X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=3c82f67e285f611ba64a11092d61c86cc890724a;hb=edd0bfa30456c0f609418e730af641835b8650aa;hp=83f4da38f52638f300a655ce88dbfc3d46a9cd8a;hpb=1b041ed87130177797758f471aa78fe55e65c9ae;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index 83f4da3..3c82f67 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -25,18 +25,18 @@ import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
 import Cardinal
 import Comparisons ((~=), (~~=))
 import qualified Face (Face(Face, v0, v1, v2, v3))
-import FunctionValues
+import FunctionValues (FunctionValues, eval, rotate)
 import Misc (all_equal, disjoint)
-import Point
+import Point (Point(..), dot)
 import Tetrahedron (Tetrahedron(..), c, volume)
 import ThreeDimensional
 
-data Cube = Cube { h :: Double,
-                   i :: Int,
-                   j :: Int,
-                   k :: Int,
-                   fv :: FunctionValues,
-                   tetrahedra_volume :: Double }
+data Cube = Cube { h  :: !Double,
+                   i  :: !Int,
+                   j  :: !Int,
+                   k  :: !Int,
+                   fv :: !FunctionValues,
+                   tetrahedra_volume :: !Double }
             deriving (Eq)
 
 
@@ -49,9 +49,13 @@ instance Arbitrary Cube where
       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)
+      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
@@ -64,8 +68,7 @@ instance Show Cube where
         " ymin: " ++ (show (ymin cube)) ++ "\n" ++
         " ymax: " ++ (show (ymax cube)) ++ "\n" ++
         " zmin: " ++ (show (zmin cube)) ++ "\n" ++
-        " zmax: " ++ (show (zmax cube)) ++ "\n" ++
-        " fv: " ++ (show (Cube.fv cube)) ++ "\n"
+        " zmax: " ++ (show (zmax cube)) ++ "\n"
         where
           subscript =
               (show (i cube)) ++ "," ++ (show (j cube)) ++ "," ++ (show (k cube))
@@ -74,7 +77,7 @@ instance Show Cube where
 -- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 xmin :: Cube -> Double
-xmin cube = (2*i' - 1)*delta / 2
+xmin cube = (i' - 1/2)*delta
     where
       i' = fromIntegral (i cube) :: Double
       delta = h cube
@@ -82,7 +85,7 @@ xmin cube = (2*i' - 1)*delta / 2
 -- | The right-side boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 xmax :: Cube -> Double
-xmax cube = (2*i' + 1)*delta / 2
+xmax cube = (i' + 1/2)*delta
     where
       i' = fromIntegral (i cube) :: Double
       delta = h cube
@@ -90,7 +93,7 @@ xmax cube = (2*i' + 1)*delta / 2
 -- | The front boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 ymin :: Cube -> Double
-ymin cube = (2*j' - 1)*delta / 2
+ymin cube = (j' - 1/2)*delta
     where
       j' = fromIntegral (j cube) :: Double
       delta = h cube
@@ -98,7 +101,7 @@ ymin cube = (2*j' - 1)*delta / 2
 -- | The back boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 ymax :: Cube -> Double
-ymax cube = (2*j' + 1)*delta / 2
+ymax cube = (j' + 1/2)*delta
     where
       j' = fromIntegral (j cube) :: Double
       delta = h cube
@@ -106,7 +109,7 @@ ymax cube = (2*j' + 1)*delta / 2
 -- | The bottom boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 zmin :: Cube -> Double
-zmin cube = (2*k' - 1)*delta / 2
+zmin cube = (k' - 1/2)*delta
     where
       k' = fromIntegral (k cube) :: Double
       delta = h cube
@@ -114,7 +117,7 @@ zmin cube = (2*k' - 1)*delta / 2
 -- | The top boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 zmax :: Cube -> Double
-zmax cube = (2*k' + 1)*delta / 2
+zmax cube = (k' + 1/2)*delta
     where
       k' = fromIntegral (k cube) :: Double
       delta = h cube
@@ -122,7 +125,7 @@ zmax cube = (2*k' + 1)*delta / 2
 instance ThreeDimensional Cube where
     -- | The center of Cube_ijk coincides with v_ijk at
     --   (ih, jh, kh). See Sorokina and Zeilfelder, p. 76.
-    center cube = (x, y, z)
+    center cube = Point x y z
            where
              delta = h cube
              i' = fromIntegral (i cube) :: Double
@@ -134,7 +137,7 @@ instance ThreeDimensional Cube where
 
