src/Tetrahedron.hs

   1 module Tetrahedron (
   2   Tetrahedron(..),
   3   b0, -- Cube test
   4   b1, -- Cube test
   5   b2, -- Cube test
   6   b3, -- Cube test
   7   c,
   8   polynomial,
   9   tetrahedron_properties,
  10   tetrahedron_tests,
  11   volume -- Cube test
  12   )
  13 where
  14
  15 import qualified Data.Vector as V (
  16   singleton,
  17   snoc,
  18   sum
  19   )
  20
  21 import Test.Framework (Test, testGroup)
  22 import Test.Framework.Providers.HUnit (testCase)
  23 import Test.Framework.Providers.QuickCheck2 (testProperty)
  24 import Test.HUnit (Assertion, assertEqual)
  25 import Test.QuickCheck (Arbitrary(..), Gen, Property, (==>))
  26
  27 import Comparisons ((~=), nearly_ge)
  28 import FunctionValues (FunctionValues(..), empty_values)
  29 import Misc (factorial)
  30 import Point (Point, scale)
  31 import RealFunction (RealFunction, cmult, fexp)
  32 import ThreeDimensional (ThreeDimensional(..))
  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 instance ThreeDimensional Tetrahedron where
  70     center (Tetrahedron _ v0' v1' v2' v3' _) =
  71         (v0' + v1' + v2' + v3') `scale` (1/4)
  72
  73     contains_point t p0 =
  74       b0_unscaled `nearly_ge` 0 &&
  75       b1_unscaled `nearly_ge` 0 &&
  76       b2_unscaled `nearly_ge` 0 &&
  77       b3_unscaled `nearly_ge` 0
  78       where
  79         -- Drop the useless division and volume calculation that we
  80         -- would do if we used the regular b0,..b3 functions.
  81         b0_unscaled :: Double
  82         b0_unscaled = volume inner_tetrahedron
  83           where inner_tetrahedron = t { v0 = p0 }
  84
  85         b1_unscaled :: Double
  86         b1_unscaled = volume inner_tetrahedron
  87           where inner_tetrahedron = t { v1 = p0 }
  88
  89         b2_unscaled :: Double
  90         b2_unscaled = volume inner_tetrahedron
  91           where inner_tetrahedron = t { v2 = p0 }
  92
  93         b3_unscaled :: Double
  94         b3_unscaled = volume inner_tetrahedron
  95           where inner_tetrahedron = t { v3 = p0 }
  96
  97
  98 polynomial :: Tetrahedron -> (RealFunction Point)
  99 polynomial t =
 100     V.sum $ V.singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `V.snoc`
 101             ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `V.snoc`
 102             ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `V.snoc`
 103             ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `V.snoc`
 104             ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `V.snoc`
 105             ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `V.snoc`
 106             ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `V.snoc`
 107             ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `V.snoc`
 108             ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `V.snoc`
 109             ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `V.snoc`
 110             ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `V.snoc`
 111             ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `V.snoc`
 112             ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `V.snoc`
 113             ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `V.snoc`
 114             ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `V.snoc`
 115             ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `V.snoc`
 116             ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `V.snoc`
 117             ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `V.snoc`
 118             ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `V.snoc`
 119             ((c t 3 0 0 0) `cmult` (beta t 3 0 0 0))
 120
 121
 122
 123 -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
 124 --   capital 'B' in the Sorokina/Zeilfelder paper.
 125 beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
 126 beta t i j k l
 127   | (i + j + k + l == 3) =
 128       coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
 129   | otherwise = error "basis function index out of bounds"
 130   where
 131     denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
 132     coefficient = 6 / (fromIntegral denominator)
 133     b0_term = (b0 t) `fexp` i
 134     b1_term = (b1 t) `fexp` j
 135     b2_term = (b2 t) `fexp` k
 136     b3_term = (b3 t) `fexp` l
 137
 138
 139 -- | The coefficient function. c t i j k l returns the coefficient
 140 --   c_ijkl with respect to the tetrahedron t. The definition uses
 141 --   pattern matching to mimic the definitions given in Sorokina and
 142 --   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
 143 --   function will simply error.
