src/Tests/Cube.hs

   1 module Tests.Cube
   2 where
   3
   4 import Prelude hiding (LT)
   5
   6 import Cardinal
   7 import Comparisons
   8 import Cube hiding (i, j, k)
   9 import FunctionValues
  10 import Misc (all_equal, disjoint)
  11 import Tetrahedron (b0, b1, b2, b3, c, fv,
  12                     v0, v1, v2, v3, volume)
  13
  14
  15 -- Quickcheck tests.
  16
  17 -- | The 'front_half_tetrahedra' and 'back_half_tetrahedra' should
  18 --   have no tetrahedra in common.
  19 prop_front_back_tetrahedra_disjoint :: Cube -> Bool
  20 prop_front_back_tetrahedra_disjoint c =
  21     disjoint (front_half_tetrahedra c) (back_half_tetrahedra c)
  22
  23
  24 -- | The 'top_half_tetrahedra' and 'down_half_tetrahedra' should
  25 --   have no tetrahedra in common.
  26 prop_top_down_tetrahedra_disjoint :: Cube -> Bool
  27 prop_top_down_tetrahedra_disjoint c =
  28     disjoint (top_half_tetrahedra c) (down_half_tetrahedra c)
  29
  30
  31 -- | The 'left_half_tetrahedra' and 'right_half_tetrahedra' should
  32 --   have no tetrahedra in common.
  33 prop_left_right_tetrahedra_disjoint :: Cube -> Bool
  34 prop_left_right_tetrahedra_disjoint c =
  35     disjoint (left_half_tetrahedra c) (right_half_tetrahedra c)
  36
  37
  38 -- | Since the grid size is necessarily positive, all tetrahedra
  39 --   (which comprise cubes of positive volume) must have positive volume
  40 --   as well.
  41 prop_all_volumes_positive :: Cube -> Bool
  42 prop_all_volumes_positive cube =
  43     null nonpositive_volumes
  44     where
  45       ts = tetrahedra cube
  46       volumes = map volume ts
  47       nonpositive_volumes = filter (<= 0) volumes
  48
  49 -- | In fact, since all of the tetrahedra are identical, we should
  50 --   already know their volumes. There's 24 tetrahedra to a cube, so
  51 --   we'd expect the volume of each one to be (1/24)*h^3.
  52 prop_all_volumes_exact :: Cube -> Bool
  53 prop_all_volumes_exact cube =
  54     and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
  55     where
  56       delta = h cube
  57
  58 -- | All tetrahedron should have their v0 located at the center of the cube.
  59 prop_v0_all_equal :: Cube -> Bool
  60 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
  61     where
  62       t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
  63       t1 = head $ tail (tetrahedra cube)
  64
  65
  66 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
  67 --   third and fourth indices of c-t1 have been switched. This is
  68 --   because we store the triangles oriented such that their volume is
  69 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
  70 --   in opposite directions, one of them has to have negative volume!
  71 prop_c0120_identity1 :: Cube -> Bool
  72 prop_c0120_identity1 cube =
  73    c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
  74      where
  75        t0 = tetrahedron cube 0
  76        t3 = tetrahedron cube 3
  77
  78
  79 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
  80 --   'prop_c0120_identity1' with tetrahedrons 1 and 2.
  81 prop_c0120_identity2 :: Cube -> Bool
  82 prop_c0120_identity2 cube =
  83    c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
  84      where
  85        t0 = tetrahedron cube 0
  86        t1 = tetrahedron cube 1
  87
  88 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
  89 --   'prop_c0120_identity1' with tetrahedrons 1 and 2.
  90 prop_c0120_identity3 :: Cube -> Bool
  91 prop_c0120_identity3 cube =
  92    c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
  93      where
  94        t1 = tetrahedron cube 1
  95        t2 = tetrahedron cube 2
  96
  97 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
  98 --   'prop_c0120_identity1' with tetrahedrons 2 and 3.
  99 prop_c0120_identity4 :: Cube -> Bool
 100 prop_c0120_identity4 cube =
 101    c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
 102      where
 103        t2 = tetrahedron cube 2
 104        t3 = tetrahedron cube 3
 105
 106
 107 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 108 --   'prop_c0120_identity1' with tetrahedrons 4 and 5.
 109 prop_c0120_identity5 :: Cube -> Bool
 110 prop_c0120_identity5 cube =
 111     c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
 112     where
 113       t4 = tetrahedron cube 4
 114       t5 = tetrahedron cube 5
 115
 116 -- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 117 -- --   'prop_c0120_identity1' with tetrahedrons 5 and 6.
