src/Tetrahedron.hs

   1 {-# LANGUAGE BangPatterns #-}
   2
   3 module Tetrahedron (
   4   Tetrahedron(..),
   5   b0, -- Cube test
   6   b1, -- Cube test
   7   b2, -- Cube test
   8   b3, -- Cube test
   9   barycenter,
  10   c,
  11   polynomial,
  12   tetrahedron_properties,
  13   tetrahedron_tests,
  14   volume ) -- Cube test
  15 where
  16
  17 import Data.Vector ( singleton, snoc )
  18 import qualified Data.Vector as V ( sum )
  19 import Test.Tasty ( TestTree, testGroup )
  20 import Test.Tasty.HUnit ( Assertion, assertEqual, testCase )
  21 import Test.Tasty.QuickCheck (
  22   Arbitrary(..),
  23   Gen,
  24   Property,
  25   (==>),
  26   testProperty )
  27
  28 import Comparisons ( (~=) )
  29 import FunctionValues ( FunctionValues(..), empty_values )
  30 import Misc ( factorial )
  31 import Point ( Point(..), scale )
  32 import RealFunction ( RealFunction, cmult, fexp )
  33
  34 data Tetrahedron =
  35   Tetrahedron { function_values :: FunctionValues,
  36                 v0 :: !Point,
  37                 v1 :: !Point,
  38                 v2 :: !Point,
  39                 v3 :: !Point,
  40                 precomputed_volume :: !Double
  41               }
  42     deriving (Eq)
  43
  44
  45 instance Arbitrary Tetrahedron where
  46     arbitrary = do
  47       rnd_v0 <- arbitrary :: Gen Point
  48       rnd_v1 <- arbitrary :: Gen Point
  49       rnd_v2 <- arbitrary :: Gen Point
  50       rnd_v3 <- arbitrary :: Gen Point
  51       rnd_fv <- arbitrary :: Gen FunctionValues
  52
  53       -- We can't assign an incorrect precomputed volume,
  54       -- so we have to calculate the correct one here.
  55       let t' = Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 0
  56       let vol = volume t'
  57       return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol)
  58
  59
  60 instance Show Tetrahedron where
  61     show t = "Tetrahedron:\n" ++
  62              "  function_values: " ++ (show (function_values t)) ++ "\n" ++
  63              "  v0: " ++ (show (v0 t)) ++ "\n" ++
  64              "  v1: " ++ (show (v1 t)) ++ "\n" ++
  65              "  v2: " ++ (show (v2 t)) ++ "\n" ++
  66              "  v3: " ++ (show (v3 t)) ++ "\n"
  67
  68
  69 -- | Find the barycenter of the given tetrahedron.
  70 --   We just average the four vertices.
  71 barycenter :: Tetrahedron -> Point
  72 barycenter (Tetrahedron _ v0' v1' v2' v3' _) =
  73   (v0' + v1' + v2' + v3') `scale` (1/4)
  74
  75
  76
  77 {-# INLINE polynomial #-}
  78 polynomial :: Tetrahedron -> (RealFunction Point)
  79 polynomial t =
  80     V.sum $ singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `snoc`
  81             ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `snoc`
  82             ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `snoc`
  83             ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `snoc`
  84             ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `snoc`
  85             ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `snoc`
  86             ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `snoc`
  87             ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `snoc`
  88             ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `snoc`
  89             ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `snoc`
  90             ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `snoc`
  91             ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `snoc`
  92             ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `snoc`
  93             ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `snoc`
  94             ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `snoc`
  95             ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `snoc`
  96             ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `snoc`
  97             ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `snoc`
  98             ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `snoc`
  99             ((c t 3 0 0 0) `cmult` (beta t 3 0 0 0))
 100
 101
 102
 103 -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
 104 --   capital 'B' in the Sorokina/Zeilfelder paper.
 105 beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
 106 beta t i j k l =
 107   coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
 108   where
 109     denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
 110     coefficient = (6 / (fromIntegral denominator)) :: Double
 111     b0_term = (b0 t) `fexp` i
 112     b1_term = (b1 t) `fexp` j
 113     b2_term = (b2 t) `fexp` k
 114     b3_term = (b3 t) `fexp` l
 115
 116
 117 -- | The coefficient function. c t i j k l returns the coefficient
 118 --   c_ijkl with respect to the tetrahedron t. The definition uses
 119 --   pattern matching to mimic the definitions given in Sorokina and
 120 --   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the world
 121 --   will end. This is for performance reasons.
