doc/spline3.lyx

   1 #LyX 1.6.8 created this file. For more info see http://www.lyx.org/
   2 \lyxformat 345
   3 \begin_document
   4 \begin_header
   5 \textclass amsart
   6 \use_default_options false
   7 \begin_modules
   8 theorems-ams
   9 eqs-within-sections
  10 figs-within-sections
  11 \end_modules
  12 \language english
  13 \inputencoding auto
  14 \font_roman ae
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  23 \graphics default
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  25 \spacing single
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  27 \papersize letterpaper
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  29 \use_amsmath 1
  30 \use_esint 1
  31 \cite_engine basic
  32 \use_bibtopic false
  33 \paperorientation portrait
  34 \secnumdepth 3
  35 \tocdepth 3
  36 \paragraph_separation skip
  37 \defskip bigskip
  38 \quotes_language english
  39 \papercolumns 1
  40 \papersides 1
  41 \paperpagestyle default
  42 \tracking_changes false
  43 \output_changes false
  44 \author ""
  45 \author ""
  46 \end_header
  47
  48 \begin_body
  49
  50 \begin_layout Title
  51 Spline3
  52 \end_layout
  53
  54 \begin_layout Author
  55 Michael Orlitzky
  56 \end_layout
  57
  58 \begin_layout Example*
  59 First, will will define our grid
  60 \begin_inset Formula $G$
  61 \end_inset
  62
  63  to be over a
  64 \begin_inset Formula $3\times3\times3$
  65 \end_inset
  66
  67  cube having grid size
  68 \begin_inset Formula $h=1$
  69 \end_inset
  70
  71 .
  72 \end_layout
  73
  74 \begin_layout Example*
  75 To test the reproduction of trilinears, we will take the function (per Sorokina
  76  and Zeilfelder, p.
  77  88),
  78 \end_layout
  79
  80 \begin_layout Example*
  81 \begin_inset Formula \[
  82 f\left(x,y,z\right)=1+x+xy+xyz\]
  83
  84 \end_inset
  85
  86
  87 \end_layout
  88
  89 \begin_layout Example*
  90 and compute its value at the 27 points,
  91 \begin_inset Formula $\left(0,0,0\right),\left(0,0,1\right),\dots\left(2,2,2\right)$
  92 \end_inset
  93
  94 .
  95 \end_layout
  96
  97 \begin_layout Example*
  98 \begin_inset listings
  99 inline false
 100 status open
 101
 102 \begin_layout Plain Layout
 103
 104 sage: x,y,z = var('x,y,z')
 105 \end_layout
 106
 107 \begin_layout Plain Layout
 108
 109 sage: f = 1 + x + x*y + x*y*z
 110 \end_layout
 111
 112 \begin_layout Plain Layout
 113
 114 sage: f(x=0, y=0, z=0)
 115 \end_layout
 116
 117 \begin_layout Plain Layout
 118
 119 1
 120 \end_layout
 121
 122 \begin_layout Plain Layout
 123
 124 sage: f(x=1, y=0, z=0)
 125 \end_layout
 126
 127 \begin_layout Plain Layout
 128
 129 2
 130 \end_layout
 131
 132 \begin_layout Plain Layout
 133
 134 sage: f(x=2, y=0, z=0)
 135 \end_layout
 136
 137 \begin_layout Plain Layout
 138
 139 3
 140 \end_layout
 141
 142 \begin_layout Plain Layout
 143
 144 sage: f(x=0, y=1, z=0)
 145 \end_layout
 146
 147 \begin_layout Plain Layout
 148
 149 1
 150 \end_layout
 151
 152 \begin_layout Plain Layout
 153
 154 sage: f(x=1, y=1, z=0)
 155 \end_layout
 156
 157 \begin_layout Plain Layout
 158
 159 3
 160 \end_layout
 161
 162 \begin_layout Plain Layout
 163
 164 sage: f(x=2, y=1, z=0)
 165 \end_layout
 166
 167 \begin_layout Plain Layout
 168
 169 5
 170 \end_layout
 171
 172 \begin_layout Plain Layout
 173
 174 sage: f(x=0, y=2, z=0)
 175 \end_layout
 176
 177 \begin_layout Plain Layout
 178
 179 1
 180 \end_layout
 181
