X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=d3f5151260905827254885c579f1f208f2586bd3;hb=3a954903101eca7594a65824868517b9758e188d;hp=6e31f1911cdef2ed70fe7fcdf836d0bed541c3d8;hpb=e5151f5050e80027f69640813db618aaea54946e;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index 6e31f19..d3f5151 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -1,14 +1,42 @@
-module Cube
+module Cube (
+  Cube(..),
+  cube_properties,
+  find_containing_tetrahedron,
+  tetrahedra,
+  tetrahedron
+  )
 where
 
-import Data.List ( (\\) )
+import Data.Maybe (fromJust)
+import qualified Data.Vector as V (
+  Vector,
+  findIndex,
+  map,
+  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 hiding (c)
+import Tetrahedron (
+  Tetrahedron(..),
+  c,
+  b0,
+  b1,
+  b2,
+  b3,
+  volume
+  )
 import ThreeDimensional
 
 data Cube = Cube { h :: Double,
@@ -202,7 +230,7 @@ right_face c = Face.Face v0' v1' v2' v3'
 tetrahedron :: Cube -> Int -> Tetrahedron
 
 tetrahedron c 0 =
-    Tetrahedron (Cube.fv c) v0' v1' v2' v3' vol 0
+    Tetrahedron (fv c) v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (front_face c)
@@ -211,57 +239,57 @@ tetrahedron c 0 =
       vol = tetrahedra_volume c
 
 tetrahedron c 1 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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)
+      fv' = rotate ccwx (fv c)
       vol = tetrahedra_volume c
 
 tetrahedron c 2 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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
+      fv' = rotate ccwx $ rotate ccwx $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 3 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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)
+      fv' = rotate cwx (fv c)
       vol = tetrahedra_volume c
 
 tetrahedron c 4 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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)
+      fv' = rotate cwy (fv c)
       vol = tetrahedra_volume c
 
 tetrahedron c 5 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate cwy $ rotate cwz $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 6 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 6
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (top_face c)
@@ -269,31 +297,31 @@ tetrahedron c 6 =
       v3' = Face.v3 (top_face c)
       fv' = rotate cwy $ rotate cwz
                        $ rotate cwz
-                       $ Tetrahedron.fv (tetrahedron c 0)
+                       $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 7 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate cwy $ rotate ccwz $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 8 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate cwy $ rotate cwy $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 9 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 9
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (back_face c)
@@ -301,11 +329,11 @@ tetrahedron c 9 =
       v3' = Face.v2 (back_face c)
       fv' = rotate cwy $ rotate cwy
                        $ rotate cwx
-                       $ Tetrahedron.fv (tetrahedron c 0)
+                       $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 10 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 10
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (back_face c)
@@ -314,12 +342,12 @@ tetrahedron c 10 =
       fv' = rotate cwy $ rotate cwy
                        $ rotate cwx
                        $ rotate cwx
-                       $ Tetrahedron.fv (tetrahedron c 0)
+                       $ fv c
 
       vol = tetrahedra_volume c
 
 tetrahedron c 11 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 11
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (back_face c)
@@ -327,31 +355,31 @@ tetrahedron c 11 =
       v3' = Face.v0 (back_face c)
       fv' = rotate cwy $ rotate cwy
                        $ rotate ccwx
-                       $ Tetrahedron.fv (tetrahedron c 0)
+                       $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 12 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0))
+      fv' = rotate ccwy $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 13 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate ccwy $ rotate ccwz $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 14 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 14
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (down_face c)
@@ -359,41 +387,41 @@ tetrahedron c 14 =
       v3' = Face.v3 (down_face c)
       fv' = rotate ccwy $ rotate ccwz
                         $ rotate ccwz
-                        $ Tetrahedron.fv (tetrahedron c 0)
+                        $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 15 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate ccwy $ rotate cwz $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 16 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0))
+      fv' = rotate ccwz $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 17 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate ccwz $ rotate cwy $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 18 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 18
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (right_face c)
@@ -401,42 +429,42 @@ tetrahedron c 18 =
       v3' = Face.v3 (right_face c)
       fv' = rotate ccwz $ rotate cwy
                         $ rotate cwy
-                        $ Tetrahedron.fv (tetrahedron c 0)
+                        $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 19 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+                        $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 20 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0))
+      fv' = rotate cwz $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 21 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+      fv' = rotate cwz $ rotate ccwy $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 22 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 22
+    Tetrahedron fv' v0' v1' v2' v3' vol
     where
       v0' = center c
       v1' = center (left_face c)
@@ -444,57 +472,100 @@ tetrahedron c 22 =
       v3' = Face.v3 (left_face c)
       fv' = rotate cwz $ rotate ccwy
                        $ rotate ccwy
-                       $ Tetrahedron.fv (tetrahedron c 0)
+                       $ fv c
       vol = tetrahedra_volume c
 
