X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTetrahedron.hs;h=e71151a1e6a1d370a1057010415976a332cc15fe;hb=db0cb44e128f8dc20625bc0888000df262cbe2ac;hp=4606761b94b3df7986fd07ce18b9d32277677d8b;hpb=3d197ab1a23d654d60617db6559daed195f1e016;p=spline3.git

diff --git a/src/Tetrahedron.hs b/src/Tetrahedron.hs
index 4606761..e71151a 100644
--- a/src/Tetrahedron.hs
+++ b/src/Tetrahedron.hs
@@ -1,25 +1,45 @@
-module Tetrahedron
+module Tetrahedron (
+  Tetrahedron(..),
+  b0, -- Cube test
+  b1, -- Cube test
+  b2, -- Cube test
+  b3, -- Cube test
+  c,
+  polynomial,
+  tetrahedron_properties,
+  tetrahedron_tests,
+  volume -- Cube test
+  )
 where
 
-import Numeric.LinearAlgebra hiding (i, scale)
-import Prelude hiding (LT)
-import Test.QuickCheck (Arbitrary(..), Gen, Positive(..))
+import qualified Data.Vector as V (
+  singleton,
+  snoc,
+  sum
+  )
 
-import Cardinal
-import Comparisons (nearly_ge)
-import FunctionValues
+import Test.Framework (Test, testGroup)
+import Test.Framework.Providers.HUnit (testCase)
+import Test.Framework.Providers.QuickCheck2 (testProperty)
+import Test.HUnit (Assertion, assertEqual)
+import Test.QuickCheck (Arbitrary(..), Gen, Property, (==>))
+
+import Comparisons ((~=), nearly_ge)
+import FunctionValues (FunctionValues(..), empty_values)
 import Misc (factorial)
-import Point
-import RealFunction
-import ThreeDimensional
+import Point (Point, scale)
+import RealFunction (RealFunction, cmult, fexp)
+import ThreeDimensional (ThreeDimensional(..))
 
-data Tetrahedron = Tetrahedron { fv :: FunctionValues,
-                                 v0 :: Point,
-                                 v1 :: Point,
-                                 v2 :: Point,
-                                 v3 :: Point,
-                                 precomputed_volume :: Double }
-                   deriving (Eq)
+data Tetrahedron =
+  Tetrahedron { function_values :: FunctionValues,
+                v0 :: Point,
+                v1 :: Point,
+                v2 :: Point,
+                v3 :: Point,
+                precomputed_volume :: Double
+              }
+    deriving (Eq)
 
 
 instance Arbitrary Tetrahedron where
@@ -29,13 +49,17 @@ instance Arbitrary Tetrahedron where
       rnd_v2 <- arbitrary :: Gen Point
       rnd_v3 <- arbitrary :: Gen Point
       rnd_fv <- arbitrary :: Gen FunctionValues
-      (Positive rnd_vol) <- arbitrary :: Gen (Positive Double)
-      return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 rnd_vol)
+
+      -- We can't assign an incorrect precomputed volume,
+      -- so we have to calculate the correct one here.
+      let t' = Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 0
+      let vol = volume t'
+      return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol)
 
 
 instance Show Tetrahedron where
     show t = "Tetrahedron:\n" ++
-             "  fv: " ++ (show (fv t)) ++ "\n" ++
+             "  function_values: " ++ (show (function_values t)) ++ "\n" ++
              "  v0: " ++ (show (v0 t)) ++ "\n" ++
              "  v1: " ++ (show (v1 t)) ++ "\n" ++
              "  v2: " ++ (show (v2 t)) ++ "\n" ++
@@ -43,8 +67,10 @@ instance Show Tetrahedron where
 
 
 instance ThreeDimensional Tetrahedron where
-    center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
-    contains_point t p =
+    center (Tetrahedron _ v0' v1' v2' v3' _) =
+        (v0' + v1' + v2' + v3') `scale` (1/4)
+
+    contains_point t p0 =
       b0_unscaled `nearly_ge` 0 &&
       b1_unscaled `nearly_ge` 0 &&
       b2_unscaled `nearly_ge` 0 &&
@@ -54,46 +80,44 @@ instance ThreeDimensional Tetrahedron where
         -- would do if we used the regular b0,..b3 functions.