 122 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
 123 c !t !i !j !k !l =
 124   coefficient i j k l
 125   where
 126     fvs = function_values t
 127     f = front fvs
 128     b = back fvs
 129     r = right fvs
 130     l' = left fvs
 131     t' = top fvs
 132     d  = down fvs
 133     fl = front_left fvs
 134     fr = front_right fvs
 135     fd = front_down fvs
 136     ft = front_top fvs
 137     bl = back_left fvs
 138     br = back_right fvs
 139     bd = back_down fvs
 140     bt = back_top fvs
 141     ld = left_down fvs
 142     lt = left_top fvs
 143     rd = right_down fvs
 144     rt = right_top fvs
 145     fld = front_left_down fvs
 146     flt = front_left_top fvs
 147     frd = front_right_down fvs
 148     frt = front_right_top fvs
 149     i' = interior fvs
 150
 151     coefficient :: Int -> Int -> Int -> Int -> Double
 152     coefficient 0 0 3 0 =
 153       (1/8) * (i' + f + l' + t' + lt + fl + ft + flt)
 154
 155     coefficient 0 0 0 3 =
 156       (1/8) * (i' + f + r + t' + rt + fr + ft + frt)
 157
 158     coefficient 0 0 2 1 =
 159       (5/24)*(i' + f + t' + ft) + (1/24)*(l' + fl + lt + flt)
 160
 161     coefficient 0 0 1 2 =
 162       (5/24)*(i' + f + t' + ft) + (1/24)*(r + fr + rt + frt)
 163
 164     coefficient 0 1 2 0 =
 165       (5/24)*(i' + f) + (1/8)*(l' + t' + fl + ft)
 166                       + (1/24)*(lt + flt)
 167
 168     coefficient 0 1 0 2 =
 169       (5/24)*(i' + f) + (1/8)*(r + t' + fr + ft)
 170                       + (1/24)*(rt + frt)
 171
 172     coefficient 0 1 1 1 =
 173       (13/48)*(i' + f) + (7/48)*(t' + ft)
 174                        + (1/32)*(l' + r + fl + fr)
 175                        + (1/96)*(lt + rt + flt + frt)
 176
 177     coefficient 0 2 1 0 =
 178       (13/48)*(i' + f) + (17/192)*(l' + t' + fl + ft)
 179                        + (1/96)*(lt + flt)
 180                        + (1/64)*(r + d + fr + fd)
 181                        + (1/192)*(rt + ld + frt + fld)
 182
 183     coefficient 0 2 0 1 =
 184       (13/48)*(i' + f) + (17/192)*(r + t' + fr + ft)
 185                        + (1/96)*(rt + frt)
 186                        + (1/64)*(l' + d + fl + fd)
 187                        + (1/192)*(rd + lt + flt + frd)
 188
 189     coefficient 0 3 0 0 =
 190      (13/48)*(i' + f) + (5/96)*(l' + r + t' + d + fl + fr + ft + fd)
 191                       + (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld)
 192
 193     coefficient 1 0 2 0 =
 194       (1/4)*i' + (1/6)*(f + l' + t')
 195                + (1/12)*(lt + fl + ft)
 196
 197     coefficient 1 0 0 2 =
 198       (1/4)*i' + (1/6)*(f + r + t')
 199                + (1/12)*(rt + fr + ft)
 200
 201     coefficient 1 0 1 1 =
 202       (1/3)*i' + (5/24)*(f + t')
 203                + (1/12)*ft
 204                + (1/24)*(l' + r)
 205                + (1/48)*(lt + rt + fl + fr)
 206
 207     coefficient 1 1 1 0 =
 208       (1/3)*i' + (5/24)*f
 209                + (1/8)*(l' + t')
 210                + (5/96)*(fl + ft)
 211                + (1/48)*(d + r + lt)
 212                + (1/96)*(fd + ld + rt + fr)
 213
 214     coefficient 1 1 0 1 =
 215       (1/3)*i' + (5/24)*f
 216                + (1/8)*(r + t')
 217                + (5/96)*(fr + ft)
 218                + (1/48)*(d + l' + rt)
 219                + (1/96)*(fd + lt + rd + fl)
 220
 221     coefficient 1 2 0 0 =
 222       (1/3)*i' + (5/24)*f
 223                + (7/96)*(l' + r + t' + d)
 224                + (1/32)*(fl + fr + ft + fd)