 182 \begin_layout Plain Layout
 183
 184 sage: f(x=1, y=2, z=0)
 185 \end_layout
 186
 187 \begin_layout Plain Layout
 188
 189 4
 190 \end_layout
 191
 192 \begin_layout Plain Layout
 193
 194 sage: f(x=2, y=2, z=0)
 195 \end_layout
 196
 197 \begin_layout Plain Layout
 198
 199 7
 200 \end_layout
 201
 202 \begin_layout Plain Layout
 203
 204 sage: f(x=0, y=0, z=1)
 205 \end_layout
 206
 207 \begin_layout Plain Layout
 208
 209 1
 210 \end_layout
 211
 212 \begin_layout Plain Layout
 213
 214 sage: f(x=1, y=0, z=1)
 215 \end_layout
 216
 217 \begin_layout Plain Layout
 218
 219 2
 220 \end_layout
 221
 222 \begin_layout Plain Layout
 223
 224 sage: f(x=2, y=0, z=1)
 225 \end_layout
 226
 227 \begin_layout Plain Layout
 228
 229 3
 230 \end_layout
 231
 232 \begin_layout Plain Layout
 233
 234 sage: f(x=0, y=1, z=1)
 235 \end_layout
 236
 237 \begin_layout Plain Layout
 238
 239 1
 240 \end_layout
 241
 242 \begin_layout Plain Layout
 243
 244 sage: f(x=1, y=1, z=1)
 245 \end_layout
 246
 247 \begin_layout Plain Layout
 248
 249 4
 250 \end_layout
 251
 252 \begin_layout Plain Layout
 253
 254 sage: f(x=2, y=1, z=1)
 255 \end_layout
 256
 257 \begin_layout Plain Layout
 258
 259 7
 260 \end_layout
 261
 262 \begin_layout Plain Layout
 263
 264 sage: f(x=0, y=2, z=1)
 265 \end_layout
 266
 267 \begin_layout Plain Layout
 268
 269 1
 270 \end_layout
 271
 272 \begin_layout Plain Layout
 273
 274 sage: f(x=1, y=2, z=1)
 275 \end_layout
 276
 277 \begin_layout Plain Layout
 278
 279 6
 280 \end_layout
 281
 282 \begin_layout Plain Layout
 283
 284 sage: f(x=2, y=2, z=1)
 285 \end_layout
 286
 287 \begin_layout Plain Layout
 288
 289 11
 290 \end_layout
 291
 292 \begin_layout Plain Layout
 293
 294 sage: f(x=0, y=0, z=2)
 295 \end_layout
 296
 297 \begin_layout Plain Layout
 298
 299 1
 300 \end_layout
 301
 302 \begin_layout Plain Layout
 303
 304 sage: f(x=1, y=0, z=2)
 305 \end_layout
 306
 307 \begin_layout Plain Layout
 308
 309 2
 310 \end_layout
 311
 312 \begin_layout Plain Layout
 313
 314 sage: f(x=2, y=0, z=2)
 315 \end_layout
 316
 317 \begin_layout Plain Layout
 318
 319 3
 320 \end_layout
 321
 322 \begin_layout Plain Layout
 323
 324 sage: f(x=0, y=1, z=2)
 325 \end_layout
 326
 327 \begin_layout Plain Layout
 328
 329 1
 330 \end_layout
 331
 332 \begin_layout Plain Layout
 333
 334 sage: f(x=1, y=1, z=2)
 335 \end_layout
 336
 337 \begin_layout Plain Layout
 338
 339 5
 340 \end_layout
 341
 342 \begin_layout Plain Layout
 343
 344 sage: f(x=2, y=1, z=2)
 345 \end_layout
 346
 347 \begin_layout Plain Layout
 348
 349 9
 350 \end_layout
 351
 352 \begin_layout Plain Layout
 353
 354 sage: f(x=0, y=2, z=2)
 355 \end_layout
 356
 357 \begin_layout Plain Layout
 358
 359 1
 360 \end_layout
 361
 362 \begin_layout Plain Layout
 363
 364 sage: f(x=1, y=2, z=2)
 365 \end_layout
 366
 367 \begin_layout Plain Layout
 368
 369 8
 370 \end_layout
 371
 372 \begin_layout Plain Layout
 373
 374 sage: f(x=2, y=2, z=2)
 375 \end_layout
 376
 377 \begin_layout Plain Layout
 378
 379 15
 380 \end_layout
 381
 382 \end_inset
 383
 384
 385 \end_layout
 386
 387 \begin_layout Example*
 388 We are most interested in the
 389 \begin_inset Quotes eld
 390 \end_inset
 391
 392 interior
 393 \begin_inset Quotes erd
 394 \end_inset
 395
 396  cube centered at
 397 \begin_inset Formula $\left(1,1,1\right)$
 398 \end_inset
 399
 400  with datum
 401 \begin_inset Formula $4$
 402 \end_inset
 403
 404 .