 144 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
 145 c t i j k l =
 146   coefficient i j k l
 147   where
 148     fvs = function_values t
 149     f = front fvs
 150     b = back fvs
 151     r = right fvs
 152     l' = left fvs
 153     t' = top fvs
 154     d  = down fvs
 155     fl = front_left fvs
 156     fr = front_right fvs
 157     fd = front_down fvs
 158     ft = front_top fvs
 159     bl = back_left fvs
 160     br = back_right fvs
 161     bd = back_down fvs
 162     bt = back_top fvs
 163     ld = left_down fvs
 164     lt = left_top fvs
 165     rd = right_down fvs
 166     rt = right_top fvs
 167     fld = front_left_down fvs
 168     flt = front_left_top fvs
 169     frd = front_right_down fvs
 170     frt = front_right_top fvs
 171     i' = interior fvs
 172
 173     coefficient :: Int -> Int -> Int -> Int -> Double
 174     coefficient 0 0 3 0 =
 175       (1/8) * (i' + f + l' + t' + lt + fl + ft + flt)
 176
 177     coefficient 0 0 0 3 =
 178       (1/8) * (i' + f + r + t' + rt + fr + ft + frt)
 179
 180     coefficient 0 0 2 1 =
 181       (5/24)*(i' + f + t' + ft) + (1/24)*(l' + fl + lt + flt)
 182
 183     coefficient 0 0 1 2 =
 184       (5/24)*(i' + f + t' + ft) + (1/24)*(r + fr + rt + frt)
 185
 186     coefficient 0 1 2 0 =
 187       (5/24)*(i' + f) + (1/8)*(l' + t' + fl + ft)
 188                       + (1/24)*(lt + flt)
 189
 190     coefficient 0 1 0 2 =
 191       (5/24)*(i' + f) + (1/8)*(r + t' + fr + ft)
 192                       + (1/24)*(rt + frt)
 193
 194     coefficient 0 1 1 1 =
 195       (13/48)*(i' + f) + (7/48)*(t' + ft)
 196                        + (1/32)*(l' + r + fl + fr)
 197                        + (1/96)*(lt + rt + flt + frt)
 198
 199     coefficient 0 2 1 0 =
 200       (13/48)*(i' + f) + (17/192)*(l' + t' + fl + ft)
 201                        + (1/96)*(lt + flt)
 202                        + (1/64)*(r + d + fr + fd)
 203                        + (1/192)*(rt + ld + frt + fld)
 204
 205     coefficient 0 2 0 1 =
 206       (13/48)*(i' + f) + (17/192)*(r + t' + fr + ft)
 207                        + (1/96)*(rt + frt)
 208                        + (1/64)*(l' + d + fl + fd)
 209                        + (1/192)*(rd + lt + flt + frd)
 210
 211     coefficient 0 3 0 0 =
 212      (13/48)*(i' + f) + (5/96)*(l' + r + t' + d + fl + fr + ft + fd)
 213                       + (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld)
 214
 215     coefficient 1 0 2 0 =
 216       (1/4)*i' + (1/6)*(f + l' + t')
 217                + (1/12)*(lt + fl + ft)
 218
 219     coefficient 1 0 0 2 =
 220       (1/4)*i' + (1/6)*(f + r + t')
 221                + (1/12)*(rt + fr + ft)
 222
 223     coefficient 1 0 1 1 =
 224       (1/3)*i' + (5/24)*(f + t')
 225                + (1/12)*ft
 226                + (1/24)*(l' + r)
 227                + (1/48)*(lt + rt + fl + fr)
 228
 229     coefficient 1 1 1 0 =
 230       (1/3)*i' + (5/24)*f
 231                + (1/8)*(l' + t')
 232                + (5/96)*(fl + ft)
 233                + (1/48)*(d + r + lt)
 234                + (1/96)*(fd + ld + rt + fr)
 235
 236     coefficient 1 1 0 1 =
 237       (1/3)*i' + (5/24)*f
 238                + (1/8)*(r + t')
 239                + (5/96)*(fr + ft)
 240                + (1/48)*(d + l' + rt)
 241                + (1/96)*(fd + lt + rd + fl)
 242
 243     coefficient 1 2 0 0 =
 244       (1/3)*i' + (5/24)*f
 245                + (7/96)*(l' + r + t' + d)
 246                + (1/32)*(fl + fr + ft + fd)
 247                + (1/96)*(rt + rd + lt + ld)
 248
 249     coefficient 2 0 1 0 =
 250       (3/8)*i' + (7/48)*(f + t' + l')