         b0_unscaled :: Double
         b0_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v0 = p }
+          where inner_tetrahedron = t { v0 = p0 }
 
         b1_unscaled :: Double
         b1_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v1 = p }
+          where inner_tetrahedron = t { v1 = p0 }
 
         b2_unscaled :: Double
         b2_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v2 = p }
+          where inner_tetrahedron = t { v2 = p0 }
 
         b3_unscaled :: Double
         b3_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v3 = p }
+          where inner_tetrahedron = t { v3 = p0 }
 
 
 polynomial :: Tetrahedron -> (RealFunction Point)
 polynomial t =
-    sum [ (c t i j k l) `cmult` (beta t i j k l) | i <- [0..3],
-                                                   j <- [0..3],
-                                                   k <- [0..3],
-                                                   l <- [0..3],
-                                                   i + j + k + l == 3]
-
-
--- | Returns the domain point of t with indices i,j,k,l.
---   Simply an alias for the domain_point function.
-xi :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
-xi = domain_point
-
--- | Returns the domain point of t with indices i,j,k,l.
-domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
-domain_point t i j k l
-   | i + j + k + l == 3 = weighted_sum `scale` (1/3)
-   | otherwise = error "domain point index out of bounds"
-   where
-     v0' = (v0 t) `scale` (fromIntegral i)
-     v1' = (v1 t) `scale` (fromIntegral j)
-     v2' = (v2 t) `scale` (fromIntegral k)
-     v3' = (v3 t) `scale` (fromIntegral l)
-     weighted_sum = v0' + v1' + v2' + v3'
+    V.sum $ V.singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `V.snoc`
+            ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `V.snoc`
+            ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `V.snoc`
+            ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `V.snoc`
+            ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `V.snoc`
+            ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `V.snoc`
+            ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `V.snoc`
+            ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `V.snoc`
+            ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `V.snoc`
+            ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `V.snoc`
+            ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `V.snoc`
+            ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `V.snoc`
+            ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `V.snoc`
+            ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `V.snoc`
+            ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `V.snoc`
+            ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `V.snoc`
+            ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `V.snoc`
+            ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `V.snoc`
+            ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `V.snoc`
+            ((c t 3 0 0 0) `cmult` (beta t 3 0 0 0))
+
 
 
 -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
@@ -118,151 +142,184 @@ beta t i j k l
 --   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
 --   function will simply error.
 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
-c t 0 0 3 0 = eval (fv t) $
-              (1/8) * (I + F + L + T + LT + FL + FT + FLT)
-
-c t 0 0 0 3 = eval (fv t) $
-              (1/8) * (I + F + R + T + RT + FR + FT + FRT)
-
-c t 0 0 2 1 = eval (fv t) $
-              (5/24)*(I + F + T + FT) +
-              (1/24)*(L + FL + LT + FLT)