 tetrahedron c 23 =
-    Tetrahedron fv' v0' v1' v2' v3' vol 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 (tetrahedron c 0)
+                       $ 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] ]
-
--- | All completely contained in the front half of the cube.
-front_half_tetrahedra :: Cube -> [Tetrahedron]
-front_half_tetrahedra c =
-  [ tetrahedron c n | n <- [0,1,2,3,6,12,19,21] ]
-
--- | All tetrahedra completely contained in the top half of the cube.
-top_half_tetrahedra :: Cube -> [Tetrahedron]
-top_half_tetrahedra c =
-  [ tetrahedron c n | n <- [4,5,6,7,0,10,16,20] ]
-
--- | All tetrahedra completely contained in the back half of the cube.
-back_half_tetrahedra :: Cube -> [Tetrahedron]
-back_half_tetrahedra c =
-  [ tetrahedron c n | n <- [8,9,10,11,4,14,17,23] ]
-
--- | All tetrahedra completely contained in the down half of the cube.
-down_half_tetrahedra :: Cube -> [Tetrahedron]
-down_half_tetrahedra c =
-  [ tetrahedron c n | n <- [12,13,14,15,2,8,18,22] ]
-
--- | All tetrahedra completely contained in the right half of the cube.
-right_half_tetrahedra :: Cube -> [Tetrahedron]
-right_half_tetrahedra c =
-  [ tetrahedron c n | n <- [16,17,18,19,1,5,9,13] ]
-
--- | All tetrahedra completely contained in the left half of the cube.
-left_half_tetrahedra :: Cube -> [Tetrahedron]
-left_half_tetrahedra c =
-  [ tetrahedron c n | n <- [20,21,22,23,3,7,11,15] ]
+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) =
@@ -531,33 +602,639 @@ in_left_half c (_,y,_) =
 --
 find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
 find_containing_tetrahedron c p =
-  head containing_tetrahedra
+  candidates `V.unsafeIndex` (fromJust lucky_idx)
   where
-    candidates = tetrahedra c
-    non_candidates_x =
-        if (in_front_half c p) then
-          back_half_tetrahedra c
+    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
-          front_half_tetrahedra c
+          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
+
+
+
+
+
+
+-- Tests
+
+-- Quickcheck tests.
 
-    candidates_x = candidates \\ non_candidates_x
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 c =
+  disjoint (front_left_top_tetrahedra c) (front_right_down_tetrahedra c)
 
-    non_candidates_y =
-      if (in_left_half c p) then
-        right_half_tetrahedra c
-      else
-        left_half_tetrahedra c
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 c =
+  disjoint (back_left_top_tetrahedra c) (back_right_down_tetrahedra c)
 
-    candidates_xy = candidates_x \\ non_candidates_y
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 c =
+  disjoint (front_left_top_tetrahedra c) (back_right_top_tetrahedra c)
 
-    non_candidates_z =
-      if (in_top_half c p) then
-        down_half_tetrahedra c
-      else
-        top_half_tetrahedra c
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 c =
+  disjoint (front_left_down_tetrahedra c) (back_right_down_tetrahedra c)
 
-    candidates_xyz = candidates_xy \\ non_candidates_z
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 c =
+  disjoint (front_left_top_tetrahedra c) (back_left_down_tetrahedra c)
 
-    contains_our_point = flip contains_point p
-    containing_tetrahedra = filter contains_our_point candidates_xyz
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 c =
+  disjoint (front_right_top_tetrahedra c) (back_right_down_tetrahedra c)
 
