' v2' v3'
       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
-left_face c = Face v0' v1' v2' v3'
+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)
+      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
-right_face c = Face v0' v1' v2' v3'
+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)
+      v0' = (center c) + (-delta, delta, delta)
+      v1' = (center c) + (delta, delta, delta)
+      v2' = (center c) + (delta, delta, -delta)
+      v3' = (center c) + (-delta, delta, -delta)
 
 
+tetrahedron :: Cube -> Int -> Tetrahedron
 
-tetrahedron0 :: Cube -> Tetrahedron
-tetrahedron0 c =
-    Tetrahedron (Cube.fv c) v0' v1' v2' v3'
+tetrahedron c 0 =
+    Tetrahedron (Cube.fv c) v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (front_face c)
-      v2' = v0 (front_face c)
-      v3' = v1 (front_face c)
+      v2' = Face.v0 (front_face c)
+      v3' = Face.v1 (front_face c)
+      vol = tetrahedra_volume c
 
-tetrahedron1 :: Cube -> Tetrahedron
-tetrahedron1 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 1 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (front_face c)
-      v2' = v1 (front_face c)
-      v3' = v2 (front_face c)
-      fv' = rotate (Cube.fv c) ccwx
+      v2' = Face.v1 (front_face c)
+      v3' = Face.v2 (front_face c)
+      fv' = rotate ccwx (Cube.fv c)
+      vol = tetrahedra_volume c
 
-tetrahedron2 :: Cube -> Tetrahedron
-tetrahedron2 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 2 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (front_face c)
-      v2' = v2 (front_face c)
-      v3' = v3 (front_face c)
-      fv' = rotate (Cube.fv c) (ccwx . ccwx)
+      v2' = Face.v2 (front_face c)
+      v3' = Face.v3 (front_face c)
+      fv' = rotate ccwx $ rotate ccwx $ Cube.fv c
+      vol = tetrahedra_volume c
 
-tetrahedron3 :: Cube -> Tetrahedron
-tetrahedron3 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 3 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (front_face c)
-      v2' = v3 (front_face c)
-      v3' = v1 (front_face c)
-      fv' = rotate (Cube.fv c) cwx
+      v2' = Face.v3 (front_face c)
+      v3' = Face.v0 (front_face c)
+      fv' = rotate cwx (Cube.fv c)
+      vol = tetrahedra_volume c
 
-tetrahedron4 :: Cube -> Tetrahedron
-tetrahedron4 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 4 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (top_face c)
-      v2' = v0 (front_face c)
-      v3' = v1 (front_face c)
-      fv' = rotate (Cube.fv c) cwy
+      v2' = Face.v0 (top_face c)
+      v3' = Face.v1 (top_face c)
+      fv' = rotate cwy (Cube.fv c)
+      vol = tetrahedra_volume c
 
-tetrahedron5 :: Cube -> Tetrahedron
-tetrahedron5 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 5 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (top_face c)
-      v2' = v1 (top_face c)
-      v3' = v2 (top_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron0 c)) ccwx
+      v2' = Face.v1 (top_face c)
+      v3' = Face.v2 (top_face c)
+      fv' = rotate cwy $ rotate cwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron6 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 6 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (top_face c)
-      v2' = v2 (top_face c)
-      v3' = v3 (top_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron0 c)) (ccwx . ccwx)
-
-tetrahedron7 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+      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' = v3 (top_face c)
-      v3' = v1 (top_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron0 c)) cwx
-
-tetrahedrons :: Cube -> [Tetrahedron]
-tetrahedrons 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,
-                --  tetrahedron21 c,
-                --  tetrahedron22 c,
-                --  tetrahedron23 c,
-                --  tetrahedron24 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