 405  We can enter the data above into a list,
 406 \end_layout
 407
 408 \begin_layout Example*
 409 \begin_inset listings
 410 inline false
 411 status open
 412
 413 \begin_layout Plain Layout
 414
 415 sage: g = [[[ f(x=a, y=b, z=c) for a in range(0,3) ] for b in range(0,3)
 416  ] for c in range(0,3) ]
 417 \end_layout
 418
 419 \begin_layout Plain Layout
 420
 421 sage: g
 422 \end_layout
 423
 424 \begin_layout Plain Layout
 425
 426 [[[1, 2, 3], [1, 3, 5], [1, 4, 7]],
 427 \end_layout
 428
 429 \begin_layout Plain Layout
 430
 431  [[1, 2, 3], [1, 4, 7], [1, 6, 11]],
 432 \end_layout
 433
 434 \begin_layout Plain Layout
 435
 436  [[1, 2, 3], [1, 5, 9], [1, 8, 15]]]
 437 \end_layout
 438
 439 \end_inset
 440
 441
 442 \end_layout
 443
 444 \begin_layout Example*
 445 although the list will be indexed by
 446 \begin_inset Formula $\left(z,y,x\right)$
 447 \end_inset
 448
 449  so we define a function to access it by
 450 \begin_inset Formula $\left(x,y,z\right)$
 451 \end_inset
 452
 453 ,
 454 \end_layout
 455
 456 \begin_layout Example*
 457 \begin_inset listings
 458 inline false
 459 status open
 460
 461 \begin_layout Plain Layout
 462
 463 sage: def grid(x,y,z):
 464 \end_layout
 465
 466 \begin_layout Plain Layout
 467
 468 ....:     return g[z][y][x]
 469 \end_layout
 470
 471 \end_inset
 472
 473
 474 \end_layout
 475
 476 \begin_layout Example*
 477 and define directional functions according to Sorokina and Zeilfelder, p.
 478  81.
 479 \end_layout
 480
 481 \begin_layout Example*
 482 \begin_inset listings
 483 inline false
 484 status open
 485
 486 \begin_layout Plain Layout
 487
 488 sage: def I(x,y,z):
 489 \end_layout
 490
 491 \begin_layout Plain Layout
 492
 493 ....:     return grid(x,y,z)
 494 \end_layout
 495
 496 \begin_layout Plain Layout
 497
 498 \end_layout
 499
 500 \begin_layout Plain Layout
 501
 502 sage: def F(x,y,z):
 503 \end_layout
 504
 505 \begin_layout Plain Layout
 506
 507 ....:     return grid(x-1, y, z)
 508 \end_layout
 509
 510 \begin_layout Plain Layout
 511
 512 \end_layout
 513
 514 \begin_layout Plain Layout
 515
 516 sage: def B(x,y,z):
 517 \end_layout
 518
 519 \begin_layout Plain Layout
 520
 521 ....:     return grid(x+1, y, z)
 522 \end_layout
 523
 524 \begin_layout Plain Layout
 525
 526 \end_layout
 527
 528 \begin_layout Plain Layout
 529
 530 sage: def L(x,y,z):
 531 \end_layout
 532
 533 \begin_layout Plain Layout
 534
 535 ....:     return grid(x, y-1, z)
 536 \end_layout
 537
 538 \begin_layout Plain Layout
 539
 540 \end_layout
 541
 542 \begin_layout Plain Layout
 543
 544 sage: def R(x,y,z):
 545 \end_layout
 546
 547 \begin_layout Plain Layout
 548
 549 ....:     return grid(x, y+1, z)
 550 \end_layout
 551
 552 \begin_layout Plain Layout
 553
 554 \end_layout