 225                + (1/96)*(rt + rd + lt + ld)
 226
 227     coefficient 2 0 1 0 =
 228       (3/8)*i' + (7/48)*(f + t' + l')
 229                + (1/48)*(r + d + b + lt + fl + ft)
 230                + (1/96)*(rt + bt + fr + fd + ld + bl)
 231
 232     coefficient 2 0 0 1 =
 233       (3/8)*i' + (7/48)*(f + t' + r)
 234                + (1/48)*(l' + d + b + rt + fr + ft)
 235                + (1/96)*(lt + bt + fl + fd + rd + br)
 236
 237     coefficient 2 1 0 0 =
 238       (3/8)*i' + (1/12)*(t' + r + l' + d)
 239                + (1/64)*(ft + fr + fl + fd)
 240                + (7/48)*f
 241                + (1/48)*b
 242                + (1/96)*(rt + ld + lt + rd)
 243                + (1/192)*(bt + br + bl + bd)
 244
 245     coefficient 3 0 0 0 =
 246       (3/8)*i' + (1/12)*(t' + f + l' + r + d + b)
 247                + (1/96)*(lt + fl + ft + rt + bt + fr)
 248                + (1/96)*(fd + ld + bd + br + rd + bl)
 249
 250
 251
 252 -- | Compute the determinant of the 4x4 matrix,
 253 --
 254 --   [1]
 255 --   [x]
 256 --   [y]
 257 --   [z]
 258 --
 259 --   where [1] = [1, 1, 1, 1],
 260 --         [x] = [x1,x2,x3,x4],
 261 --
 262 --   et cetera.
 263 --
 264 --   The termX nonsense is an attempt to prevent Double overflow.
 265 --   which has been observed to happen with large coordinates.
 266 --
 267 det :: Point -> Point -> Point -> Point -> Double
 268 det p0 p1 p2 p3 =
 269   term5 + term6
 270   where
 271     Point x1 y1 z1 = p0
 272     Point x2 y2 z2 = p1
 273     Point x3 y3 z3 = p2
 274     Point x4 y4 z4 = p3
 275     term1 = ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3
 276     term2 = ((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4
 277     term3 = ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1
 278     term4 = ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2
 279     term5 = term1 - term2
 280     term6 = term3 - term4
 281
 282
 283 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
 284 --   page 436.
 285 {-# INLINE volume #-}
 286 volume :: Tetrahedron -> Double
 287 volume (Tetrahedron _ v0' v1' v2' v3' _) =
 288   (1/6)*(det v0' v1' v2' v3')
 289
 290 -- | The barycentric coordinates of a point with respect to v0.
 291 {-# INLINE b0 #-}
 292 b0 :: Tetrahedron -> (RealFunction Point)
 293 b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 294              where
 295                inner_tetrahedron = t { v0 = point }
 296
 297
 298 -- | The barycentric coordinates of a point with respect to v1.
 299 {-# INLINE b1 #-}
 300 b1 :: Tetrahedron -> (RealFunction Point)
 301 b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 302              where
 303                inner_tetrahedron = t { v1 = point }
 304
 305
 306 -- | The barycentric coordinates of a point with respect to v2.
 307 {-# INLINE b2 #-}
 308 b2 :: Tetrahedron -> (RealFunction Point)
 309 b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 310              where
 311                inner_tetrahedron = t { v2 = point }
 312
 313
 314 -- | The barycentric coordinates of a point with respect to v3.
 315 {-# INLINE b3 #-}
 316 b3 :: Tetrahedron -> (RealFunction Point)
 317 b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 318              where
 319                inner_tetrahedron = t { v3 = point }
 320
 321
 322
 323
 324 -- | Check the volume of a particular tetrahedron (computed by hand)
 325 --   Its vertices are in clockwise order, so the volume should be
 326 --   negative.
