X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=6e33423d2da3a828257b21ab85055db58b681af8;hb=7cee33b2fa4789525a12685923edf1f38924a7f4;hp=d3f5151260905827254885c579f1f208f2586bd3;hpb=3a954903101eca7594a65824868517b9758e188d;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index d3f5151..6e33423 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -3,11 +3,10 @@ module Cube (
   cube_properties,
   find_containing_tetrahedron,
   tetrahedra,
-  tetrahedron
-  )
+  tetrahedron )
 where
 
-import Data.Maybe (fromJust)
+import Data.Maybe ( fromJust )
 import qualified Data.Vector as V (
   Vector,
   findIndex,
@@ -15,579 +14,578 @@ import qualified Data.Vector as V (
   minimum,
   singleton,
   snoc,
-  unsafeIndex
-  )
-import Prelude hiding (LT)
-import Test.Framework (Test, testGroup)
-import Test.Framework.Providers.QuickCheck2 (testProperty)
-import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
-
-import Cardinal
-import Comparisons ((~=), (~~=))
-import qualified Face (Face(Face, v0, v1, v2, v3))
-import FunctionValues
-import Misc (all_equal, disjoint)
-import Point
-import Tetrahedron (
-  Tetrahedron(..),
-  c,
-  b0,
-  b1,
-  b2,
-  b3,
-  volume
-  )
-import ThreeDimensional
-
-data Cube = Cube { h :: Double,
-                   i :: Int,
-                   j :: Int,
-                   k :: Int,
-                   fv :: FunctionValues,
-                   tetrahedra_volume :: Double }
+  unsafeIndex)
+import Prelude hiding ( LT )
+import Test.Framework ( Test, testGroup )
+import Test.Framework.Providers.QuickCheck2 ( testProperty )
+import Test.QuickCheck ( Arbitrary(..), Gen, Positive(..), choose )
+
+import Cardinal (
+  Cardinal(..),
+  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(..), dot )
+import Tetrahedron ( Tetrahedron(..), 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
       (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)
+      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 -- 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 0
+              (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
+zmax cube = (k' + 1/2)
     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
+      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
+      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
+      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
+      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
+      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
+      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'
+right_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
+      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 c 0 =
-    Tetrahedron (fv c) v0' v1' v2' v3' vol
+tetrahedron cube 0 =
+    Tetrahedron (fv cube) 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 =
+      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 c
-      v1' = center (front_face c)
-      v2' = Face.v1 (front_face c)
-      v3' = Face.v2 (front_face c)
-      fv' = rotate ccwx (fv c)
-      vol = tetrahedra_volume c
-
-tetrahedron c 2 =
+      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 c
-      v1' = center (front_face c)
-      v2' = Face.v2 (front_face c)
-      v3' = Face.v3 (front_face c)
-      fv' = rotate ccwx $ rotate ccwx $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 3 =
+      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 c
-      v1' = center (front_face c)
-      v2' = Face.v3 (front_face c)
-      v3' = Face.v0 (front_face c)
-      fv' = rotate cwx (fv c)
-      vol = tetrahedra_volume c
-
-tetrahedron c 4 =
+      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 c
-      v1' = center (top_face c)
-      v2' = Face.v0 (top_face c)
-      v3' = Face.v1 (top_face c)
-      fv' = rotate cwy (fv c)
-      vol = tetrahedra_volume c
-
-tetrahedron c 5 =
+      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 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 =
+      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 c
-      v1' = center (top_face c)
-      v2' = Face.v2 (top_face c)
-      v3' = Face.v3 (top_face c)
+      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 c
-      vol = tetrahedra_volume c
+                       $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 7 =
+tetrahedron cube 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 =
+      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 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 =
+      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 c
-      v1' = center (back_face c)
-      v2' = Face.v1 (back_face c)
-      v3' = Face.v2 (back_face c)
+      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 c
-      vol = tetrahedra_volume c
+                       $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 10 =
+tetrahedron cube 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)
+      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
                        $ rotate cwx
-                       $ fv c
+                       $ fv cube
 