     -- | 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 cube (x, y, z)
+    contains_point cube (Point x y z)
         | x < (xmin cube) = False
         | x > (xmax cube) = False
         | y < (ymin cube) = False
@@ -152,10 +155,11 @@ top_face :: Cube -> Face.Face
 top_face cube = Face.Face v0' v1' v2' v3'
     where
       delta = (1/2)*(h cube)
-      v0' = (center cube) + (delta, -delta, delta)
-      v1' = (center cube) + (delta, delta, delta)
-      v2' = (center cube) + (-delta, delta, delta)
-      v3' = (center cube) + (-delta, -delta, delta)
+      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 )
 
 
 
@@ -164,10 +168,11 @@ back_face :: Cube -> Face.Face
 back_face cube = Face.Face v0' v1' v2' v3'
     where
       delta = (1/2)*(h cube)
-      v0' = (center cube) + (delta, -delta, -delta)
-      v1' = (center cube) + (delta, delta, -delta)
-      v2' = (center cube) + (delta, delta, delta)
-      v3' = (center cube) + (delta, -delta, delta)
+      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.
@@ -175,10 +180,11 @@ down_face :: Cube -> Face.Face
 down_face cube = Face.Face v0' v1' v2' v3'
     where
       delta = (1/2)*(h cube)
-      v0' = (center cube) + (-delta, -delta, -delta)
-      v1' = (center cube) + (-delta, delta, -delta)
-      v2' = (center cube) + (delta, delta, -delta)
-      v3' = (center cube) + (delta, -delta, -delta)
+      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) )
 
 
 
@@ -187,20 +193,22 @@ front_face :: Cube -> Face.Face
 front_face cube = Face.Face v0' v1' v2' v3'
     where
       delta = (1/2)*(h cube)
-      v0' = (center cube) + (-delta, -delta, delta)
-      v1' = (center cube) + (-delta, delta, delta)
-      v2' = (center cube) + (-delta, delta, -delta)
-      v3' = (center cube) + (-delta, -delta, -delta)
+      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 cube = Face.Face v0' v1' v2' v3'
     where
       delta = (1/2)*(h cube)
-      v0' = (center cube) + (delta, -delta, delta)
-      v1' = (center cube) + (-delta, -delta, delta)
-      v2' = (center cube) + (-delta, -delta, -delta)
-      v3' = (center cube) + (delta, -delta, -delta)
+      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.
@@ -208,10 +216,11 @@ right_face :: Cube -> Face.Face
 right_face cube = Face.Face v0' v1' v2' v3'
     where
       delta = (1/2)*(h cube)
-      v0' = (center cube) + (-delta, delta, delta)
-      v1' = (center cube) + (delta, delta, delta)
-      v2' = (center cube) + (delta, delta, -delta)
-      v3' = (center cube) + (-delta, delta, -delta)
+      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
@@ -220,18 +229,20 @@ tetrahedron cube 0 =
     Tetrahedron (fv cube) v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (front_face cube)
-      v2' = Face.v0 (front_face cube)
-      v3' = Face.v1 (front_face cube)
+      ff  = front_face cube
+      v1' = center ff
+      v2' = Face.v0 ff
+      v3' = Face.v1 ff
       vol = tetrahedra_volume cube
 
 tetrahedron cube 1 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (front_face cube)
-      v2' = Face.v1 (front_face cube)
-      v3' = Face.v2 (front_face cube)
+      ff  = front_face cube
+      v1' = center ff
+      v2' = Face.v1 ff
+      v3' = Face.v2 ff
       fv' = rotate ccwx (fv cube)
       vol = tetrahedra_volume cube
 
@@ -239,9 +250,10 @@ tetrahedron cube 2 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (front_face cube)
-      v2' = Face.v2 (front_face cube)
-      v3' = Face.v3 (front_face cube)
+      ff  = front_face cube
+      v1' = center ff
+      v2' = Face.v2 ff
+      v3' = Face.v3 ff
       fv' = rotate ccwx $ rotate ccwx $ fv cube
       vol = tetrahedra_volume cube
 
@@ -249,9 +261,10 @@ tetrahedron cube 3 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (front_face cube)
-      v2' = Face.v3 (front_face cube)
-      v3' = Face.v0 (front_face cube)
+      ff  = front_face cube
+      v1' = center ff
+      v2' = Face.v3 ff
+      v3' = Face.v0 ff
       fv' = rotate cwx (fv cube)
       vol = tetrahedra_volume cube
 
@@ -259,9 +272,10 @@ tetrahedron cube 4 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (top_face cube)
-      v2' = Face.v0 (top_face cube)
-      v3' = Face.v1 (top_face cube)
+      tf  = top_face cube
+      v1' = center tf
+      v2' = Face.v0 tf
+      v3' = Face.v1 tf
       fv' = rotate cwy (fv cube)
       vol = tetrahedra_volume cube
 