-
-c t 0 0 1 2 = eval (fv t) $
-              (5/24)*(I + F + T + FT) +
-              (1/24)*(R + FR + RT + FRT)
-
-c t 0 1 2 0 = eval (fv t) $
-              (5/24)*(I + F) +
-              (1/8)*(L + T + FL + FT) +
-              (1/24)*(LT + FLT)
-
-c t 0 1 0 2 = eval (fv t) $
-              (5/24)*(I + F) +
-              (1/8)*(R + T + FR + FT) +
-              (1/24)*(RT + FRT)
-
-c t 0 1 1 1 = eval (fv t) $
-          (13/48)*(I + F) +
-          (7/48)*(T + FT) +
-          (1/32)*(L + R + FL + FR) +
-          (1/96)*(LT + RT + FLT + FRT)
-
-c t 0 2 1 0 = eval (fv t) $
-          (13/48)*(I + F) +
-          (17/192)*(L + T + FL + FT) +
-          (1/96)*(LT + FLT) +
-          (1/64)*(R + D + FR + FD) +
-          (1/192)*(RT + LD + FRT + FLD)
-
-c t 0 2 0 1 = eval (fv t) $
-          (13/48)*(I + F) +
-          (17/192)*(R + T + FR + FT) +
-          (1/96)*(RT + FRT) +
-          (1/64)*(L + D + FL + FD) +
-          (1/192)*(RD + LT + FLT + FRD)
-
-c t 0 3 0 0 = eval (fv t) $
-          (13/48)*(I + F) +
-          (5/96)*(L + R + T + D + FL + FR + FT + FD) +
-          (1/192)*(RT + RD + LT + LD + FRT + FRD + FLT + FLD)
-
-c t 1 0 2 0 = eval (fv t) $
-              (1/4)*I +
-              (1/6)*(F + L + T) +
-              (1/12)*(LT + FL + FT)
-
-c t 1 0 0 2 = eval (fv t) $
-              (1/4)*I +
-              (1/6)*(F + R + T) +
-              (1/12)*(RT + FR + FT)
-
-c t 1 0 1 1 = eval (fv t) $
-          (1/3)*I +
-          (5/24)*(F + T) +
-          (1/12)*FT +
-          (1/24)*(L + R) +
-          (1/48)*(LT + RT + FL + FR)
-
-c t 1 1 1 0 = eval (fv t) $
-          (1/3)*I +
-          (5/24)*F +
-          (1/8)*(L + T) +
-          (5/96)*(FL + FT) +
-          (1/48)*(D + R + LT) +
-          (1/96)*(FD + LD + RT + FR)
-
-c t 1 1 0 1 = eval (fv t) $
-          (1/3)*I +
-          (5/24)*F +
-          (1/8)*(R + T) +
-          (5/96)*(FR + FT) +
-          (1/48)*(D + L + RT) +
-          (1/96)*(FD + LT + RD + FL)
-
-c t 1 2 0 0 = eval (fv t) $
-          (1/3)*I +
-          (5/24)*F +
-          (7/96)*(L + R + T + D) +
-          (1/32)*(FL + FR + FT + FD) +
-          (1/96)*(RT + RD + LT + LD)
-
-c t 2 0 1 0 = eval (fv t) $
-          (3/8)*I +
-          (7/48)*(F + T + L) +
-          (1/48)*(R + D + B + LT + FL + FT) +
-          (1/96)*(RT + BT + FR + FD + LD + BL)
-
-c t 2 0 0 1 = eval (fv t) $
-          (3/8)*I +
-          (7/48)*(F + T + R) +
-          (1/48)*(L + D + B + RT + FR + FT) +
-          (1/96)*(LT + BT + FL + FD + RD + BR)
-
-c t 2 1 0 0 = eval (fv t) $
-          (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)
-
-c t 3 0 0 0 = eval (fv t) $
-          (3/8)*I +
-          (1/12)*(T + F + L + R + D + B) +
-          (1/96)*(LT + FL + FT + RT + BT + FR) +
-          (1/96)*(FD + LD + BD + BR + RD + BL)
-
-c _ _ _ _ _ = error "coefficient index out of bounds"
-
-
-
--- | The matrix used in the tetrahedron volume calculation as given in
---   Lai & Schumaker, Definition 15.4, page 436.
-vol_matrix :: Tetrahedron -> Matrix Double
-vol_matrix t = (4><4)
-               [1,  1,  1,  1,
-                x1, x2, x3, x4,
-                y1, y2, y3, y4,
-                z1, z2, z3, z4 ]
-    where
-      (x1, y1, z1) = v0 t
-      (x2, y2, z2) = v1 t
-      (x3, y3, z3) = v2 t
-      (x4, y4, z4) = v3 t
+c t i j k l =
+  coefficient i j k l
+  where
+    fvs = function_values t
+    f = front fvs
+    b = back fvs
+    r = right fvs
+    l' = left fvs
+    t' = top fvs
+    d  = down fvs
+    fl = front_left fvs
+    fr = front_right fvs
+    fd = front_down fvs
+    ft = front_top fvs
+    bl = back_left fvs
+    br = back_right fvs
+    bd = back_down fvs
+    bt = back_top fvs