-      vol = tetrahedra_volume c
+      vol = tetrahedra_volume cube
 
-tetrahedron c 11 =
+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
-                       $ fv c
-      vol = tetrahedra_volume c
+                       $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 12 =
+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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 13 =
+      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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 14 =
+      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
-                        $ fv c
-      vol = tetrahedra_volume c
+                        $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 15 =
+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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 16 =
+      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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 17 =
+      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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 18 =
+      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
-                        $ fv c
-      vol = tetrahedra_volume c
+                        $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 19 =
+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
-                        $ fv c
-      vol = tetrahedra_volume c
+                        $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 20 =
+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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 21 =
+      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 $ fv c
-      vol = tetrahedra_volume c
-
-tetrahedron c 22 =
+      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
-                       $ fv c
-      vol = tetrahedra_volume c
+                       $ fv cube
+      vol = tetrahedra_volume cube
 
-tetrahedron c 23 =
+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
-                       $ fv c
-      vol = tetrahedra_volume c
-
--- Feels dirty, but whatever.
-tetrahedron _ _ = error "asked for a nonexistent tetrahedron"
+                       $ 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 = [ tetrahedron c n | n <- [0..23] ]
+tetrahedra cube = [ tetrahedron cube 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_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  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_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  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_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  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)
+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  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_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  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_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  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_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  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)
+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 c (_,_,z) =
+in_top_half cube (Point _ _ z) =
   distance_from_top <= distance_from_bottom
   where
-    distance_from_top = abs $ (zmax c) - z
-    distance_from_bottom = abs $ (zmin c) - z
+    distance_from_top = abs $ (zmax cube) - z
+    distance_from_bottom = abs $ (zmin cube) - z
 
 in_front_half :: Cube -> Point -> Bool
-in_front_half c (x,_,_) =
+in_front_half cube (Point x _ _) =
     distance_from_front <= distance_from_back
   where
-    distance_from_front = abs $ (xmin c) - x
-    distance_from_back = abs $ (xmax c) - x
+    distance_from_front = abs $ (xmin cube) - x
+    distance_from_back = abs $ (xmax cube) - x
 
 
 in_left_half :: Cube -> Point -> Bool
-in_left_half c (_,y,_) =
+in_left_half cube (Point _ y _) =
     distance_from_left <= distance_from_right
   where
-    distance_from_left = abs $ (ymin c) - y
-    distance_from_right = abs $ (ymax c) - y
+    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
@@ -600,47 +598,58 @@ in_left_half c (_,y,_) =
 --   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 c p =
+find_containing_tetrahedron cube 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
+    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 c
+            front_left_top_tetrahedra cube
           else
-            front_left_down_tetrahedra c
+            front_left_down_tetrahedra cube
         else
           if top_half then
-            front_right_top_tetrahedra c
+            front_right_top_tetrahedra cube
           else
-            front_right_down_tetrahedra c
-
-      else -- bottom half
+            front_right_down_tetrahedra cube
 
+      | otherwise = -- back half
         if left_half then
           if top_half then
-            back_left_top_tetrahedra c
+            back_left_top_tetrahedra cube
           else
-            back_left_down_tetrahedra c
+            back_left_down_tetrahedra cube
         else
           if top_half then
-            back_right_top_tetrahedra c
+            back_right_top_tetrahedra cube
           else
-            back_right_down_tetrahedra c
+            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
 
-    -- 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 :: 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 -> (center t) `dot` p == shortest_distance)
+                  (\t -> (barycenter t) `dot` p == shortest_distance)
                   candidates
 
 
@@ -653,51 +662,50 @@ find_containing_tetrahedron c p =
 -- Quickcheck tests.
 
 prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
-prop_opposite_octant_tetrahedra_disjoint1 c =
-  disjoint (front_left_top_tetrahedra c) (front_right_down_tetrahedra c)
+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 c =
-  disjoint (back_left_top_tetrahedra c) (back_right_down_tetrahedra c)
+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 c =
-  disjoint (front_left_top_tetrahedra c) (back_right_top_tetrahedra c)
+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 c =
-  disjoint (front_left_down_tetrahedra c) (back_right_down_tetrahedra c)
+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 c =
-  disjoint (front_left_top_tetrahedra c) (back_left_down_tetrahedra c)
+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 c =
-  disjoint (front_right_top_tetrahedra c) (back_right_down_tetrahedra c)
+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.
+--   (which comprise cubes of positive volume) must have positive
+--   volume as well.
 prop_all_volumes_positive :: Cube -> Bool
 prop_all_volumes_positive cube =
-    null nonpositive_volumes
+    all (>= 0) volumes
     where
       ts = tetrahedra cube
       volumes = map volume ts
-      nonpositive_volumes = filter (<= 0) volumes
+
 
 -- | In fact, since all of the tetrahedra are identical, we should
 --   already know their volumes. There's 24 tetrahedra to a cube, so
---   we'd expect the volume of each one to be (1/24)*h^3.
+--   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)*(delta^(3::Int)) | t <- tetrahedra cube]
-    where
-      delta = h cube
+    and [volume t ~~= 1/24 | t <- tetrahedra cube]
 
--- | All tetrahedron should have their v0 located at the center of the 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
@@ -706,7 +714,7 @@ prop_v0_all_equal cube = (v0 t0) == (v0 t1)
 
 
 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
---   third and fourth indices of c-t1 have been switched. This is
+--   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!
@@ -755,8 +763,8 @@ prop_c0120_identity5 cube =
       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.
+-- | 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
@@ -765,8 +773,8 @@ prop_c0120_identity6 cube =
        t6 = tetrahedron cube 6
 
 
--- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
--- --   'prop_c0120_identity1' with tetrahedrons 6 and 7.
+-- | 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
@@ -960,23 +968,6 @@ prop_c1011_identity cube =
         t6 = tetrahedron cube 6
 
 
-
--- | Given in Sorokina and Zeilfelder, p. 78.
-prop_cijk1_identity :: Cube -> Bool
-prop_cijk1_identity cube =
-     and [ c t0 i j k 1 ~=
-                 (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
-                 (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
-                 (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
-                 (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
-                                                      j <- [0..2],
-                                                      k <- [0..2],
-                                                      i + j + k == 2]
-      where
-        t0 = tetrahedron cube 0
-        t1 = tetrahedron cube 1
-
-
 -- | The function values at the interior should be the same for all
 --   tetrahedra.
 prop_interior_values_all_identical :: Cube -> Bool
@@ -988,7 +979,7 @@ prop_interior_values_all_identical cube =
 --   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
+    expr1 ~= expr2
     where
       t0 = tetrahedron cube 0
       t6 = tetrahedron cube 6
@@ -1022,7 +1013,7 @@ prop_c_tilde_2100_rotation_correct cube =
 --   even meaningful!
 prop_c_tilde_2100_correct :: Cube -> Bool
 prop_c_tilde_2100_correct cube =
-    c t6 2 1 0 0 == expected
+    c t6 2 1 0 0 ~= expected
     where
       t0 = tetrahedron cube 0
       t6 = tetrahedron cube 6
@@ -1144,14 +1135,6 @@ prop_t7_shares_edge_with_t20 cube =
         t20 = tetrahedron cube 20
 
 
-
-
-
-p78_25_properties :: Test.Framework.Test
-p78_25_properties =
-    testGroup "p. 78, Section (2.5) Properties" [
-      testProperty "c_ijk1 identity" prop_cijk1_identity ]
-
 p79_26_properties :: Test.Framework.Test
 p79_26_properties =
     testGroup "p. 79, Section (2.6) Properties" [
@@ -1212,7 +1195,6 @@ edge_incidence_tests =
 cube_properties :: Test.Framework.Test
 cube_properties =
   testGroup "Cube Properties" [
-    p78_25_properties,
     p79_26_properties,
     p79_27_properties,
     p79_28_properties,