 251                + (1/48)*(r + d + b + lt + fl + ft)
 252                + (1/96)*(rt + bt + fr + fd + ld + bl)
 253
 254     coefficient 2 0 0 1 =
 255       (3/8)*i' + (7/48)*(f + t' + r)
 256                + (1/48)*(l' + d + b + rt + fr + ft)
 257                + (1/96)*(lt + bt + fl + fd + rd + br)
 258
 259     coefficient 2 1 0 0 =
 260       (3/8)*i' + (1/12)*(t' + r + l' + d)
 261                + (1/64)*(ft + fr + fl + fd)
 262                + (7/48)*f
 263                + (1/48)*b
 264                + (1/96)*(rt + ld + lt + rd)
 265                + (1/192)*(bt + br + bl + bd)
 266
 267     coefficient 3 0 0 0 =
 268       (3/8)*i' + (1/12)*(t' + f + l' + r + d + b)
 269                + (1/96)*(lt + fl + ft + rt + bt + fr)
 270                + (1/96)*(fd + ld + bd + br + rd + bl)
 271
 272     coefficient _ _ _ _ = error "coefficient index out of bounds"
 273
 274
 275
 276 -- | Compute the determinant of the 4x4 matrix,
 277 --
 278 --   [1]
 279 --   [x]
 280 --   [y]
 281 --   [z]
 282 --
 283 --   where [1] = [1, 1, 1, 1],
 284 --         [x] = [x1,x2,x3,x4],
 285 --
 286 --   et cetera.
 287 --
 288 --   The termX nonsense is an attempt to prevent Double overflow.
 289 --   which has been observed to happen with large coordinates.
 290 --
 291 det :: Point -> Point -> Point -> Point -> Double
 292 det p0 p1 p2 p3 =
 293   term5 + term6
 294   where
 295     (x1, y1, z1) = p0
 296     (x2, y2, z2) = p1
 297     (x3, y3, z3) = p2
 298     (x4, y4, z4) = p3
 299     term1 = ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3
 300     term2 = ((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4
 301     term3 = ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1
 302     term4 = ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2
 303     term5 = term1 - term2
 304     term6 = term3 - term4
 305
 306
 307 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
 308 --   page 436.
 309 volume :: Tetrahedron -> Double
 310 volume t
 311        | v0' == v1' = 0
 312        | v0' == v2' = 0
 313        | v0' == v3' = 0
 314        | v1' == v2' = 0
 315        | v1' == v3' = 0
 316        | v2' == v3' = 0
 317        | otherwise = (1/6)*(det v0' v1' v2' v3')
 318   where
 319     v0' = v0 t
 320     v1' = v1 t
 321     v2' = v2 t
 322     v3' = v3 t
 323
 324
 325 -- | The barycentric coordinates of a point with respect to v0.
 326 b0 :: Tetrahedron -> (RealFunction Point)
 327 b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 328              where
 329                inner_tetrahedron = t { v0 = point }
 330
 331
 332 -- | The barycentric coordinates of a point with respect to v1.
 333 b1 :: Tetrahedron -> (RealFunction Point)
 334 b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 335              where
 336                inner_tetrahedron = t { v1 = point }
 337
 338
 339 -- | The barycentric coordinates of a point with respect to v2.
 340 b2 :: Tetrahedron -> (RealFunction Point)
 341 b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 342              where
 343                inner_tetrahedron = t { v2 = point }
 344
 345
 346 -- | The barycentric coordinates of a point with respect to v3.
 347 b3 :: Tetrahedron -> (RealFunction Point)
 348 b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 349              where
 350                inner_tetrahedron = t { v3 = point }
 351
 352
 353
 354
 355 -- Tests
 356
 357
 358 -- | Check the volume of a particular tetrahedron (computed by hand)
 359 --   and whether or not it contains a specific point chosen to be
 360 --   outside of it. Its vertices are in clockwise order, so the volume
 361 --   should be negative.