+    ld = left_down fvs
+    lt = left_top fvs
+    rd = right_down fvs
+    rt = right_top fvs
+    fld = front_left_down fvs
+    flt = front_left_top fvs
+    frd = front_right_down fvs
+    frt = front_right_top fvs
+    i' = interior fvs
+
+    coefficient :: Int -> Int -> Int -> Int -> Double
+    coefficient 0 0 3 0 =
+      (1/8) * (i' + f + l' + t' + lt + fl + ft + flt)
+
+    coefficient 0 0 0 3 =
+      (1/8) * (i' + f + r + t' + rt + fr + ft + frt)
+
+    coefficient 0 0 2 1 =
+      (5/24)*(i' + f + t' + ft) + (1/24)*(l' + fl + lt + flt)
+
+    coefficient 0 0 1 2 =
+      (5/24)*(i' + f + t' + ft) + (1/24)*(r + fr + rt + frt)
+
+    coefficient 0 1 2 0 =
+      (5/24)*(i' + f) + (1/8)*(l' + t' + fl + ft)
+                      + (1/24)*(lt + flt)
+
+    coefficient 0 1 0 2 =
+      (5/24)*(i' + f) + (1/8)*(r + t' + fr + ft)
+                      + (1/24)*(rt + frt)
+
+    coefficient 0 1 1 1 =
+      (13/48)*(i' + f) + (7/48)*(t' + ft)
+                       + (1/32)*(l' + r + fl + fr)
+                       + (1/96)*(lt + rt + flt + frt)
+
+    coefficient 0 2 1 0 =
+      (13/48)*(i' + f) + (17/192)*(l' + t' + fl + ft)
+                       + (1/96)*(lt + flt)
+                       + (1/64)*(r + d + fr + fd)
+                       + (1/192)*(rt + ld + frt + fld)
+
+    coefficient 0 2 0 1 =
+      (13/48)*(i' + f) + (17/192)*(r + t' + fr + ft)
+                       + (1/96)*(rt + frt)
+                       + (1/64)*(l' + d + fl + fd)
+                       + (1/192)*(rd + lt + flt + frd)
+
+    coefficient 0 3 0 0 =
+     (13/48)*(i' + f) + (5/96)*(l' + r + t' + d + fl + fr + ft + fd)
+                      + (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld)
+
+    coefficient 1 0 2 0 =
+      (1/4)*i' + (1/6)*(f + l' + t')
+               + (1/12)*(lt + fl + ft)
+
+    coefficient 1 0 0 2 =
+      (1/4)*i' + (1/6)*(f + r + t')
+               + (1/12)*(rt + fr + ft)
+
+    coefficient 1 0 1 1 =
+      (1/3)*i' + (5/24)*(f + t')
+               + (1/12)*ft
+               + (1/24)*(l' + r)
+               + (1/48)*(lt + rt + fl + fr)
+
+    coefficient 1 1 1 0 =
+      (1/3)*i' + (5/24)*f
+               + (1/8)*(l' + t')
+               + (5/96)*(fl + ft)
+               + (1/48)*(d + r + lt)
+               + (1/96)*(fd + ld + rt + fr)
+
+    coefficient 1 1 0 1 =
+      (1/3)*i' + (5/24)*f
+               + (1/8)*(r + t')
+               + (5/96)*(fr + ft)
+               + (1/48)*(d + l' + rt)
+               + (1/96)*(fd + lt + rd + fl)
+
+    coefficient 1 2 0 0 =
+      (1/3)*i' + (5/24)*f
+               + (7/96)*(l' + r + t' + d)
+               + (1/32)*(fl + fr + ft + fd)
+               + (1/96)*(rt + rd + lt + ld)
+
+    coefficient 2 0 1 0 =
+      (3/8)*i' + (7/48)*(f + t' + l')
+               + (1/48)*(r + d + b + lt + fl + ft)
+               + (1/96)*(rt + bt + fr + fd + ld + bl)
+
+    coefficient 2 0 0 1 =
+      (3/8)*i' + (7/48)*(f + t' + r)
+               + (1/48)*(l' + d + b + rt + fr + ft)
+               + (1/96)*(lt + bt + fl + fd + rd + br)
+
+    coefficient 2 1 0 0 =
+      (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)
+
+    coefficient 3 0 0 0 =
+      (3/8)*i' + (1/12)*(t' + f + l' + r + d + b)
+               + (1/96)*(lt + fl + ft + rt + bt + fr)
+               + (1/96)*(fd + ld + bd + br + rd + bl)
+
+    coefficient _ _ _ _ = error "coefficient index out of bounds"
+
+
+
+-- | Compute the determinant of the 4x4 matrix,
+--
+--   [1]
+--   [x]
+--   [y]
+--   [z]
+--
+--   where [1] = [1, 1, 1, 1],
+--         [x] = [x1,x2,x3,x4],
+--
+--   et cetera.
+--
+--   The termX nonsense is an attempt to prevent Double overflow.
+--   which has been observed to happen with large coordinates.
