X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTetrahedron.hs;h=87bfd5e8a65b8621179d480bd636a1feeed6854b;hb=fe1984d347cee4262a3af2117fd402bb603fd0ce;hp=1f7c22b355c0392f26988a1d9c315e63fee2d68e;hpb=b2e1c440b9b1bb99ae564d6600230bbd1f7d204c;p=spline3.git

diff --git a/src/Tetrahedron.hs b/src/Tetrahedron.hs
index 1f7c22b..87bfd5e 100644
--- a/src/Tetrahedron.hs
+++ b/src/Tetrahedron.hs
@@ -1,25 +1,55 @@
-module Tetrahedron
+-- The local "coefficient" function defined within the "c" function
+-- pattern matches on a bunch of integers, but doesn't handle the
+-- "otherwise" case for performance reasons.
+{-# OPTIONS_GHC -Wno-incomplete-patterns #-}
+{-# LANGUAGE BangPatterns #-}
+
+module Tetrahedron (
+  Tetrahedron(..),
+  b0, -- Cube test
+  b1, -- Cube test
+  b2, -- Cube test
+  b3, -- Cube test
+  barycenter,
+  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)
-
-import Cardinal
-import Comparisons (nearly_ge)
-import FunctionValues
-import Misc (factorial)
-import Point
-import RealFunction
-import ThreeDimensional
-
-data Tetrahedron = Tetrahedron { fv :: FunctionValues,
-                                 v0 :: Point,
-                                 v1 :: Point,
-                                 v2 :: Point,
-                                 v3 :: Point,
-                                 precomputed_volume :: Double }
-                   deriving (Eq)
+import Data.Vector ( singleton, snoc )
+import qualified Data.Vector as V ( sum )
+import Test.Tasty ( TestTree, testGroup )
+import Test.Tasty.HUnit ( Assertion, assertEqual, testCase )
+import Test.Tasty.QuickCheck (
+  Arbitrary( arbitrary ),
+  Gen,
+  Property,
+  (==>),
+  testProperty )
+
+import Comparisons ( (~=) )
+import FunctionValues (
+  FunctionValues( front, back, left, right, top, down, front_left,
+                  front_right, front_down, front_top, back_left, back_right,
+                  back_down, back_top, left_down, left_top, right_down,
+                  right_top, front_left_down, front_left_top,
+                  front_right_down, front_right_top, interior ),
+  empty_values )
+import Misc ( factorial )
+import Point ( Point(Point), scale )
+import RealFunction ( RealFunction, cmult, fexp )
+
+data Tetrahedron =
+  Tetrahedron { function_values :: FunctionValues,
+                v0 :: !Point,
+                v1 :: !Point,
+                v2 :: !Point,
+                v3 :: !Point,
+                precomputed_volume :: !Double
+              }
+    deriving (Eq)
 
 
 instance Arbitrary Tetrahedron where
@@ -29,6 +59,7 @@ instance Arbitrary Tetrahedron where
       rnd_v2 <- arbitrary :: Gen Point
       rnd_v3 <- arbitrary :: Gen Point
       rnd_fv <- arbitrary :: Gen FunctionValues
+
       -- 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
@@ -38,77 +69,55 @@ instance Arbitrary Tetrahedron where
 
 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" ++
              "  v3: " ++ (show (v3 t)) ++ "\n"
 
 
-instance ThreeDimensional Tetrahedron where
-    center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
-    contains_point t p =
-      b0_unscaled `nearly_ge` 0 &&
-      b1_unscaled `nearly_ge` 0 &&
-      b2_unscaled `nearly_ge` 0 &&
-      b3_unscaled `nearly_ge` 0
-      where
-        -- Drop the useless division and volume calculation that we
-        -- would do if we used the regular b0,..b3 functions.
-        b0_unscaled :: Double
-        b0_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v0 = p }
-
-        b1_unscaled :: Double
-        b1_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v1 = p }
-
-        b2_unscaled :: Double
-        b2_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v2 = p }
+-- | Find the barycenter of the given tetrahedron.
+--   We just average the four vertices.
+barycenter :: Tetrahedron -> Point
+barycenter (Tetrahedron _ v0' v1' v2' v3' _) =
+  (v0' + v1' + v2' + v3') `scale` (1/4)
 
