X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=3c202a7f08b23ab188a2e62166a553bf01a96d8e;hb=993490fd9d940f5e8dea4f934c07c1a5a6c1f8ff;hp=2ec9e4848318fb12135309900e5505df20f3614e;hpb=574a15e7ccfa3109f06fa05ed09e2282a03a7dad;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index 2ec9e48..3c202a7 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -1,21 +1,48 @@
 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 qualified Face (Face(Face, v0, v1, v2, v3))
 import FunctionValues
 import Point
-import Tetrahedron hiding (c)
+import Tetrahedron (Tetrahedron(Tetrahedron))
 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_" ++ subscript ++ "\n" ++
@@ -35,7 +62,7 @@ instance Show Cube where
 
 -- | 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,
@@ -88,7 +115,7 @@ zmax c = (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.    
+    --   (ih, jh, kh). See Sorokina and Zeilfelder, p. 76.
     center c = (x, y, z)
            where
              delta = h c
@@ -101,13 +128,13 @@ 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 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
+    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
 
 
@@ -181,108 +208,113 @@ right_face c = Face.Face v0' v1' v2' v3'
       v3' = (center c) + (-delta, delta, -delta)
 
 
-tetrahedron0 :: Cube -> Tetrahedron
-tetrahedron0 c =
-    Tetrahedron (Cube.fv c) v0' v1' v2' v3'
+tetrahedron :: Cube -> Int -> Tetrahedron
+
+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
 
-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' = 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' = 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' = 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' = 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' = Face.v1 (top_face c)
       v3' = Face.v2 (top_face c)
-      fv' = rotate cwy $ rotate cwz $ Tetrahedron.fv (tetrahedron0 c)
+      fv' = rotate cwy $ rotate cwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron6 :: Cube -> Tetrahedron
-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' = Face.v2 (top_face c)
       v3' = Face.v3 (top_face c)
-      fv' = rotate cwy $ rotate cwz $ rotate cwz $ Tetrahedron.fv (tetrahedron0 c)
+      fv' = rotate cwy $ rotate cwz
+                       $ rotate cwz
+                       $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron7 :: Cube -> Tetrahedron
-tetrahedron7 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ Tetrahedron.fv (tetrahedron0 c)
+      fv' = rotate cwy $ rotate ccwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron8 :: Cube -> Tetrahedron
-tetrahedron8 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ (Tetrahedron.fv (tetrahedron0 c))
+      fv' = rotate cwy $ rotate cwy $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron9 :: Cube -> Tetrahedron
-tetrahedron9 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ Tetrahedron.fv (tetrahedron0 c)
+      fv' = rotate cwy $ rotate cwy
+                       $ rotate cwx
+                       $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron10 :: Cube -> Tetrahedron
-tetrahedron10 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 10 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (back_face c)
@@ -291,12 +323,12 @@ tetrahedron10 c =
       fv' = rotate cwy $ rotate cwy
                        $ rotate cwx
                        $ rotate cwx
-                       $ Tetrahedron.fv (tetrahedron0 c)
+                       $ fv c
 
+      vol = tetrahedra_volume c
 
-tetrahedron11 :: Cube -> Tetrahedron
-tetrahedron11 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 11 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (back_face c)
@@ -304,34 +336,31 @@ tetrahedron11 c =
       v3' = Face.v0 (back_face c)
       fv' = rotate cwy $ rotate cwy
                        $ rotate ccwx
-                       $ Tetrahedron.fv (tetrahedron0 c)
-
+                       $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron12 :: Cube -> Tetrahedron
-tetrahedron12 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 (Tetrahedron.fv (tetrahedron0 c))
+      fv' = rotate ccwy $ fv c
+      vol = tetrahedra_volume c
 
-
-tetrahedron13 :: Cube -> Tetrahedron
-tetrahedron13 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ Tetrahedron.fv (tetrahedron0 c)
-
+      fv' = rotate ccwy $ rotate ccwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron14 :: Cube -> Tetrahedron
-tetrahedron14 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 14 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (down_face c)
@@ -339,45 +368,41 @@ tetrahedron14 c =
       v3' = Face.v3 (down_face c)
       fv' = rotate ccwy $ rotate ccwz
                         $ rotate ccwz
-                        $ Tetrahedron.fv (tetrahedron0 c)
+                        $ fv c
+      vol = tetrahedra_volume c
 
-
-tetrahedron15 :: Cube -> Tetrahedron
-tetrahedron15 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ Tetrahedron.fv (tetrahedron0 c)
-
+      fv' = rotate ccwy $ rotate cwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron16 :: Cube -> Tetrahedron
-tetrahedron16 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 (Tetrahedron.fv (tetrahedron0 c))
-
+      fv' = rotate ccwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron17 :: Cube -> Tetrahedron
-tetrahedron17 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ Tetrahedron.fv (tetrahedron0 c)
+      fv' = rotate ccwz $ rotate cwy $ fv c
+      vol = tetrahedra_volume c
 
-
-tetrahedron18 :: Cube -> Tetrahedron
-tetrahedron18 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 18 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (right_face c)
@@ -385,46 +410,42 @@ tetrahedron18 c =
       v3' = Face.v3 (right_face c)
       fv' = rotate ccwz $ rotate cwy
                         $ rotate cwy
-                        $ Tetrahedron.fv (tetrahedron0 c)
-
+                        $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron19 :: Cube -> Tetrahedron
-tetrahedron19 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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
-                        $ Tetrahedron.fv (tetrahedron0 c)
+                        $ fv c
+      vol = tetrahedra_volume c
 
-
-tetrahedron20 :: Cube -> Tetrahedron
-tetrahedron20 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 (Tetrahedron.fv (tetrahedron0 c))
-
+      fv' = rotate cwz $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron21 :: Cube -> Tetrahedron
-tetrahedron21 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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 $ Tetrahedron.fv (tetrahedron0 c)
+      fv' = rotate cwz $ rotate ccwy $ fv c
+      vol = tetrahedra_volume c
 
-
-tetrahedron22 :: Cube -> Tetrahedron
-tetrahedron22 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+tetrahedron c 22 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (left_face c)
@@ -432,54 +453,173 @@ tetrahedron22 c =
       v3' = Face.v3 (left_face c)
       fv' = rotate cwz $ rotate ccwy
                        $ rotate ccwy
-                       $ Tetrahedron.fv (tetrahedron0 c)
-
+                       $ fv c
+      vol = tetrahedra_volume c
 
-tetrahedron23 :: Cube -> Tetrahedron
-tetrahedron23 c =
-    Tetrahedron fv' v0' v1' v2' v3'
+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
-                       $ Tetrahedron.fv (tetrahedron0 c)
-
-
-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,
-     tetrahedron22 c,
-     tetrahedron23 c]
-
-
--- | Takes a 'Cube', and returns all Tetrahedra belonging to it that
---   contain the given 'Point'.
-find_containing_tetrahedra :: Cube -> Point -> [Tetrahedron]
-find_containing_tetrahedra c p =
-    filter contains_our_point all_tetrahedra
-    where
-      contains_our_point = flip contains_point p
-      all_tetrahedra = tetrahedrons c
+                       $ 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