 362 tetrahedron1_geometry_tests :: Test.Framework.Test
 363 tetrahedron1_geometry_tests =
 364   testGroup "tetrahedron1 geometry"
 365               [ testCase "volume1" volume1,
 366                 testCase "doesn't contain point1" doesnt_contain_point1]
 367   where
 368     p0 = (0, -0.5, 0)
 369     p1 = (0, 0.5, 0)
 370     p2 = (2, 0, 0)
 371     p3 = (1, 0, 1)
 372     t = Tetrahedron { v0 = p0,
 373                       v1 = p1,
 374                       v2 = p2,
 375                       v3 = p3,
 376                       function_values = empty_values,
 377                       precomputed_volume = 0 }
 378
 379     volume1 :: Assertion
 380     volume1 =
 381       assertEqual "volume is correct" True (vol ~= (-1/3))
 382       where
 383         vol = volume t
 384
 385     doesnt_contain_point1 :: Assertion
 386     doesnt_contain_point1 =
 387       assertEqual "doesn't contain an exterior point" False contained
 388       where
 389         exterior_point = (5, 2, -9.0212)
 390         contained = contains_point t exterior_point
 391
 392
 393 -- | Check the volume of a particular tetrahedron (computed by hand)
 394 --   and whether or not it contains a specific point chosen to be
 395 --   inside of it. Its vertices are in counter-clockwise order, so the
 396 --   volume should be positive.
 397 tetrahedron2_geometry_tests :: Test.Framework.Test
 398 tetrahedron2_geometry_tests =
 399   testGroup "tetrahedron2 geometry"
 400               [ testCase "volume1" volume1,
 401                 testCase "contains point1" contains_point1]
 402   where
 403     p0 = (0, -0.5, 0)
 404     p1 = (2, 0, 0)
 405     p2 = (0, 0.5, 0)
 406     p3 = (1, 0, 1)
 407     t = Tetrahedron { v0 = p0,
 408                       v1 = p1,
 409                       v2 = p2,
 410                       v3 = p3,
 411                       function_values = empty_values,
 412                       precomputed_volume = 0 }
 413
 414     volume1 :: Assertion
 415     volume1 = assertEqual "volume1 is correct" True (vol ~= (1/3))
 416       where
 417         vol = volume t
 418
 419     contains_point1 :: Assertion
 420     contains_point1 = assertEqual "contains an inner point" True contained
 421         where
 422           inner_point = (1, 0, 0.5)
 423           contained = contains_point t inner_point
 424
 425
 426 -- | Ensure that tetrahedra do not contain a particular point chosen to
 427 --   be outside of them.
 428 containment_tests :: Test.Framework.Test
 429 containment_tests =
 430   testGroup "containment tests"
 431               [ testCase "doesn't contain point2" doesnt_contain_point2,
 432                 testCase "doesn't contain point3" doesnt_contain_point3,
 433                 testCase "doesn't contain point4" doesnt_contain_point4,
 434                 testCase "doesn't contain point5" doesnt_contain_point5]
 435   where
 436     p2 = (0.5, 0.5, 1)
 437     p3 = (0.5, 0.5, 0.5)
 438     exterior_point = (0, 0, 0)
 439
 440     doesnt_contain_point2 :: Assertion
 441     doesnt_contain_point2 =
 442       assertEqual "doesn't contain an exterior point" False contained
 443       where
 444         p0 = (0, 1, 1)
 445         p1 = (1, 1, 1)
 446         t = Tetrahedron { v0 = p0,
 447                           v1 = p1,
 448                           v2 = p2,
 449                           v3 = p3,
 450                           function_values = empty_values,
 451                           precomputed_volume = 0 }
 452         contained = contains_point t exterior_point
 453
 454
 455     doesnt_contain_point3 :: Assertion
 456     doesnt_contain_point3 =
 457       assertEqual "doesn't contain an exterior point" False contained
 458       where
 459         p0 = (1, 1, 1)
 460         p1 = (1, 0, 1)
 461         t = Tetrahedron { v0 = p0,
 462                           v1 = p1,
 463                           v2 = p2,
 464                           v3 = p3,
 465                           function_values = empty_values,
 466                           precomputed_volume = 0 }
 467         contained = contains_point t exterior_point
 468
 469
 470     doesnt_contain_point4 :: Assertion
 471     doesnt_contain_point4 =
 472       assertEqual "doesn't contain an exterior point" False contained
 473       where
 474         p0 = (1, 0, 1)
 475         p1 = (0, 0, 1)
 476         t = Tetrahedron { v0 = p0,
 477                           v1 = p1,
 478                           v2 = p2,
 479                           v3 = p3,
 480                           function_values = empty_values,
 481                           precomputed_volume = 0 }
 482         contained = contains_point t exterior_point
 483
 484
 485     doesnt_contain_point5 :: Assertion
 486     doesnt_contain_point5 =
 487       assertEqual "doesn't contain an exterior point" False contained
 488       where
 489         p0 = (0, 0, 1)
 490         p1 = (0, 1, 1)
 491         t = Tetrahedron { v0 = p0,
 492                           v1 = p1,
 493                           v2 = p2,
 494                           v3 = p3,
 495                           function_values = empty_values,
 496                           precomputed_volume = 0 }
 497         contained = contains_point t exterior_point
 498
 499
 500 -- | The barycentric coordinate of v0 with respect to itself should
 501 --   be one.