-        b3_unscaled :: Double
-        b3_unscaled = volume inner_tetrahedron
-          where inner_tetrahedron = t { v3 = p }
 
 
+{-# INLINE polynomial #-}
 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 $ singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `snoc`
+            ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `snoc`
+            ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `snoc`
+            ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `snoc`
+            ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `snoc`
+            ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `snoc`
+            ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `snoc`
+            ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `snoc`
+            ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `snoc`
+            ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `snoc`
+            ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `snoc`
+            ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `snoc`
+            ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `snoc`
+            ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `snoc`
+            ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `snoc`
+            ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `snoc`
+            ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `snoc`
+            ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `snoc`
+            ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `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
 --   capital 'B' in the Sorokina/Zeilfelder paper.
 beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
-beta t i j k l
-  | (i + j + k + l == 3) =
-      coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
-  | otherwise = error "basis function index out of bounds"
+beta t i j k l =
+  coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
   where
     denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
-    coefficient = 6 / (fromIntegral denominator)
+    coefficient = (6 / (fromIntegral denominator)) :: Double
     b0_term = (b0 t) `fexp` i
     b1_term = (b1 t) `fexp` j
     b2_term = (b2 t) `fexp` k
@@ -118,157 +127,178 @@ beta t i j k l
 -- | The coefficient function. c t i j k l returns the coefficient
 --   c_ijkl with respect to the tetrahedron t. The definition uses
 --   pattern matching to mimic the definitions given in Sorokina and
---   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
---   function will simply error.
+--   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the world
+--   will end. This is for performance reasons.
 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)
+
+
+
+-- | 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
+    Point x1 y1 z1 = p0
+    Point x2 y2 z2 = p1
+    Point x3 y3 z3 = p2
+    Point 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.
+{-# INLINE volume #-}
 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))
-
+volume (Tetrahedron _ v0' v1' v2' v3' _) =
+  (1/6)*(det v0' v1' v2' v3')
 
 -- | The barycentric coordinates of a point with respect to v0.
+{-# INLINE b0 #-}
 b0 :: Tetrahedron -> (RealFunction Point)
 b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
@@ -276,6 +306,7 @@ b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 
 
 -- | The barycentric coordinates of a point with respect to v1.
+{-# INLINE b1 #-}
 b1 :: Tetrahedron -> (RealFunction Point)
 b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
@@ -283,6 +314,7 @@ b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 
 
 -- | The barycentric coordinates of a point with respect to v2.
+{-# INLINE b2 #-}
 b2 :: Tetrahedron -> (RealFunction Point)
 b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
@@ -290,7 +322,278 @@ b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 
 
 -- | The barycentric coordinates of a point with respect to v3.
+{-# INLINE b3 #-}
 b3 :: Tetrahedron -> (RealFunction Point)
 b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v3 = point }
+
+
+
+
+-- | Check the volume of a particular tetrahedron (computed by hand)
+--   Its vertices are in clockwise order, so the volume should be
+--   negative.
+tetrahedron1_geometry_tests :: TestTree
+tetrahedron1_geometry_tests =
+  testGroup "tetrahedron1 geometry"
+              [ testCase "volume1" volume1 ]
+  where
+    p0 = Point 0 (-0.5) 0
+    p1 = Point 0 0.5 0
+    p2 = Point 2 0 0
+    p3 = Point 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
+
+
+-- | Check the volume of a particular tetrahedron (computed by hand)
+--   Its vertices are in counter-clockwise order, so the volume should
+--   be positive.
+tetrahedron2_geometry_tests :: TestTree
+tetrahedron2_geometry_tests =
+  testGroup "tetrahedron2 geometry"
+              [ testCase "volume1" volume1 ]
+  where
+    p0 = Point 0 (-0.5) 0
+    p1 = Point 2 0 0
+    p2 = Point 0 0.5 0
+    p3 = Point 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
+
+
+
+-- | 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 :: TestTree
+tetrahedron_tests =
+    testGroup "Tetrahedron tests" [
+      tetrahedron1_geometry_tests,
+      tetrahedron2_geometry_tests ]
+
+
+
+p78_24_properties :: TestTree
+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 =
+      weighted_sum `scale` (1/3)
+      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 :: TestTree
+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 ]