 118 prop_c0120_identity6 :: Cube -> Bool
 119 prop_c0120_identity6 cube =
 120    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
 121      where
 122        t5 = tetrahedron cube 5
 123        t6 = tetrahedron cube 6
 124
 125
 126 -- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 127 -- --   'prop_c0120_identity1' with tetrahedrons 6 and 7.
 128 prop_c0120_identity7 :: Cube -> Bool
 129 prop_c0120_identity7 cube =
 130    c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
 131      where
 132        t6 = tetrahedron cube 6
 133        t7 = tetrahedron cube 7
 134
 135
 136 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 137 --   'prop_c0120_identity1'.
 138 prop_c0210_identity1 :: Cube -> Bool
 139 prop_c0210_identity1 cube =
 140     c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
 141       where
 142         t0 = tetrahedron cube 0
 143         t3 = tetrahedron cube 3
 144
 145
 146 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 147 --   'prop_c0120_identity1'.
 148 prop_c0300_identity1 :: Cube -> Bool
 149 prop_c0300_identity1 cube =
 150     c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
 151       where
 152         t0 = tetrahedron cube 0
 153         t3 = tetrahedron cube 3
 154
 155
 156 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 157 --   'prop_c0120_identity1'.
 158 prop_c1110_identity :: Cube -> Bool
 159 prop_c1110_identity cube =
 160     c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
 161       where
 162         t0 = tetrahedron cube 0
 163         t3 = tetrahedron cube 3
 164
 165
 166 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 167 --   'prop_c0120_identity1'.
 168 prop_c1200_identity1 :: Cube -> Bool
 169 prop_c1200_identity1 cube =
 170     c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
 171       where
 172         t0 = tetrahedron cube 0
 173         t3 = tetrahedron cube 3
 174
 175
 176 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 177 --   'prop_c0120_identity1'.
 178 prop_c2100_identity1 :: Cube -> Bool
 179 prop_c2100_identity1 cube =
 180     c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
 181       where
 182         t0 = tetrahedron cube 0
 183         t3 = tetrahedron cube 3
 184
 185
 186
 187 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
 188 --   third and fourth indices of c-t3 have been switched. This is
 189 --   because we store the triangles oriented such that their volume is
 190 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
 191 --   point in opposite directions, one of them has to have negative
 192 --   volume!
 193 prop_c0102_identity1 :: Cube -> Bool
 194 prop_c0102_identity1 cube =
 195     c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
 196       where
 197         t0 = tetrahedron cube 0
 198         t1 = tetrahedron cube 1
 199
 200
 201 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 202 --   'prop_c0102_identity1'.
 203 prop_c0201_identity1 :: Cube -> Bool
 204 prop_c0201_identity1 cube =
 205     c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
 206       where
 207         t0 = tetrahedron cube 0
 208         t1 = tetrahedron cube 1
 209
 210
 211 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 212 --   'prop_c0102_identity1'.
 213 prop_c0300_identity2 :: Cube -> Bool
 214 prop_c0300_identity2 cube =
 215     c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
 216       where
 217         t0 = tetrahedron cube 0
 218         t1 = tetrahedron cube 1
 219
 220
 221 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 222 --   'prop_c0102_identity1'.
 223 prop_c1101_identity :: Cube -> Bool
 224 prop_c1101_identity cube =
 225     c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
 226       where
 227         t0 = tetrahedron cube 0
 228         t1 = tetrahedron cube 1
 229
 230
 231 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 232 --   'prop_c0102_identity1'.
 233 prop_c1200_identity2 :: Cube -> Bool
 234 prop_c1200_identity2 cube =
 235     c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
 236       where
 237         t0 = tetrahedron cube 0
 238         t1 = tetrahedron cube 1
 239
 240
 241 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 242 --   'prop_c0102_identity1'.
 243 prop_c2100_identity2 :: Cube -> Bool
 244 prop_c2100_identity2 cube =
 245     c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
 246       where
 247         t0 = tetrahedron cube 0
 248         t1 = tetrahedron cube 1
 249
 250
 251 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
 252 --   fourth indices of c-t6 have been switched. This is because we
 253 --   store the triangles oriented such that their volume is
 254 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
 255 --   point in opposite directions, one of them has to have negative
 256 --   volume!