@@ -269,9 +283,10 @@ tetrahedron cube 5 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (top_face cube)
-      v2' = Face.v1 (top_face cube)
-      v3' = Face.v2 (top_face cube)
+      tf  = top_face cube
+      v1' = center tf
+      v2' = Face.v1 tf
+      v3' = Face.v2 tf
       fv' = rotate cwy $ rotate cwz $ fv cube
       vol = tetrahedra_volume cube
 
@@ -279,9 +294,10 @@ tetrahedron cube 6 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (top_face cube)
-      v2' = Face.v2 (top_face cube)
-      v3' = Face.v3 (top_face cube)
+      tf  = top_face cube
+      v1' = center tf
+      v2' = Face.v2 tf
+      v3' = Face.v3 tf
       fv' = rotate cwy $ rotate cwz
                        $ rotate cwz
                        $ fv cube
@@ -291,9 +307,10 @@ tetrahedron cube 7 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (top_face cube)
-      v2' = Face.v3 (top_face cube)
-      v3' = Face.v0 (top_face cube)
+      tf  = top_face cube
+      v1' = center tf
+      v2' = Face.v3 tf
+      v3' = Face.v0 tf
       fv' = rotate cwy $ rotate ccwz $ fv cube
       vol = tetrahedra_volume cube
 
@@ -301,9 +318,10 @@ tetrahedron cube 8 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (back_face cube)
-      v2' = Face.v0 (back_face cube)
-      v3' = Face.v1 (back_face cube)
+      bf  = back_face cube
+      v1' = center bf
+      v2' = Face.v0 bf
+      v3' = Face.v1 bf
       fv' = rotate cwy $ rotate cwy $ fv cube
       vol = tetrahedra_volume cube
 
@@ -311,9 +329,10 @@ tetrahedron cube 9 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (back_face cube)
-      v2' = Face.v1 (back_face cube)
-      v3' = Face.v2 (back_face cube)
+      bf  = back_face cube
+      v1' = center bf
+      v2' = Face.v1 bf
+      v3' = Face.v2 bf
       fv' = rotate cwy $ rotate cwy
                        $ rotate cwx
                        $ fv cube
@@ -323,9 +342,10 @@ tetrahedron cube 10 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (back_face cube)
-      v2' = Face.v2 (back_face cube)
-      v3' = Face.v3 (back_face cube)
+      bf  = back_face cube
+      v1' = center bf
+      v2' = Face.v2 bf
+      v3' = Face.v3 bf
       fv' = rotate cwy $ rotate cwy
                        $ rotate cwx
                        $ rotate cwx
@@ -337,9 +357,10 @@ tetrahedron cube 11 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (back_face cube)
-      v2' = Face.v3 (back_face cube)
-      v3' = Face.v0 (back_face cube)
+      bf  = back_face cube
+      v1' = center bf
+      v2' = Face.v3 bf
+      v3' = Face.v0 bf
       fv' = rotate cwy $ rotate cwy
                        $ rotate ccwx
                        $ fv cube
@@ -349,9 +370,10 @@ tetrahedron cube 12 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (down_face cube)
-      v2' = Face.v0 (down_face cube)
-      v3' = Face.v1 (down_face cube)
+      df  = down_face cube
+      v1' = center df
+      v2' = Face.v0 df
+      v3' = Face.v1 df
       fv' = rotate ccwy $ fv cube
       vol = tetrahedra_volume cube
 
@@ -359,9 +381,10 @@ tetrahedron cube 13 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (down_face cube)
-      v2' = Face.v1 (down_face cube)
-      v3' = Face.v2 (down_face cube)
+      df  = down_face cube
+      v1' = center df
+      v2' = Face.v1 df
+      v3' = Face.v2 df
       fv' = rotate ccwy $ rotate ccwz $ fv cube
       vol = tetrahedra_volume cube
 
@@ -369,9 +392,10 @@ tetrahedron cube 14 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (down_face cube)
-      v2' = Face.v2 (down_face cube)
-      v3' = Face.v3 (down_face cube)
+      df  = down_face cube
+      v1' = center df
+      v2' = Face.v2 df
+      v3' = Face.v3 df
       fv' = rotate ccwy $ rotate ccwz
                         $ rotate ccwz
                         $ fv cube
@@ -381,9 +405,10 @@ tetrahedron cube 15 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (down_face cube)
-      v2' = Face.v3 (down_face cube)
-      v3' = Face.v0 (down_face cube)
+      df  = down_face cube
+      v1' = center df
+      v2' = Face.v3 df
+      v3' = Face.v0 df
       fv' = rotate ccwy $ rotate cwz $ fv cube
       vol = tetrahedra_volume cube
 