 327 tetrahedron1_geometry_tests :: TestTree
 328 tetrahedron1_geometry_tests =
 329   testGroup "tetrahedron1 geometry"
 330               [ testCase "volume1" volume1 ]
 331   where
 332     p0 = Point 0 (-0.5) 0
 333     p1 = Point 0 0.5 0
 334     p2 = Point 2 0 0
 335     p3 = Point 1 0 1
 336     t = Tetrahedron { v0 = p0,
 337                       v1 = p1,
 338                       v2 = p2,
 339                       v3 = p3,
 340                       function_values = empty_values,
 341                       precomputed_volume = 0 }
 342
 343     volume1 :: Assertion
 344     volume1 =
 345       assertEqual "volume is correct" True (vol ~= (-1/3))
 346       where
 347         vol = volume t
 348
 349
 350 -- | Check the volume of a particular tetrahedron (computed by hand)
 351 --   Its vertices are in counter-clockwise order, so the volume should
 352 --   be positive.
 353 tetrahedron2_geometry_tests :: TestTree
 354 tetrahedron2_geometry_tests =
 355   testGroup "tetrahedron2 geometry"
 356               [ testCase "volume1" volume1 ]
 357   where
 358     p0 = Point 0 (-0.5) 0
 359     p1 = Point 2 0 0
 360     p2 = Point 0 0.5 0
 361     p3 = Point 1 0 1
 362     t = Tetrahedron { v0 = p0,
 363                       v1 = p1,
 364                       v2 = p2,
 365                       v3 = p3,
 366                       function_values = empty_values,
 367                       precomputed_volume = 0 }
 368
 369     volume1 :: Assertion
 370     volume1 = assertEqual "volume1 is correct" True (vol ~= (1/3))
 371       where
 372         vol = volume t
 373
 374
 375
 376 -- | The barycentric coordinate of v0 with respect to itself should
 377 --   be one.
 378 prop_b0_v0_always_unity :: Tetrahedron -> Property
 379 prop_b0_v0_always_unity t =
 380     (volume t) > 0 ==> (b0 t) (v0 t) ~= 1.0
 381
 382 -- | The barycentric coordinate of v1 with respect to v0 should
 383 --   be zero.
 384 prop_b0_v1_always_zero :: Tetrahedron -> Property
 385 prop_b0_v1_always_zero t =
 386     (volume t) > 0 ==> (b0 t) (v1 t) ~= 0
 387
 388 -- | The barycentric coordinate of v2 with respect to v0 should
 389 --   be zero.
 390 prop_b0_v2_always_zero :: Tetrahedron -> Property
 391 prop_b0_v2_always_zero t =
 392     (volume t) > 0 ==> (b0 t) (v2 t) ~= 0
 393
 394 -- | The barycentric coordinate of v3 with respect to v0 should
 395 --   be zero.
 396 prop_b0_v3_always_zero :: Tetrahedron -> Property
 397 prop_b0_v3_always_zero t =
 398     (volume t) > 0 ==> (b0 t) (v3 t) ~= 0
 399
 400 -- | The barycentric coordinate of v1 with respect to itself should
 401 --   be one.
 402 prop_b1_v1_always_unity :: Tetrahedron -> Property
 403 prop_b1_v1_always_unity t =
 404     (volume t) > 0 ==> (b1 t) (v1 t) ~= 1.0
 405
 406 -- | The barycentric coordinate of v0 with respect to v1 should
 407 --   be zero.
 408 prop_b1_v0_always_zero :: Tetrahedron -> Property
 409 prop_b1_v0_always_zero t =
 410     (volume t) > 0 ==> (b1 t) (v0 t) ~= 0
 411
 412 -- | The barycentric coordinate of v2 with respect to v1 should
 413 --   be zero.
 414 prop_b1_v2_always_zero :: Tetrahedron -> Property
 415 prop_b1_v2_always_zero t =
 416     (volume t) > 0 ==> (b1 t) (v2 t) ~= 0
 417
 418 -- | The barycentric coordinate of v3 with respect to v1 should
 419 --   be zero.
 420 prop_b1_v3_always_zero :: Tetrahedron -> Property
 421 prop_b1_v3_always_zero t =
 422     (volume t) > 0 ==> (b1 t) (v3 t) ~= 0
 423
 424 -- | The barycentric coordinate of v2 with respect to itself should
 425 --   be one.
 426 prop_b2_v2_always_unity :: Tetrahedron -> Property
 427 prop_b2_v2_always_unity t =
 428     (volume t) > 0 ==> (b2 t) (v2 t) ~= 1.0
 429
 430 -- | The barycentric coordinate of v0 with respect to v2 should
 431 --   be zero.