+--
+det :: Point -> Point -> Point -> Point -> Double
+det p0 p1 p2 p3 =
+  term5 + term6
+  where
+    (x1, y1, z1) = p0
+    (x2, y2, z2) = p1
+    (x3, y3, z3) = p2
+    (x4, y4, z4) = p3
+    term1 = ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3
+    term2 = ((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4
+    term3 = ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1
+    term4 = ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2
+    term5 = term1 - term2
+    term6 = term3 - term4
+
 
 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
 --   page 436.
 volume :: Tetrahedron -> Double
 volume t
-       | (v0 t) == (v1 t) = 0
-       | (v0 t) == (v2 t) = 0
-       | (v0 t) == (v3 t) = 0
-       | (v1 t) == (v2 t) = 0
-       | (v1 t) == (v3 t) = 0
-       | (v2 t) == (v3 t) = 0
-       | otherwise = (1/6)*(det (vol_matrix t))
+       | v0' == v1' = 0
+       | v0' == v2' = 0
+       | v0' == v3' = 0
+       | v1' == v2' = 0
+       | v1' == v3' = 0
+       | v2' == v3' = 0
+       | otherwise = (1/6)*(det v0' v1' v2' v3')
+  where
+    v0' = v0 t
+    v1' = v1 t
+    v2' = v2 t
+    v3' = v3 t
 
 
 -- | The barycentric coordinates of a point with respect to v0.
@@ -291,3 +348,368 @@ b3 :: Tetrahedron -> (RealFunction Point)
 b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v3 = point }
+
+
+
+
+-- Tests
+
+
+-- | Check the volume of a particular tetrahedron (computed by hand)
+--   and whether or not it contains a specific point chosen to be
+--   outside of it. Its vertices are in clockwise order, so the volume
+--   should be negative.
+tetrahedron1_geometry_tests :: Test.Framework.Test
+tetrahedron1_geometry_tests =
+  testGroup "tetrahedron1 geometry"
+              [ testCase "volume1" volume1,
+                testCase "doesn't contain point1" doesnt_contain_point1]
+  where
+    p0 = (0, -0.5, 0)
+    p1 = (0, 0.5, 0)
+    p2 = (2, 0, 0)
+    p3 = (1, 0, 1)
+    t = Tetrahedron { v0 = p0,
+                      v1 = p1,
+                      v2 = p2,
+                      v3 = p3,
+                      function_values = empty_values,
+                      precomputed_volume = 0 }
+
+    volume1 :: Assertion
+    volume1 =
+      assertEqual "volume is correct" True (vol ~= (-1/3))
+      where
+        vol = volume t
+
+    doesnt_contain_point1 :: Assertion
+    doesnt_contain_point1 =
+      assertEqual "doesn't contain an exterior point" False contained
+      where
+        exterior_point = (5, 2, -9.0212)
+        contained = contains_point t exterior_point
+
+
+-- | Check the volume of a particular tetrahedron (computed by hand)
+--   and whether or not it contains a specific point chosen to be
+--   inside of it. Its vertices are in counter-clockwise order, so the
+--   volume should be positive.
+tetrahedron2_geometry_tests :: Test.Framework.Test
+tetrahedron2_geometry_tests =
+  testGroup "tetrahedron2 geometry"
+              [ testCase "volume1" volume1,
+                testCase "contains point1" contains_point1]
+  where
+    p0 = (0, -0.5, 0)
+    p1 = (2, 0, 0)
+    p2 = (0, 0.5, 0)
+    p3 = (1, 0, 1)
+    t = Tetrahedron { v0 = p0,
+                      v1 = p1,
+                      v2 = p2,
+                      v3 = p3,
+                      function_values = empty_values,
+                      precomputed_volume = 0 }
+
+    volume1 :: Assertion
+    volume1 = assertEqual "volume1 is correct" True (vol ~= (1/3))
+      where
+        vol = volume t
+
+    contains_point1 :: Assertion
+    contains_point1 = assertEqual "contains an inner point" True contained
+        where
+          inner_point = (1, 0, 0.5)
+          contained = contains_point t inner_point
+
+
+-- | Ensure that tetrahedra do not contain a particular point chosen to
+--   be outside of them.