 502 prop_b0_v0_always_unity :: Tetrahedron -> Property
 503 prop_b0_v0_always_unity t =
 504     (volume t) > 0 ==> (b0 t) (v0 t) ~= 1.0
 505
 506 -- | The barycentric coordinate of v1 with respect to v0 should
 507 --   be zero.
 508 prop_b0_v1_always_zero :: Tetrahedron -> Property
 509 prop_b0_v1_always_zero t =
 510     (volume t) > 0 ==> (b0 t) (v1 t) ~= 0
 511
 512 -- | The barycentric coordinate of v2 with respect to v0 should
 513 --   be zero.
 514 prop_b0_v2_always_zero :: Tetrahedron -> Property
 515 prop_b0_v2_always_zero t =
 516     (volume t) > 0 ==> (b0 t) (v2 t) ~= 0
 517
 518 -- | The barycentric coordinate of v3 with respect to v0 should
 519 --   be zero.
 520 prop_b0_v3_always_zero :: Tetrahedron -> Property
 521 prop_b0_v3_always_zero t =
 522     (volume t) > 0 ==> (b0 t) (v3 t) ~= 0
 523
 524 -- | The barycentric coordinate of v1 with respect to itself should
 525 --   be one.
 526 prop_b1_v1_always_unity :: Tetrahedron -> Property
 527 prop_b1_v1_always_unity t =
 528     (volume t) > 0 ==> (b1 t) (v1 t) ~= 1.0
 529
 530 -- | The barycentric coordinate of v0 with respect to v1 should
 531 --   be zero.
 532 prop_b1_v0_always_zero :: Tetrahedron -> Property
 533 prop_b1_v0_always_zero t =
 534     (volume t) > 0 ==> (b1 t) (v0 t) ~= 0
 535
 536 -- | The barycentric coordinate of v2 with respect to v1 should
 537 --   be zero.
 538 prop_b1_v2_always_zero :: Tetrahedron -> Property
 539 prop_b1_v2_always_zero t =
 540     (volume t) > 0 ==> (b1 t) (v2 t) ~= 0
 541
 542 -- | The barycentric coordinate of v3 with respect to v1 should
 543 --   be zero.
 544 prop_b1_v3_always_zero :: Tetrahedron -> Property
 545 prop_b1_v3_always_zero t =
 546     (volume t) > 0 ==> (b1 t) (v3 t) ~= 0
 547
 548 -- | The barycentric coordinate of v2 with respect to itself should
 549 --   be one.
 550 prop_b2_v2_always_unity :: Tetrahedron -> Property
 551 prop_b2_v2_always_unity t =
 552     (volume t) > 0 ==> (b2 t) (v2 t) ~= 1.0
 553
 554 -- | The barycentric coordinate of v0 with respect to v2 should
 555 --   be zero.
 556 prop_b2_v0_always_zero :: Tetrahedron -> Property
 557 prop_b2_v0_always_zero t =
 558     (volume t) > 0 ==> (b2 t) (v0 t) ~= 0
 559
 560 -- | The barycentric coordinate of v1 with respect to v2 should
 561 --   be zero.