 555
 556 \begin_layout Plain Layout
 557
 558 sage: def T(x,y,z):
 559 \end_layout
 560
 561 \begin_layout Plain Layout
 562
 563 ....:     return grid(x, y, z+1)
 564 \end_layout
 565
 566 \begin_layout Plain Layout
 567
 568 \end_layout
 569
 570 \begin_layout Plain Layout
 571
 572 sage: def D(x,y,z):
 573 \end_layout
 574
 575 \begin_layout Plain Layout
 576
 577 ....:     return grid(x, y, z-1)
 578 \end_layout
 579
 580 \begin_layout Plain Layout
 581
 582 \end_layout
 583
 584 \begin_layout Plain Layout
 585
 586 sage: def FL(x,y,z):
 587 \end_layout
 588
 589 \begin_layout Plain Layout
 590
 591 ....:     return grid(x-1, y-1, z)
 592 \end_layout
 593
 594 \begin_layout Plain Layout
 595
 596 \end_layout
 597
 598 \begin_layout Plain Layout
 599
 600 sage: def FR(x,y,z):
 601 \end_layout
 602
 603 \begin_layout Plain Layout
 604
 605 ....:     return grid(x-1, y+1, z)
 606 \end_layout
 607
 608 \begin_layout Plain Layout
 609
 610 \end_layout
 611
 612 \begin_layout Plain Layout
 613
 614 sage: def FD(x,y,z):
 615 \end_layout
 616
 617 \begin_layout Plain Layout
 618
 619 ....:     return grid(x-1, y, z-1)
 620 \end_layout
 621
 622 \begin_layout Plain Layout
 623
 624 \end_layout
 625
 626 \begin_layout Plain Layout
 627
 628 sage: def FT(x,y,z):
 629 \end_layout
 630
 631 \begin_layout Plain Layout
 632
 633 ....:     return grid(x-1, y, z+1)
 634 \end_layout
 635
 636 \begin_layout Plain Layout
 637
 638 \end_layout
 639
 640 \begin_layout Plain Layout
 641
 642 sage: def BL(x,y,z):
 643 \end_layout
 644
 645 \begin_layout Plain Layout
 646
 647 ....:     return grid(x+1, y-1, z)
 648 \end_layout
 649
 650 \begin_layout Plain Layout
 651
 652 \end_layout
 653
 654 \begin_layout Plain Layout
 655
 656 sage: def BR(x,y,z):
 657 \end_layout
 658
 659 \begin_layout Plain Layout
 660
 661 ....:     return grid(x+1, y+1, z)
 662 \end_layout
 663
 664 \begin_layout Plain Layout
 665
 666 \end_layout
 667
 668 \begin_layout Plain Layout
 669
 670 sage: def BD(x,y,z):
 671 \end_layout
 672
 673 \begin_layout Plain Layout
 674
 675 ....:     return grid(x+1, y, z-1)
 676 \end_layout
 677
 678 \begin_layout Plain Layout
 679
 680 \end_layout
 681
 682 \begin_layout Plain Layout
 683
 684 sage: def BT(x,y,z):
 685 \end_layout
 686
 687 \begin_layout Plain Layout
 688
 689 ....:     return grid(x+1, y, z+1)
 690 \end_layout
 691
 692 \begin_layout Plain Layout
 693
 694 \end_layout
 695
 696 \begin_layout Plain Layout
 697
 698 sage: def LD(x,y,z):
 699 \end_layout
 700
 701 \begin_layout Plain Layout
 702
 703 ....:     return grid(x, y-1, z-1)
 704 \end_layout
 705
 706 \begin_layout Plain Layout
 707
 708 \end_layout
 709
 710 \begin_layout Plain Layout
 711
 712 sage: def LT(x,y,z):
 713 \end_layout
 714
 715 \begin_layout Plain Layout
