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