+
+-- | Since the grid size is necessarily positive, all tetrahedra
+--   (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
+    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.
+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
+
+-- | 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
+      t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
+      t1 = head $ tail (tetrahedra cube)
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
+--   third and fourth indices of c-t1 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!
+prop_c0120_identity1 :: Cube -> Bool
+prop_c0120_identity1 cube =
+   c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
+     where
+       t0 = tetrahedron cube 0
+       t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 1 and 2.
+prop_c0120_identity2 :: Cube -> Bool
+prop_c0120_identity2 cube =
+   c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
+     where
+       t0 = tetrahedron cube 0
+       t1 = tetrahedron cube 1
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 1 and 2.
+prop_c0120_identity3 :: Cube -> Bool
+prop_c0120_identity3 cube =
+   c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
+     where
+       t1 = tetrahedron cube 1
+       t2 = tetrahedron cube 2
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 2 and 3.
+prop_c0120_identity4 :: Cube -> Bool
+prop_c0120_identity4 cube =
+   c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
+     where
+       t2 = tetrahedron cube 2
+       t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 4 and 5.
+prop_c0120_identity5 :: Cube -> Bool
+prop_c0120_identity5 cube =
+    c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
+    where
+      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.
+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
+     where
+       t5 = tetrahedron cube 5
+       t6 = tetrahedron cube 6
+
+
+-- -- | 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
+     where
+       t6 = tetrahedron cube 6
+       t7 = tetrahedron cube 7
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c0210_identity1 :: Cube -> Bool
+prop_c0210_identity1 cube =
+    c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c0300_identity1 :: Cube -> Bool
+prop_c0300_identity1 cube =
+    c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c1110_identity :: Cube -> Bool
+prop_c1110_identity cube =
+    c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c1200_identity1 :: Cube -> Bool
+prop_c1200_identity1 cube =
+    c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c2100_identity1 :: Cube -> Bool
+prop_c2100_identity1 cube =
+    c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
+--   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!
+prop_c0102_identity1 :: Cube -> Bool
+prop_c0102_identity1 cube =
+    c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c0201_identity1 :: Cube -> Bool
+prop_c0201_identity1 cube =
+    c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c0300_identity2 :: Cube -> Bool
+prop_c0300_identity2 cube =
+    c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c1101_identity :: Cube -> Bool
+prop_c1101_identity cube =
+    c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c1200_identity2 :: Cube -> Bool
+prop_c1200_identity2 cube =
+    c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c2100_identity2 :: Cube -> Bool
+prop_c2100_identity2 cube =
+    c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
+--   fourth indices of c-t6 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!
+prop_c3000_identity :: Cube -> Bool
+prop_c3000_identity cube =
+    c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
+                    - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c2010_identity :: Cube -> Bool
+prop_c2010_identity cube =
+    c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
+                    - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c2001_identity :: Cube -> Bool
+prop_c2001_identity cube =
+    c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
+                    - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c1020_identity :: Cube -> Bool
+prop_c1020_identity cube =
+    c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
+                    - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c1002_identity :: Cube -> Bool
+prop_c1002_identity cube =
+    c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
+                    - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c1011_identity :: Cube -> Bool
+prop_c1011_identity cube =
+    c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
