X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTetrahedron.hs;h=6da41945dd6ad01b521acb2d0349c01390c7bd54;hb=993490fd9d940f5e8dea4f934c07c1a5a6c1f8ff;hp=85ac4a372d222cea1eaa56a821f54cf340e08b82;hpb=f07f76b231a3df623aab8b6035ac6000ce2a5eb2;p=spline3.git

diff --git a/src/Tetrahedron.hs b/src/Tetrahedron.hs
index 85ac4a3..6da4194 100644
--- a/src/Tetrahedron.hs
+++ b/src/Tetrahedron.hs
@@ -1,22 +1,63 @@
-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 qualified Data.Vector as V (
+  singleton,
+  snoc,
+  sum
+  )
 import Numeric.LinearAlgebra hiding (i, scale)
 import Prelude hiding (LT)
+import Test.Framework (Test, testGroup)
+import Test.Framework.Providers.HUnit (testCase)
+import Test.Framework.Providers.QuickCheck2 (testProperty)
+import Test.HUnit
+import Test.QuickCheck (Arbitrary(..), Gen, Property, (==>))
 
 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 }
-                   deriving (Eq)
+data Tetrahedron =
+  Tetrahedron { fv :: FunctionValues,
+                v0 :: Point,
+                v1 :: Point,
+                v2 :: Point,
+                v3 :: Point,
+                precomputed_volume :: Double
+              }
+    deriving (Eq)
+
+
+instance Arbitrary Tetrahedron where
+    arbitrary = do
+      rnd_v0 <- arbitrary :: Gen Point
+      rnd_v1 <- arbitrary :: Gen Point
+      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
+      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" ++
@@ -28,18 +69,56 @@ instance Show Tetrahedron where
 
 
 instance ThreeDimensional Tetrahedron where
-    center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
+    center (Tetrahedron _ v0' v1' v2' v3' _) =
+        (v0' + v1' + v2' + v3') `scale` (1/4)
+
     contains_point t p =
-        (b0 t p) >= 0 && (b1 t p) >= 0 && (b2 t p) >= 0 && (b3 t p) >= 0
+      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 }
+
+        b3_unscaled :: Double
+        b3_unscaled = volume inner_tetrahedron
+          where inner_tetrahedron = t { v3 = p }
 
 
 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]
+    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))
 
 
 -- | Returns the domain point of t with indices i,j,k,l.
@@ -76,6 +155,11 @@ beta t i j k l
     b3_term = (b3 t) `fexp` 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.
 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)
@@ -197,6 +281,8 @@ 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,
@@ -204,21 +290,13 @@ vol_matrix t = (4><4)
                 y1, y2, y3, y4,
                 z1, z2, z3, z4 ]
     where
-      x1 = x_coord (v0 t)
-      x2 = x_coord (v1 t)
-      x3 = x_coord (v2 t)
-      x4 = x_coord (v3 t)
-      y1 = y_coord (v0 t)
-      y2 = y_coord (v1 t)
-      y3 = y_coord (v2 t)
-      y4 = y_coord (v3 t)
-      z1 = z_coord (v0 t)
-      z2 = z_coord (v1 t)
-      z3 = z_coord (v2 t)
-      z4 = z_coord (v3 t)
-
--- Computed using the formula from Lai & Schumaker, Definition 15.4,
--- page 436.
+      (x1, y1, z1) = v0 t
+      (x2, y2, z2) = v1 t
+      (x3, y3, z3) = v2 t
+      (x4, y4, z4) = v3 t
+
+-- | Computed using the formula from Lai & Schumaker, Definition 15.4,
+--   page 436.
 volume :: Tetrahedron -> Double
 volume t
        | (v0 t) == (v1 t) = 0
@@ -230,22 +308,380 @@ volume t
        | otherwise = (1/6)*(det (vol_matrix t))
 
 
+-- | The barycentric coordinates of a point with respect to v0.
 b0 :: Tetrahedron -> (RealFunction Point)
-b0 t point = (volume inner_tetrahedron) / (volume t)
+b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v0 = point }
 
+
+-- | The barycentric coordinates of a point with respect to v1.
 b1 :: Tetrahedron -> (RealFunction Point)
-b1 t point = (volume inner_tetrahedron) / (volume t)
+b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v1 = point }
 
+
+-- | The barycentric coordinates of a point with respect to v2.
 b2 :: Tetrahedron -> (RealFunction Point)
-b2 t point = (volume inner_tetrahedron) / (volume t)
+b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v2 = point }
 
+
+-- | The barycentric coordinates of a point with respect to v3.
 b3 :: Tetrahedron -> (RealFunction Point)
-b3 t point = (volume inner_tetrahedron) / (volume t)
+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,
+                      fv = 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,
+                      fv = 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,
+                          fv = 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,
+                          fv = 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,
+                          fv = 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,
+                          fv = 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
+
+
+-- | Used for convenience in the next few tests; not a test itself.
+p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
+p t i j k l = (polynomial t) (xi 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))
+
+
+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]
+
+
+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 ]