+containment_tests :: Test.Framework.Test
+containment_tests =
+  testGroup "containment tests"
+              [ testCase "doesn't contain point2" doesnt_contain_point2,
+                testCase "doesn't contain point3" doesnt_contain_point3,
+                testCase "doesn't contain point4" doesnt_contain_point4,
+                testCase "doesn't contain point5" doesnt_contain_point5]
+  where
+    p2 = (0.5, 0.5, 1)
+    p3 = (0.5, 0.5, 0.5)
+    exterior_point = (0, 0, 0)
+
+    doesnt_contain_point2 :: Assertion
+    doesnt_contain_point2 =
+      assertEqual "doesn't contain an exterior point" False contained
+      where
+        p0 = (0, 1, 1)
+        p1 = (1, 1, 1)
+        t = Tetrahedron { v0 = p0,
+                          v1 = p1,
+                          v2 = p2,
+                          v3 = p3,
+                          function_values = empty_values,
+                          precomputed_volume = 0 }
+        contained = contains_point t exterior_point
+
+
+    doesnt_contain_point3 :: Assertion
+    doesnt_contain_point3 =
+      assertEqual "doesn't contain an exterior point" False contained
+      where
+        p0 = (1, 1, 1)
+        p1 = (1, 0, 1)
+        t = Tetrahedron { v0 = p0,
+                          v1 = p1,
+                          v2 = p2,
+                          v3 = p3,
+                          function_values = empty_values,
+                          precomputed_volume = 0 }
+        contained = contains_point t exterior_point
+
+
+    doesnt_contain_point4 :: Assertion
+    doesnt_contain_point4 =
+      assertEqual "doesn't contain an exterior point" False contained
+      where
+        p0 = (1, 0, 1)
+        p1 = (0, 0, 1)
+        t = Tetrahedron { v0 = p0,
+                          v1 = p1,
+                          v2 = p2,
+                          v3 = p3,
+                          function_values = empty_values,
+                          precomputed_volume = 0 }
+        contained = contains_point t exterior_point
+
+
+    doesnt_contain_point5 :: Assertion
+    doesnt_contain_point5 =
+      assertEqual "doesn't contain an exterior point" False contained
+      where
+        p0 = (0, 0, 1)
+        p1 = (0, 1, 1)
+        t = Tetrahedron { v0 = p0,
+                          v1 = p1,
+                          v2 = p2,
+                          v3 = p3,
+                          function_values = empty_values,
+                          precomputed_volume = 0 }
+        contained = contains_point t exterior_point
+
+
+-- | The barycentric coordinate of v0 with respect to itself should
+--   be one.
+prop_b0_v0_always_unity :: Tetrahedron -> Property
+prop_b0_v0_always_unity t =
+    (volume t) > 0 ==> (b0 t) (v0 t) ~= 1.0
+
+-- | The barycentric coordinate of v1 with respect to v0 should
+--   be zero.
+prop_b0_v1_always_zero :: Tetrahedron -> Property
+prop_b0_v1_always_zero t =
+    (volume t) > 0 ==> (b0 t) (v1 t) ~= 0
+
+-- | The barycentric coordinate of v2 with respect to v0 should
+--   be zero.
+prop_b0_v2_always_zero :: Tetrahedron -> Property
+prop_b0_v2_always_zero t =
+    (volume t) > 0 ==> (b0 t) (v2 t) ~= 0
+
+-- | The barycentric coordinate of v3 with respect to v0 should
+--   be zero.
+prop_b0_v3_always_zero :: Tetrahedron -> Property
+prop_b0_v3_always_zero t =
+    (volume t) > 0 ==> (b0 t) (v3 t) ~= 0
+
+-- | The barycentric coordinate of v1 with respect to itself should
+--   be one.
+prop_b1_v1_always_unity :: Tetrahedron -> Property
+prop_b1_v1_always_unity t =
+    (volume t) > 0 ==> (b1 t) (v1 t) ~= 1.0
+
+-- | The barycentric coordinate of v0 with respect to v1 should
+--   be zero.
+prop_b1_v0_always_zero :: Tetrahedron -> Property
+prop_b1_v0_always_zero t =
+    (volume t) > 0 ==> (b1 t) (v0 t) ~= 0
+
+-- | The barycentric coordinate of v2 with respect to v1 should
+--   be zero.
+prop_b1_v2_always_zero :: Tetrahedron -> Property
+prop_b1_v2_always_zero t =
+    (volume t) > 0 ==> (b1 t) (v2 t) ~= 0
+
+-- | The barycentric coordinate of v3 with respect to v1 should
+--   be zero.