+                    ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        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
+prop_interior_values_all_identical cube =
+    all_equal [ eval (function_values tet) I | tet <- tetrahedra cube ]
+
+
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
+--   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
+    where
+      t0 = tetrahedron cube 0
+      t6 = tetrahedron cube 6
+
+      -- What gets computed for c2100 of t6.
+      expr1 = eval (function_values t6) $
+          (3/8)*I +
+          (1/12)*(T + R + L + D) +
+          (1/64)*(FT + FR + FL + FD) +
+          (7/48)*F +
+          (1/48)*B +
+          (1/96)*(RT + LD + LT + RD) +
+          (1/192)*(BT + BR + BL + BD)
+
+      -- What should be computed for c2100 of t6.
+      expr2 = eval (function_values t0) $
+              (3/8)*I +
+              (1/12)*(F + R + L + B) +
+              (1/64)*(FT + RT + LT + BT) +
+              (7/48)*T +
+              (1/48)*D +
+              (1/96)*(FR + FL + BR + BL) +
+              (1/192)*(FD + RD + LD + BD)
+
+
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and
+--   Zeilfelder, p. 87.  This test checks the actual value based on
+--   the FunctionValues of the cube.
+--
+--   If 'prop_c_tilde_2100_rotation_correct' passes, then this test is
+--   even meaningful!
+prop_c_tilde_2100_correct :: Cube -> Bool
+prop_c_tilde_2100_correct cube =
+    c t6 2 1 0 0 == expected
+    where
+      t0 = tetrahedron cube 0
+      t6 = tetrahedron cube 6
+      fvs = function_values t0
+      expected = eval fvs $
+                  (3/8)*I +
+                  (1/12)*(F + R + L + B) +
+                  (1/64)*(FT + RT + LT + BT) +
+                  (7/48)*T +
+                  (1/48)*D +
+                  (1/96)*(FR + FL + BR + BL) +
+                  (1/192)*(FD + RD + LD + BD)
+
+
+-- Tests to check that the correct edges are incidental.
+prop_t0_shares_edge_with_t1 :: Cube -> Bool
+prop_t0_shares_edge_with_t1 cube =
+    (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+prop_t0_shares_edge_with_t3 :: Cube -> Bool
+prop_t0_shares_edge_with_t3 cube =
+    (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+prop_t0_shares_edge_with_t6 :: Cube -> Bool
+prop_t0_shares_edge_with_t6 cube =
+    (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+prop_t1_shares_edge_with_t2 :: Cube -> Bool
+prop_t1_shares_edge_with_t2 cube =
+    (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
+      where
+        t1 = tetrahedron cube 1
+        t2 = tetrahedron cube 2
+
+prop_t1_shares_edge_with_t19 :: Cube -> Bool
+prop_t1_shares_edge_with_t19 cube =
+    (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
+      where
+        t1 = tetrahedron cube 1
+        t19 = tetrahedron cube 19
+
+prop_t2_shares_edge_with_t3 :: Cube -> Bool
+prop_t2_shares_edge_with_t3 cube =
+    (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
+      where
+        t1 = tetrahedron cube 1
+        t2 = tetrahedron cube 2
+
+prop_t2_shares_edge_with_t12 :: Cube -> Bool
+prop_t2_shares_edge_with_t12 cube =
+    (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
+      where
+        t2 = tetrahedron cube 2
+        t12 = tetrahedron cube 12
+
+prop_t3_shares_edge_with_t21 :: Cube -> Bool
+prop_t3_shares_edge_with_t21 cube =
+    (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
+      where
+        t3 = tetrahedron cube 3
+        t21 = tetrahedron cube 21
+
+prop_t4_shares_edge_with_t5 :: Cube -> Bool
+prop_t4_shares_edge_with_t5 cube =
+    (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
+      where
+        t4 = tetrahedron cube 4
+        t5 = tetrahedron cube 5
+
+prop_t4_shares_edge_with_t7 :: Cube -> Bool
+prop_t4_shares_edge_with_t7 cube =
+    (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
+      where
+        t4 = tetrahedron cube 4
+        t7 = tetrahedron cube 7
+
+prop_t4_shares_edge_with_t10 :: Cube -> Bool
+prop_t4_shares_edge_with_t10 cube =
+    (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
+      where
+        t4 = tetrahedron cube 4
+        t10 = tetrahedron cube 10
+
+prop_t5_shares_edge_with_t6 :: Cube -> Bool
+prop_t5_shares_edge_with_t6 cube =
+    (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
+      where
+        t5 = tetrahedron cube 5
+        t6 = tetrahedron cube 6
+
+prop_t5_shares_edge_with_t16 :: Cube -> Bool
+prop_t5_shares_edge_with_t16 cube =
+    (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
+      where
+        t5 = tetrahedron cube 5
+        t16 = tetrahedron cube 16
+
+prop_t6_shares_edge_with_t7 :: Cube -> Bool