@@ -391,9 +416,10 @@ tetrahedron cube 16 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (right_face cube)
-      v2' = Face.v0 (right_face cube)
-      v3' = Face.v1 (right_face cube)
+      rf  = right_face cube
+      v1' = center rf
+      v2' = Face.v0 rf
+      v3' = Face.v1 rf
       fv' = rotate ccwz $ fv cube
       vol = tetrahedra_volume cube
 
@@ -401,9 +427,10 @@ tetrahedron cube 17 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (right_face cube)
-      v2' = Face.v1 (right_face cube)
-      v3' = Face.v2 (right_face cube)
+      rf  = right_face cube
+      v1' = center rf
+      v2' = Face.v1 rf
+      v3' = Face.v2 rf
       fv' = rotate ccwz $ rotate cwy $ fv cube
       vol = tetrahedra_volume cube
 
@@ -411,9 +438,10 @@ tetrahedron cube 18 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (right_face cube)
-      v2' = Face.v2 (right_face cube)
-      v3' = Face.v3 (right_face cube)
+      rf  = right_face cube
+      v1' = center rf
+      v2' = Face.v2 rf
+      v3' = Face.v3 rf
       fv' = rotate ccwz $ rotate cwy
                         $ rotate cwy
                         $ fv cube
@@ -423,9 +451,10 @@ tetrahedron cube 19 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (right_face cube)
-      v2' = Face.v3 (right_face cube)
-      v3' = Face.v0 (right_face cube)
+      rf  = right_face cube
+      v1' = center rf
+      v2' = Face.v3 rf
+      v3' = Face.v0 rf
       fv' = rotate ccwz $ rotate ccwy
                         $ fv cube
       vol = tetrahedra_volume cube
@@ -434,9 +463,10 @@ tetrahedron cube 20 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (left_face cube)
-      v2' = Face.v0 (left_face cube)
-      v3' = Face.v1 (left_face cube)
+      lf  = left_face cube
+      v1' = center lf
+      v2' = Face.v0 lf
+      v3' = Face.v1 lf
       fv' = rotate cwz $ fv cube
       vol = tetrahedra_volume cube
 
@@ -444,9 +474,10 @@ tetrahedron cube 21 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (left_face cube)
-      v2' = Face.v1 (left_face cube)
-      v3' = Face.v2 (left_face cube)
+      lf  = left_face cube
+      v1' = center lf
+      v2' = Face.v1 lf
+      v3' = Face.v2 lf
       fv' = rotate cwz $ rotate ccwy $ fv cube
       vol = tetrahedra_volume cube
 
@@ -454,9 +485,10 @@ tetrahedron cube 22 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (left_face cube)
-      v2' = Face.v2 (left_face cube)
-      v3' = Face.v3 (left_face cube)
+      lf  = left_face cube
+      v1' = center lf
+      v2' = Face.v2 lf
+      v3' = Face.v3 lf
       fv' = rotate cwz $ rotate ccwy
                        $ rotate ccwy
                        $ fv cube
@@ -466,9 +498,10 @@ tetrahedron cube 23 =
     Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center cube
-      v1' = center (left_face cube)
-      v2' = Face.v3 (left_face cube)
-      v3' = Face.v0 (left_face cube)
+      lf  = left_face cube
+      v1' = center lf
+      v2' = Face.v3 lf
+      v3' = Face.v0 lf
       fv' = rotate cwz $ rotate cwy
                        $ fv cube
       vol = tetrahedra_volume cube
@@ -555,14 +588,14 @@ back_right_down_tetrahedra  cube =
     (tetrahedron cube 18)
 
 in_top_half :: Cube -> Point -> Bool
-in_top_half cube (_,_,z) =
+in_top_half cube (Point _ _ z) =
   distance_from_top <= distance_from_bottom
   where
     distance_from_top = abs $ (zmax cube) - z
     distance_from_bottom = abs $ (zmin cube) - z
 
 in_front_half :: Cube -> Point -> Bool
-in_front_half cube (x,_,_) =
+in_front_half cube (Point x _ _) =
     distance_from_front <= distance_from_back
   where
     distance_from_front = abs $ (xmin cube) - x
@@ -570,7 +603,7 @@ in_front_half cube (x,_,_) =
 
 
 in_left_half :: Cube -> Point -> Bool
-in_left_half cube (_,y,_) =
+in_left_half cube (Point _ y _) =
     distance_from_left <= distance_from_right
   where
     distance_from_left = abs $ (ymin cube) - y
@@ -622,8 +655,9 @@ find_containing_tetrahedron cube p =
           else
             back_right_down_tetrahedra cube
 
-    -- Use the dot product instead of 'distance' here to save a
-    -- sqrt(). So, "distances" below really means "distances squared."
+    -- Use the dot product instead of Euclidean 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
@@ -665,15 +699,15 @@ prop_opposite_octant_tetrahedra_disjoint6 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