 257 prop_c3000_identity :: Cube -> Bool
 258 prop_c3000_identity cube =
 259     c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
 260                     - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
 261       where
 262         t0 = tetrahedron cube 0
 263         t6 = tetrahedron cube 6
 264
 265
 266 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 267 --   'prop_c3000_identity'.
 268 prop_c2010_identity :: Cube -> Bool
 269 prop_c2010_identity cube =
 270     c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
 271                     - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
 272       where
 273         t0 = tetrahedron cube 0
 274         t6 = tetrahedron cube 6
 275
 276
 277 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 278 --   'prop_c3000_identity'.
 279 prop_c2001_identity :: Cube -> Bool
 280 prop_c2001_identity cube =
 281     c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
 282                     - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
 283       where
 284         t0 = tetrahedron cube 0
 285         t6 = tetrahedron cube 6
 286
 287
 288 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 289 --   'prop_c3000_identity'.
 290 prop_c1020_identity :: Cube -> Bool
 291 prop_c1020_identity cube =
 292     c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
 293                     - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
 294       where
 295         t0 = tetrahedron cube 0
 296         t6 = tetrahedron cube 6
 297
 298
 299 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 300 --   'prop_c3000_identity'.
 301 prop_c1002_identity :: Cube -> Bool
 302 prop_c1002_identity cube =
 303     c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
 304                     - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
 305       where
 306         t0 = tetrahedron cube 0
 307         t6 = tetrahedron cube 6
 308
 309
 310 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 311 --   'prop_c3000_identity'.
 312 prop_c1011_identity :: Cube -> Bool
 313 prop_c1011_identity cube =
 314     c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
 315                     ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
 316       where
 317         t0 = tetrahedron cube 0
 318         t6 = tetrahedron cube 6
 319
 320
 321
 322 -- | Given in Sorokina and Zeilfelder, p. 78.
 323 prop_cijk1_identity :: Cube -> Bool
 324 prop_cijk1_identity cube =
 325      and [ c t0 i j k 1 ~=
 326                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
 327                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
 328                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
 329                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
 330                                                       j <- [0..2],
 331                                                       k <- [0..2],
 332                                                       i + j + k == 2]
 333       where
 334         t0 = tetrahedron cube 0
 335         t1 = tetrahedron cube 1
 336
 337
 338 -- | The function values at the interior should be the same for all
 339 --   tetrahedra.
 340 prop_interior_values_all_identical :: Cube -> Bool
 341 prop_interior_values_all_identical cube =
 342     all_equal [ eval (Tetrahedron.fv tet) I | tet <- tetrahedra cube ]
 343
 344
 345 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 346 --   This test checks the rotation works as expected.
 347 prop_c_tilde_2100_rotation_correct :: Cube -> Bool
 348 prop_c_tilde_2100_rotation_correct cube =
 349     expr1 == expr2
 350     where
 351       t0 = tetrahedron cube 0
 352       t6 = tetrahedron cube 6
 353
 354       -- What gets computed for c2100 of t6.
 355       expr1 = eval (Tetrahedron.fv t6) $
 356           (3/8)*I +
 357           (1/12)*(T + R + L + D) +
 358           (1/64)*(FT + FR + FL + FD) +
 359           (7/48)*F +
 360           (1/48)*B +
 361           (1/96)*(RT + LD + LT + RD) +
 362           (1/192)*(BT + BR + BL + BD)
 363
 364       -- What should be computed for c2100 of t6.
 365       expr2 = eval (Tetrahedron.fv t0) $
 366               (3/8)*I +
 367               (1/12)*(F + R + L + B) +
 368               (1/64)*(FT + RT + LT + BT) +
 369               (7/48)*T +
 370               (1/48)*D +
 371               (1/96)*(FR + FL + BR + BL) +
 372               (1/192)*(FD + RD + LD + BD)
 373
 374
 375 -- | We know what (c t6 2 1 0 0) should be from Sorokina and
 376 --   Zeilfelder, p. 87.  This test checks the actual value based on
 377 --   the FunctionValues of the cube.
 378 --
 379 --   If 'prop_c_tilde_2100_rotation_correct' passes, then this test is
 380 --   even meaningful!