 562 prop_b2_v1_always_zero :: Tetrahedron -> Property
 563 prop_b2_v1_always_zero t =
 564     (volume t) > 0 ==> (b2 t) (v1 t) ~= 0
 565
 566 -- | The barycentric coordinate of v3 with respect to v2 should
 567 --   be zero.
 568 prop_b2_v3_always_zero :: Tetrahedron -> Property
 569 prop_b2_v3_always_zero t =
 570     (volume t) > 0 ==> (b2 t) (v3 t) ~= 0
 571
 572 -- | The barycentric coordinate of v3 with respect to itself should
 573 --   be one.
 574 prop_b3_v3_always_unity :: Tetrahedron -> Property
 575 prop_b3_v3_always_unity t =
 576     (volume t) > 0 ==> (b3 t) (v3 t) ~= 1.0
 577
 578 -- | The barycentric coordinate of v0 with respect to v3 should
 579 --   be zero.
 580 prop_b3_v0_always_zero :: Tetrahedron -> Property
 581 prop_b3_v0_always_zero t =
 582     (volume t) > 0 ==> (b3 t) (v0 t) ~= 0
 583
 584 -- | The barycentric coordinate of v1 with respect to v3 should
 585 --   be zero.
 586 prop_b3_v1_always_zero :: Tetrahedron -> Property
 587 prop_b3_v1_always_zero t =
 588     (volume t) > 0 ==> (b3 t) (v1 t) ~= 0
 589
 590 -- | The barycentric coordinate of v2 with respect to v3 should
 591 --   be zero.
 592 prop_b3_v2_always_zero :: Tetrahedron -> Property
 593 prop_b3_v2_always_zero t =
 594     (volume t) > 0 ==> (b3 t) (v2 t) ~= 0
 595
 596
 597 prop_swapping_vertices_doesnt_affect_coefficients1 :: Tetrahedron -> Bool
 598 prop_swapping_vertices_doesnt_affect_coefficients1 t =
 599       c t 0 0 1 2 == c t' 0 0 1 2
 600         where
 601           t' = t { v0 = (v1 t), v1 = (v0 t) }
 602
 603 prop_swapping_vertices_doesnt_affect_coefficients2 :: Tetrahedron -> Bool
 604 prop_swapping_vertices_doesnt_affect_coefficients2 t =
 605       c t 0 1 1 1 == c t' 0 1 1 1
 606         where
 607           t' = t { v2 = (v3 t), v3 = (v2 t) }
 608
 609 prop_swapping_vertices_doesnt_affect_coefficients3 :: Tetrahedron -> Bool
 610 prop_swapping_vertices_doesnt_affect_coefficients3 t =
 611       c t 2 1 0 0 == c t' 2 1 0 0
 612         where
 613           t' = t { v2 = (v3 t), v3 = (v2 t) }
 614
 615 prop_swapping_vertices_doesnt_affect_coefficients4 :: Tetrahedron -> Bool
 616 prop_swapping_vertices_doesnt_affect_coefficients4 t =
 617       c t 2 0 0 1 == c t' 2 0 0 1
 618         where
 619           t' = t { v0 = (v3 t), v3 = (v0 t) }
 620
 621
 622
 623
 624 tetrahedron_tests :: Test.Framework.Test
 625 tetrahedron_tests =
 626     testGroup "Tetrahedron Tests" [
 627       tetrahedron1_geometry_tests,
 628       tetrahedron2_geometry_tests,
 629       containment_tests ]
 630
 631
 632
 633 p78_24_properties :: Test.Framework.Test
 634 p78_24_properties =
 635   testGroup "p. 78, Section (2.4) Properties" [
 636     testProperty "c3000 identity" prop_c3000_identity,
 637     testProperty "c2100 identity" prop_c2100_identity,
 638     testProperty "c1110 identity" prop_c1110_identity]
 639   where
 640     -- | Returns the domain point of t with indices i,j,k,l.
 641     domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
 642     domain_point t i j k l
 643       | i + j + k + l == 3 = weighted_sum `scale` (1/3)
 644       | otherwise = error "domain point index out of bounds"
 645       where
 646         v0' = (v0 t) `scale` (fromIntegral i)
 647         v1' = (v1 t) `scale` (fromIntegral j)
 648         v2' = (v2 t) `scale` (fromIntegral k)
 649         v3' = (v3 t) `scale` (fromIntegral l)
 650         weighted_sum = v0' + v1' + v2' + v3'
 651
 652
 653     -- | Used for convenience in the next few tests.