+prop_t6_shares_edge_with_t7 cube =
+    (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
+      where
+        t6 = tetrahedron cube 6
+        t7 = tetrahedron cube 7
+
+prop_t7_shares_edge_with_t20 :: Cube -> Bool
+prop_t7_shares_edge_with_t20 cube =
+    (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
+      where
+        t7 = tetrahedron cube 7
+        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" [
+      testProperty "c0120 identity1" prop_c0120_identity1,
+      testProperty "c0120 identity2" prop_c0120_identity2,
+      testProperty "c0120 identity3" prop_c0120_identity3,
+      testProperty "c0120 identity4" prop_c0120_identity4,
+      testProperty "c0120 identity5" prop_c0120_identity5,
+      testProperty "c0120 identity6" prop_c0120_identity6,
+      testProperty "c0120 identity7" prop_c0120_identity7,
+      testProperty "c0210 identity1" prop_c0210_identity1,
+      testProperty "c0300 identity1" prop_c0300_identity1,
+      testProperty "c1110 identity" prop_c1110_identity,
+      testProperty "c1200 identity1" prop_c1200_identity1,
+      testProperty "c2100 identity1" prop_c2100_identity1]
+
+p79_27_properties :: Test.Framework.Test
+p79_27_properties =
+    testGroup "p. 79, Section (2.7) Properties" [
+      testProperty "c0102 identity1" prop_c0102_identity1,
+      testProperty "c0201 identity1" prop_c0201_identity1,
+      testProperty "c0300 identity2" prop_c0300_identity2,
+      testProperty "c1101 identity" prop_c1101_identity,
+      testProperty "c1200 identity2" prop_c1200_identity2,
+      testProperty "c2100 identity2" prop_c2100_identity2 ]
+
+
+p79_28_properties :: Test.Framework.Test
+p79_28_properties =
+  testGroup "p. 79, Section (2.8) Properties" [
+    testProperty "c3000 identity" prop_c3000_identity,
+    testProperty "c2010 identity" prop_c2010_identity,
+    testProperty "c2001 identity" prop_c2001_identity,
+    testProperty "c1020 identity" prop_c1020_identity,
+    testProperty "c1002 identity" prop_c1002_identity,
+    testProperty "c1011 identity" prop_c1011_identity ]
+
+
+edge_incidence_tests :: Test.Framework.Test
+edge_incidence_tests =
+    testGroup "Edge Incidence Tests" [
+      testProperty "t0 shares edge with t6" prop_t0_shares_edge_with_t6,
+      testProperty "t0 shares edge with t1" prop_t0_shares_edge_with_t1,
+      testProperty "t0 shares edge with t3" prop_t0_shares_edge_with_t3,
+      testProperty "t1 shares edge with t2" prop_t1_shares_edge_with_t2,
+      testProperty "t1 shares edge with t19" prop_t1_shares_edge_with_t19,
+      testProperty "t2 shares edge with t3" prop_t2_shares_edge_with_t3,
+      testProperty "t2 shares edge with t12" prop_t2_shares_edge_with_t12,
+      testProperty "t3 shares edge with t21" prop_t3_shares_edge_with_t21,
+      testProperty "t4 shares edge with t5" prop_t4_shares_edge_with_t5,
+      testProperty "t4 shares edge with t7" prop_t4_shares_edge_with_t7,
+      testProperty "t4 shares edge with t10" prop_t4_shares_edge_with_t10,
+      testProperty "t5 shares edge with t6" prop_t5_shares_edge_with_t6,
+      testProperty "t5 shares edge with t16" prop_t5_shares_edge_with_t16,
+      testProperty "t6 shares edge with t7" prop_t6_shares_edge_with_t7,
+      testProperty "t7 shares edge with t20" prop_t7_shares_edge_with_t20 ]
+
+cube_properties :: Test.Framework.Test
+cube_properties =
+  testGroup "Cube Properties" [
+    p78_25_properties,
+    p79_26_properties,
+    p79_27_properties,
+    p79_28_properties,
+    edge_incidence_tests,
+    testProperty "opposite octant tetrahedra are disjoint (1)"
+      prop_opposite_octant_tetrahedra_disjoint1,
+    testProperty "opposite octant tetrahedra are disjoint (2)"
+      prop_opposite_octant_tetrahedra_disjoint2,
+    testProperty "opposite octant tetrahedra are disjoint (3)"
+     prop_opposite_octant_tetrahedra_disjoint3,
+    testProperty "opposite octant tetrahedra are disjoint (4)"
+      prop_opposite_octant_tetrahedra_disjoint4,
+    testProperty "opposite octant tetrahedra are disjoint (5)"
+      prop_opposite_octant_tetrahedra_disjoint5,
+    testProperty "opposite octant tetrahedra are disjoint (6)"
+      prop_opposite_octant_tetrahedra_disjoint6,
+    testProperty "all volumes positive" prop_all_volumes_positive,
+    testProperty "all volumes exact" prop_all_volumes_exact,
+    testProperty "v0 all equal" prop_v0_all_equal,
+    testProperty "interior values all identical"
+      prop_interior_values_all_identical,
+    testProperty "c-tilde_2100 rotation correct"
+      prop_c_tilde_2100_rotation_correct,
+    testProperty "c-tilde_2100 correct"
+      prop_c_tilde_2100_correct ]