 381 prop_c_tilde_2100_correct :: Cube -> Bool
 382 prop_c_tilde_2100_correct cube =
 383     c t6 2 1 0 0 == expected
 384     where
 385       t0 = tetrahedron cube 0
 386       t6 = tetrahedron cube 6
 387       fvs = Tetrahedron.fv t0
 388       expected = eval fvs $
 389                   (3/8)*I +
 390                   (1/12)*(F + R + L + B) +
 391                   (1/64)*(FT + RT + LT + BT) +
 392                   (7/48)*T +
 393                   (1/48)*D +
 394                   (1/96)*(FR + FL + BR + BL) +
 395                   (1/192)*(FD + RD + LD + BD)
 396
 397
 398 -- Tests to check that the correct edges are incidental.
 399 prop_t0_shares_edge_with_t1 :: Cube -> Bool
 400 prop_t0_shares_edge_with_t1 cube =
 401     (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
 402       where
 403         t0 = tetrahedron cube 0
 404         t1 = tetrahedron cube 1
 405
 406 prop_t0_shares_edge_with_t3 :: Cube -> Bool
 407 prop_t0_shares_edge_with_t3 cube =
 408     (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
 409       where
 410         t0 = tetrahedron cube 0
 411         t3 = tetrahedron cube 3
 412
 413 prop_t0_shares_edge_with_t6 :: Cube -> Bool
 414 prop_t0_shares_edge_with_t6 cube =
 415     (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
 416       where
 417         t0 = tetrahedron cube 0
 418         t6 = tetrahedron cube 6
 419
 420 prop_t1_shares_edge_with_t2 :: Cube -> Bool
 421 prop_t1_shares_edge_with_t2 cube =
 422     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 423       where
 424         t1 = tetrahedron cube 1
 425         t2 = tetrahedron cube 2
 426
 427 prop_t1_shares_edge_with_t19 :: Cube -> Bool
 428 prop_t1_shares_edge_with_t19 cube =
 429     (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
 430       where
 431         t1 = tetrahedron cube 1
 432         t19 = tetrahedron cube 19
 433
 434 prop_t2_shares_edge_with_t3 :: Cube -> Bool
 435 prop_t2_shares_edge_with_t3 cube =
 436     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 437       where
 438         t1 = tetrahedron cube 1
 439         t2 = tetrahedron cube 2
 440
 441 prop_t2_shares_edge_with_t12 :: Cube -> Bool
 442 prop_t2_shares_edge_with_t12 cube =
 443     (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
 444       where
 445         t2 = tetrahedron cube 2
 446         t12 = tetrahedron cube 12
 447
 448 prop_t3_shares_edge_with_t21 :: Cube -> Bool
 449 prop_t3_shares_edge_with_t21 cube =
 450     (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
 451       where
 452         t3 = tetrahedron cube 3
 453         t21 = tetrahedron cube 21
 454
 455 prop_t4_shares_edge_with_t5 :: Cube -> Bool
 456 prop_t4_shares_edge_with_t5 cube =
 457     (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
 458       where
 459         t4 = tetrahedron cube 4
 460         t5 = tetrahedron cube 5
 461
 462 prop_t4_shares_edge_with_t7 :: Cube -> Bool
 463 prop_t4_shares_edge_with_t7 cube =
 464     (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
 465       where
 466         t4 = tetrahedron cube 4
 467         t7 = tetrahedron cube 7
 468
 469 prop_t4_shares_edge_with_t10 :: Cube -> Bool
 470 prop_t4_shares_edge_with_t10 cube =
 471     (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
 472       where
 473         t4 = tetrahedron cube 4
 474         t10 = tetrahedron cube 10
 475
 476 prop_t5_shares_edge_with_t6 :: Cube -> Bool
 477 prop_t5_shares_edge_with_t6 cube =
 478     (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
 479       where
 480         t5 = tetrahedron cube 5
 481         t6 = tetrahedron cube 6
 482
 483 prop_t5_shares_edge_with_t16 :: Cube -> Bool
 484 prop_t5_shares_edge_with_t16 cube =
 485     (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
 486       where
 487         t5 = tetrahedron cube 5
 488         t16 = tetrahedron cube 16
 489
 490 prop_t6_shares_edge_with_t7 :: Cube -> Bool
 491 prop_t6_shares_edge_with_t7 cube =
 492     (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
 493       where
 494         t6 = tetrahedron cube 6
 495         t7 = tetrahedron cube 7
 496
 497 prop_t7_shares_edge_with_t20 :: Cube -> Bool
 498 prop_t7_shares_edge_with_t20 cube =
 499     (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
 500       where
 501         t7 = tetrahedron cube 7
 502         t20 = tetrahedron cube 20