 432 prop_b2_v0_always_zero :: Tetrahedron -> Property
 433 prop_b2_v0_always_zero t =
 434     (volume t) > 0 ==> (b2 t) (v0 t) ~= 0
 435
 436 -- | The barycentric coordinate of v1 with respect to v2 should
 437 --   be zero.
 438 prop_b2_v1_always_zero :: Tetrahedron -> Property
 439 prop_b2_v1_always_zero t =
 440     (volume t) > 0 ==> (b2 t) (v1 t) ~= 0
 441
 442 -- | The barycentric coordinate of v3 with respect to v2 should
 443 --   be zero.
 444 prop_b2_v3_always_zero :: Tetrahedron -> Property
 445 prop_b2_v3_always_zero t =
 446     (volume t) > 0 ==> (b2 t) (v3 t) ~= 0
 447
 448 -- | The barycentric coordinate of v3 with respect to itself should
 449 --   be one.
 450 prop_b3_v3_always_unity :: Tetrahedron -> Property
 451 prop_b3_v3_always_unity t =
 452     (volume t) > 0 ==> (b3 t) (v3 t) ~= 1.0
 453
 454 -- | The barycentric coordinate of v0 with respect to v3 should
 455 --   be zero.
 456 prop_b3_v0_always_zero :: Tetrahedron -> Property
 457 prop_b3_v0_always_zero t =
 458     (volume t) > 0 ==> (b3 t) (v0 t) ~= 0
 459
 460 -- | The barycentric coordinate of v1 with respect to v3 should
 461 --   be zero.
 462 prop_b3_v1_always_zero :: Tetrahedron -> Property
 463 prop_b3_v1_always_zero t =
 464     (volume t) > 0 ==> (b3 t) (v1 t) ~= 0
 465
 466 -- | The barycentric coordinate of v2 with respect to v3 should
 467 --   be zero.
 468 prop_b3_v2_always_zero :: Tetrahedron -> Property
 469 prop_b3_v2_always_zero t =
 470     (volume t) > 0 ==> (b3 t) (v2 t) ~= 0
 471
 472
 473 prop_swapping_vertices_doesnt_affect_coefficients1 :: Tetrahedron -> Bool
 474 prop_swapping_vertices_doesnt_affect_coefficients1 t =
 475       c t 0 0 1 2 == c t' 0 0 1 2
 476         where
 477           t' = t { v0 = (v1 t), v1 = (v0 t) }
 478
 479 prop_swapping_vertices_doesnt_affect_coefficients2 :: Tetrahedron -> Bool
 480 prop_swapping_vertices_doesnt_affect_coefficients2 t =
 481       c t 0 1 1 1 == c t' 0 1 1 1
 482         where
 483           t' = t { v2 = (v3 t), v3 = (v2 t) }
 484
 485 prop_swapping_vertices_doesnt_affect_coefficients3 :: Tetrahedron -> Bool
 486 prop_swapping_vertices_doesnt_affect_coefficients3 t =
 487       c t 2 1 0 0 == c t' 2 1 0 0
 488         where
 489           t' = t { v2 = (v3 t), v3 = (v2 t) }
 490
 491 prop_swapping_vertices_doesnt_affect_coefficients4 :: Tetrahedron -> Bool
 492 prop_swapping_vertices_doesnt_affect_coefficients4 t =
 493       c t 2 0 0 1 == c t' 2 0 0 1
 494         where
 495           t' = t { v0 = (v3 t), v3 = (v0 t) }
 496
 497
 498
 499
 500 tetrahedron_tests :: TestTree
 501 tetrahedron_tests =
 502     testGroup "Tetrahedron tests" [
 503       tetrahedron1_geometry_tests,
 504       tetrahedron2_geometry_tests ]
 505
 506
 507
 508 p78_24_properties :: TestTree
 509 p78_24_properties =
 510   testGroup "p. 78, Section (2.4) properties" [
 511     testProperty "c3000 identity" prop_c3000_identity,
 512     testProperty "c2100 identity" prop_c2100_identity,
 513     testProperty "c1110 identity" prop_c1110_identity]
 514   where
 515     -- | Returns the domain point of t with indices i,j,k,l.