 654     p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
 655     p t i j k l = (polynomial t) (domain_point t i j k l)
 656
 657
 658     -- | Given in Sorokina and Zeilfelder, p. 78.
 659     prop_c3000_identity :: Tetrahedron -> Property
 660     prop_c3000_identity t =
 661       (volume t) > 0 ==>
 662         c t 3 0 0 0 ~= p t 3 0 0 0
 663
 664     -- | Given in Sorokina and Zeilfelder, p. 78.
 665     prop_c2100_identity :: Tetrahedron -> Property
 666     prop_c2100_identity t =
 667       (volume t) > 0 ==>
 668         c t 2 1 0 0 ~= (term1 - term2 + term3 - term4)
 669         where
 670           term1 = (1/3)*(p t 0 3 0 0)
 671           term2 = (5/6)*(p t 3 0 0 0)
 672           term3 = 3*(p t 2 1 0 0)
 673           term4 = (3/2)*(p t 1 2 0 0)
 674
 675     -- | Given in Sorokina and Zeilfelder, p. 78.
 676     prop_c1110_identity :: Tetrahedron -> Property
 677     prop_c1110_identity t =
 678       (volume t) > 0 ==>
 679         c t 1 1 1 0 ~= (term1 + term2 - term3 - term4)
 680         where
 681           term1 = (1/3)*((p t 3 0 0 0) + (p t 0 3 0 0) + (p t 0 0 3 0))
 682           term2 = (9/2)*(p t 1 1 1 0)
 683           term3 = (3/4)*((p t 2 1 0 0) + (p t 1 2 0 0) + (p t 2 0 1 0))
 684           term4 = (3/4)*((p t 1 0 2 0) + (p t 0 2 1 0) + (p t 0 1 2 0))
 685
 686
 687
 688 tetrahedron_properties :: Test.Framework.Test
 689 tetrahedron_properties =
 690   testGroup "Tetrahedron Properties" [
 691     p78_24_properties,
 692     testProperty "b0_v0_always_unity" prop_b0_v0_always_unity,
 693     testProperty "b0_v1_always_zero" prop_b0_v1_always_zero,
 694     testProperty "b0_v2_always_zero" prop_b0_v2_always_zero,
 695     testProperty "b0_v3_always_zero" prop_b0_v3_always_zero,
 696     testProperty "b1_v1_always_unity" prop_b1_v1_always_unity,
 697     testProperty "b1_v0_always_zero" prop_b1_v0_always_zero,
 698     testProperty "b1_v2_always_zero" prop_b1_v2_always_zero,
 699     testProperty "b1_v3_always_zero" prop_b1_v3_always_zero,
 700     testProperty "b2_v2_always_unity" prop_b2_v2_always_unity,
 701     testProperty "b2_v0_always_zero" prop_b2_v0_always_zero,
 702     testProperty "b2_v1_always_zero" prop_b2_v1_always_zero,
 703     testProperty "b2_v3_always_zero" prop_b2_v3_always_zero,
 704     testProperty "b3_v3_always_unity" prop_b3_v3_always_unity,
 705     testProperty "b3_v0_always_zero" prop_b3_v0_always_zero,
 706     testProperty "b3_v1_always_zero" prop_b3_v1_always_zero,
 707     testProperty "b3_v2_always_zero" prop_b3_v2_always_zero,
 708     testProperty "swapping_vertices_doesnt_affect_coefficients1" $
 709       prop_swapping_vertices_doesnt_affect_coefficients1,
 710     testProperty "swapping_vertices_doesnt_affect_coefficients2" $
 711       prop_swapping_vertices_doesnt_affect_coefficients2,
 712     testProperty "swapping_vertices_doesnt_affect_coefficients3" $
 713       prop_swapping_vertices_doesnt_affect_coefficients3,
 714     testProperty "swapping_vertices_doesnt_affect_coefficients4" $
 715       prop_swapping_vertices_doesnt_affect_coefficients4 ]