 716
 717 ....:     return grid(x, y-1, z+1)
 718 \end_layout
 719
 720 \begin_layout Plain Layout
 721
 722 \end_layout
 723
 724 \begin_layout Plain Layout
 725
 726 sage: def RD(x,y,z):
 727 \end_layout
 728
 729 \begin_layout Plain Layout
 730
 731 ....:     return grid(x, y+1, z-1)
 732 \end_layout
 733
 734 \begin_layout Plain Layout
 735
 736 \end_layout
 737
 738 \begin_layout Plain Layout
 739
 740 sage: def RT(x,y,z):
 741 \end_layout
 742
 743 \begin_layout Plain Layout
 744
 745 ....:     return grid(x, y+1, z+1)
 746 \end_layout
 747
 748 \begin_layout Plain Layout
 749
 750 \end_layout
 751
 752 \begin_layout Plain Layout
 753
 754 sage: def FLD(x,y,z):
 755 \end_layout
 756
 757 \begin_layout Plain Layout
 758
 759 ....:     return grid(x-1, y-1, z-1)
 760 \end_layout
 761
 762 \begin_layout Plain Layout
 763
 764 \end_layout
 765
 766 \begin_layout Plain Layout
 767
 768 sage: def FLT(x,y,z):
 769 \end_layout
 770
 771 \begin_layout Plain Layout
 772
 773 ....:     return grid(x-1, y-1, z+1)
 774 \end_layout
 775
 776 \begin_layout Plain Layout
 777
 778 \end_layout
 779
 780 \begin_layout Plain Layout
 781
 782 sage: def FRD(x,y,z):
 783 \end_layout
 784
 785 \begin_layout Plain Layout
 786
 787 ....:     return grid(x-1, y+1, z-1)
 788 \end_layout
 789
 790 \begin_layout Plain Layout
 791
 792 \end_layout
 793
 794 \begin_layout Plain Layout
 795
 796 sage: def FRT(x,y,z):
 797 \end_layout
 798
 799 \begin_layout Plain Layout
 800
 801 ....:     return grid(x-1, y+1, z+1)
 802 \end_layout
 803
 804 \begin_layout Plain Layout
 805
 806 \end_layout
 807
 808 \begin_layout Plain Layout
 809
 810 sage: def BLD(x,y,z):
 811 \end_layout
 812
 813 \begin_layout Plain Layout
 814
 815 ....:     return grid(x+1, y-1, z-1)
 816 \end_layout
 817
 818 \begin_layout Plain Layout
 819
 820 \end_layout
 821
 822 \begin_layout Plain Layout
 823
 824 sage: def BLT(x,y,z):
 825 \end_layout
 826
 827 \begin_layout Plain Layout
 828
 829 ....:     return grid(x+1, y-1, z+1)
 830 \end_layout
 831
 832 \begin_layout Plain Layout
 833
 834 \end_layout
 835
 836 \begin_layout Plain Layout
 837
 838 sage: def BRD(x,y,z):
 839 \end_layout
 840
 841 \begin_layout Plain Layout
 842
 843 ....:     return grid(x+1, y+1, z-1)
 844 \end_layout
 845
 846 \begin_layout Plain Layout
 847
 848 \end_layout
 849
 850 \begin_layout Plain Layout
 851
 852 sage: def BRT(x,y,z):
 853 \end_layout
 854
 855 \begin_layout Plain Layout
 856
 857 ....:     return grid(x+1, y+1, z+1)
 858 \end_layout
 859
 860 \end_inset
 861
 862
 863 \end_layout
 864
 865 \begin_layout Example*
 866 Next, we define the coefficients for the cube centered on
 867 \begin_inset Formula $\left(1,1,1\right)$
 868 \end_inset
 869
 870  based on these directional functions.