+prop_b1_v3_always_zero :: Tetrahedron -> Property
+prop_b1_v3_always_zero t =
+    (volume t) > 0 ==> (b1 t) (v3 t) ~= 0
+
+-- | The barycentric coordinate of v2 with respect to itself should
+--   be one.
+prop_b2_v2_always_unity :: Tetrahedron -> Property
+prop_b2_v2_always_unity t =
+    (volume t) > 0 ==> (b2 t) (v2 t) ~= 1.0
+
+-- | The barycentric coordinate of v0 with respect to v2 should
+--   be zero.
+prop_b2_v0_always_zero :: Tetrahedron -> Property
+prop_b2_v0_always_zero t =
+    (volume t) > 0 ==> (b2 t) (v0 t) ~= 0
+
+-- | The barycentric coordinate of v1 with respect to v2 should
+--   be zero.
+prop_b2_v1_always_zero :: Tetrahedron -> Property
+prop_b2_v1_always_zero t =
+    (volume t) > 0 ==> (b2 t) (v1 t) ~= 0
+
+-- | The barycentric coordinate of v3 with respect to v2 should
+--   be zero.
+prop_b2_v3_always_zero :: Tetrahedron -> Property
+prop_b2_v3_always_zero t =
+    (volume t) > 0 ==> (b2 t) (v3 t) ~= 0
+
+-- | The barycentric coordinate of v3 with respect to itself should
+--   be one.
+prop_b3_v3_always_unity :: Tetrahedron -> Property
+prop_b3_v3_always_unity t =
+    (volume t) > 0 ==> (b3 t) (v3 t) ~= 1.0
+
+-- | The barycentric coordinate of v0 with respect to v3 should
+--   be zero.
+prop_b3_v0_always_zero :: Tetrahedron -> Property
+prop_b3_v0_always_zero t =
+    (volume t) > 0 ==> (b3 t) (v0 t) ~= 0
+
+-- | The barycentric coordinate of v1 with respect to v3 should
+--   be zero.
+prop_b3_v1_always_zero :: Tetrahedron -> Property
+prop_b3_v1_always_zero t =
+    (volume t) > 0 ==> (b3 t) (v1 t) ~= 0
+
+-- | The barycentric coordinate of v2 with respect to v3 should
+--   be zero.
+prop_b3_v2_always_zero :: Tetrahedron -> Property
+prop_b3_v2_always_zero t =
+    (volume t) > 0 ==> (b3 t) (v2 t) ~= 0
+
+
+prop_swapping_vertices_doesnt_affect_coefficients1 :: Tetrahedron -> Bool
+prop_swapping_vertices_doesnt_affect_coefficients1 t =
+      c t 0 0 1 2 == c t' 0 0 1 2
+        where
+          t' = t { v0 = (v1 t), v1 = (v0 t) }
+
+prop_swapping_vertices_doesnt_affect_coefficients2 :: Tetrahedron -> Bool
+prop_swapping_vertices_doesnt_affect_coefficients2 t =
+      c t 0 1 1 1 == c t' 0 1 1 1
+        where
+          t' = t { v2 = (v3 t), v3 = (v2 t) }
+
+prop_swapping_vertices_doesnt_affect_coefficients3 :: Tetrahedron -> Bool
+prop_swapping_vertices_doesnt_affect_coefficients3 t =
+      c t 2 1 0 0 == c t' 2 1 0 0
+        where
+          t' = t { v2 = (v3 t), v3 = (v2 t) }
+
+prop_swapping_vertices_doesnt_affect_coefficients4 :: Tetrahedron -> Bool
+prop_swapping_vertices_doesnt_affect_coefficients4 t =
+      c t 2 0 0 1 == c t' 2 0 0 1
+        where
+          t' = t { v0 = (v3 t), v3 = (v0 t) }
+
+
+
+
+tetrahedron_tests :: Test.Framework.Test
+tetrahedron_tests =
+    testGroup "Tetrahedron Tests" [
+      tetrahedron1_geometry_tests,
+      tetrahedron2_geometry_tests,
+      containment_tests ]
+
+
+
+p78_24_properties :: Test.Framework.Test
+p78_24_properties =
+  testGroup "p. 78, Section (2.4) Properties" [
+    testProperty "c3000 identity" prop_c3000_identity,
+    testProperty "c2100 identity" prop_c2100_identity,
+    testProperty "c1110 identity" prop_c1110_identity]
+  where
+    -- | Returns the domain point of t with indices i,j,k,l.