 516     domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
 517     domain_point t i j k l =
 518       weighted_sum `scale` (1/3)
 519       where
 520         v0' = (v0 t) `scale` (fromIntegral i)
 521         v1' = (v1 t) `scale` (fromIntegral j)
 522         v2' = (v2 t) `scale` (fromIntegral k)
 523         v3' = (v3 t) `scale` (fromIntegral l)
 524         weighted_sum = v0' + v1' + v2' + v3'
 525
 526
 527     -- | Used for convenience in the next few tests.
 528     p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
 529     p t i j k l = (polynomial t) (domain_point t i j k l)
 530
 531
 532     -- | Given in Sorokina and Zeilfelder, p. 78.
 533     prop_c3000_identity :: Tetrahedron -> Property
 534     prop_c3000_identity t =
 535       (volume t) > 0 ==>
 536         c t 3 0 0 0 ~= p t 3 0 0 0
 537
 538     -- | Given in Sorokina and Zeilfelder, p. 78.
 539     prop_c2100_identity :: Tetrahedron -> Property
 540     prop_c2100_identity t =
 541       (volume t) > 0 ==>
 542         c t 2 1 0 0 ~= (term1 - term2 + term3 - term4)
 543         where
 544           term1 = (1/3)*(p t 0 3 0 0)
 545           term2 = (5/6)*(p t 3 0 0 0)
 546           term3 = 3*(p t 2 1 0 0)
 547           term4 = (3/2)*(p t 1 2 0 0)
 548
 549     -- | Given in Sorokina and Zeilfelder, p. 78.
 550     prop_c1110_identity :: Tetrahedron -> Property
 551     prop_c1110_identity t =
 552       (volume t) > 0 ==>
 553         c t 1 1 1 0 ~= (term1 + term2 - term3 - term4)
 554         where
 555           term1 = (1/3)*((p t 3 0 0 0) + (p t 0 3 0 0) + (p t 0 0 3 0))
 556           term2 = (9/2)*(p t 1 1 1 0)
 557           term3 = (3/4)*((p t 2 1 0 0) + (p t 1 2 0 0) + (p t 2 0 1 0))
 558           term4 = (3/4)*((p t 1 0 2 0) + (p t 0 2 1 0) + (p t 0 1 2 0))
 559
 560
 561
 562 tetrahedron_properties :: TestTree
 563 tetrahedron_properties =
 564   testGroup "Tetrahedron properties" [
 565     p78_24_properties,
 566     testProperty "b0_v0_always_unity" prop_b0_v0_always_unity,
 567     testProperty "b0_v1_always_zero" prop_b0_v1_always_zero,
 568     testProperty "b0_v2_always_zero" prop_b0_v2_always_zero,
 569     testProperty "b0_v3_always_zero" prop_b0_v3_always_zero,
 570     testProperty "b1_v1_always_unity" prop_b1_v1_always_unity,
 571     testProperty "b1_v0_always_zero" prop_b1_v0_always_zero,
 572     testProperty "b1_v2_always_zero" prop_b1_v2_always_zero,
 573     testProperty "b1_v3_always_zero" prop_b1_v3_always_zero,
 574     testProperty "b2_v2_always_unity" prop_b2_v2_always_unity,
 575     testProperty "b2_v0_always_zero" prop_b2_v0_always_zero,
 576     testProperty "b2_v1_always_zero" prop_b2_v1_always_zero,
 577     testProperty "b2_v3_always_zero" prop_b2_v3_always_zero,
 578     testProperty "b3_v3_always_unity" prop_b3_v3_always_unity,
 579     testProperty "b3_v0_always_zero" prop_b3_v0_always_zero,
 580     testProperty "b3_v1_always_zero" prop_b3_v1_always_zero,
 581     testProperty "b3_v2_always_zero" prop_b3_v2_always_zero,
 582     testProperty "swapping_vertices_doesnt_affect_coefficients1"
 583       prop_swapping_vertices_doesnt_affect_coefficients1,
 584     testProperty "swapping_vertices_doesnt_affect_coefficients2"
 585       prop_swapping_vertices_doesnt_affect_coefficients2,
 586     testProperty "swapping_vertices_doesnt_affect_coefficients3"
 587       prop_swapping_vertices_doesnt_affect_coefficients3,
 588     testProperty "swapping_vertices_doesnt_affect_coefficients4"
 589       prop_swapping_vertices_doesnt_affect_coefficients4 ]