+    domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
+    domain_point t i j k l
+      | i + j + k + l == 3 = weighted_sum `scale` (1/3)
+      | otherwise = error "domain point index out of bounds"
+      where
+        v0' = (v0 t) `scale` (fromIntegral i)
+        v1' = (v1 t) `scale` (fromIntegral j)
+        v2' = (v2 t) `scale` (fromIntegral k)
+        v3' = (v3 t) `scale` (fromIntegral l)
+        weighted_sum = v0' + v1' + v2' + v3'
+
+
+    -- | Used for convenience in the next few tests.
+    p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
+    p t i j k l = (polynomial t) (domain_point t i j k l)
+
+
+    -- | Given in Sorokina and Zeilfelder, p. 78.
+    prop_c3000_identity :: Tetrahedron -> Property
+    prop_c3000_identity t =
+      (volume t) > 0 ==>
+        c t 3 0 0 0 ~= p t 3 0 0 0
+
+    -- | Given in Sorokina and Zeilfelder, p. 78.
+    prop_c2100_identity :: Tetrahedron -> Property
+    prop_c2100_identity t =
+      (volume t) > 0 ==>
+        c t 2 1 0 0 ~= (term1 - term2 + term3 - term4)
+        where
+          term1 = (1/3)*(p t 0 3 0 0)
+          term2 = (5/6)*(p t 3 0 0 0)
+          term3 = 3*(p t 2 1 0 0)
+          term4 = (3/2)*(p t 1 2 0 0)
+
+    -- | Given in Sorokina and Zeilfelder, p. 78.
+    prop_c1110_identity :: Tetrahedron -> Property
+    prop_c1110_identity t =
+      (volume t) > 0 ==>
+        c t 1 1 1 0 ~= (term1 + term2 - term3 - term4)
+        where
+          term1 = (1/3)*((p t 3 0 0 0) + (p t 0 3 0 0) + (p t 0 0 3 0))
+          term2 = (9/2)*(p t 1 1 1 0)
+          term3 = (3/4)*((p t 2 1 0 0) + (p t 1 2 0 0) + (p t 2 0 1 0))
+          term4 = (3/4)*((p t 1 0 2 0) + (p t 0 2 1 0) + (p t 0 1 2 0))
+
+
+
+tetrahedron_properties :: Test.Framework.Test
+tetrahedron_properties =
+  testGroup "Tetrahedron Properties" [
+    p78_24_properties,
+    testProperty "b0_v0_always_unity" prop_b0_v0_always_unity,
+    testProperty "b0_v1_always_zero" prop_b0_v1_always_zero,
+    testProperty "b0_v2_always_zero" prop_b0_v2_always_zero,
+    testProperty "b0_v3_always_zero" prop_b0_v3_always_zero,
+    testProperty "b1_v1_always_unity" prop_b1_v1_always_unity,
+    testProperty "b1_v0_always_zero" prop_b1_v0_always_zero,
+    testProperty "b1_v2_always_zero" prop_b1_v2_always_zero,
+    testProperty "b1_v3_always_zero" prop_b1_v3_always_zero,
+    testProperty "b2_v2_always_unity" prop_b2_v2_always_unity,
+    testProperty "b2_v0_always_zero" prop_b2_v0_always_zero,
+    testProperty "b2_v1_always_zero" prop_b2_v1_always_zero,
+    testProperty "b2_v3_always_zero" prop_b2_v3_always_zero,
+    testProperty "b3_v3_always_unity" prop_b3_v3_always_unity,
+    testProperty "b3_v0_always_zero" prop_b3_v0_always_zero,
+    testProperty "b3_v1_always_zero" prop_b3_v1_always_zero,
+    testProperty "b3_v2_always_zero" prop_b3_v2_always_zero,
+    testProperty "swapping_vertices_doesnt_affect_coefficients1" $
+      prop_swapping_vertices_doesnt_affect_coefficients1,
+    testProperty "swapping_vertices_doesnt_affect_coefficients2" $
+      prop_swapping_vertices_doesnt_affect_coefficients2,
+    testProperty "swapping_vertices_doesnt_affect_coefficients3" $
+      prop_swapping_vertices_doesnt_affect_coefficients3,
+    testProperty "swapping_vertices_doesnt_affect_coefficients4" $
+      prop_swapping_vertices_doesnt_affect_coefficients4 ]