 871 \end_layout
 872
 873 \begin_layout Example*
 874 \begin_inset listings
 875 inline false
 876 status open
 877
 878 \begin_layout Plain Layout
 879
 880 sage: c0030 = (1/8)*( I(1,1,1) + F(1,1,1) + L(1,1,1) + T(1,1,1) +
 881 \end_layout
 882
 883 \begin_layout Plain Layout
 884
 885                       LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + FLT(1,1,1) )
 886 \end_layout
 887
 888 \begin_layout Plain Layout
 889
 890 \end_layout
 891
 892 \begin_layout Plain Layout
 893
 894 sage: c0003 = (1/8)*( I(1,1,1) + F(1,1,1) + R(1,1,1) + T(1,1,1) +
 895 \end_layout
 896
 897 \begin_layout Plain Layout
 898
 899                 RT(1,1,1) + FR(1,1,1) + FT(1,1,1) + FRT(1,1,1) )
 900 \end_layout
 901
 902 \begin_layout Plain Layout
 903
 904 \end_layout
 905
 906 \begin_layout Plain Layout
 907
 908 sage: c0021 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) +
 909 \end_layout
 910
 911 \begin_layout Plain Layout
 912
 913               (1/24)*(L(1,1,1) + FL(1,1,1) + LT(1,1,1) + FLT(1,1,1))
 914 \end_layout
 915
 916 \begin_layout Plain Layout
 917
 918 \end_layout
 919
 920 \begin_layout Plain Layout
 921
 922 sage: c0012 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) +
 923 \end_layout
 924
 925 \begin_layout Plain Layout
 926
 927               (1/24)*(R(1,1,1) + FR(1,1,1) + RT(1,1,1) + FRT(1,1,1))
 928 \end_layout
 929
 930 \begin_layout Plain Layout
 931
 932 \end_layout
 933
 934 \begin_layout Plain Layout
 935
 936 sage: c0120 = (5/24)*(I(1,1,1) + F(1,1,1)) +
 937 \end_layout
 938
 939 \begin_layout Plain Layout
 940
 941               (1/8)*(L(1,1,1) + T(1,1,1) + FL(1,1,1) + FT(1,1,1)) +
 942 \end_layout
 943
 944 \begin_layout Plain Layout
 945
 946               (1/24)*(LT(1,1,1) + FLT(1,1,1))
 947 \end_layout
 948
 949 \begin_layout Plain Layout
 950
 951 \end_layout
 952
 953 \begin_layout Plain Layout
 954
 955 sage: c0102 = (5/24)*(I(1,1,1) + F(1,1,1)) +
 956 \end_layout
 957
 958 \begin_layout Plain Layout
 959
 960               (1/8)*(R(1,1,1) + T(1,1,1) + FR(1,1,1) + FT(1,1,1)) +
 961 \end_layout
 962
 963 \begin_layout Plain Layout
 964
 965               (1/24)*(RT(1,1,1) + FRT(1,1,1))
 966 \end_layout
 967
 968 \begin_layout Plain Layout
 969
 970 \end_layout
 971
 972 \begin_layout Plain Layout
 973
 974 sage: c0111 = (13/48)*(I(1,1,1) + F(1,1,1)) +
 975 \end_layout
 976
 977 \begin_layout Plain Layout
 978
 979               (7/48)*(T(1,1,1) + FT(1,1,1)) +
 980 \end_layout
 981
 982 \begin_layout Plain Layout
 983
 984               (1/32)*(L(1,1,1) + R(1,1,1) + FL(1,1,1) + FR(1,1,1)) +
 985 \end_layout
 986
 987 \begin_layout Plain Layout
 988
 989               (1/96)*(LT(1,1,1) + RT(1,1,1) + FLT(1,1,1) + FRT(1,1,1))
 990 \end_layout
 991
 992 \begin_layout Plain Layout
 993
 994 \end_layout
 995
 996 \begin_layout Plain Layout
 997
 998 sage: c0210 = (13/48)*(I(1,1,1) + F(1,1,1)) +
 999 \end_layout
1000
1001 \begin_layout Plain Layout
1002
1003               (17/192)*(L(1,1,1) + T(1,1,1) + FL(1,1,1) + FT(1,1,1)) +
1004 \end_layout
1005
1006 \begin_layout Plain Layout
1007
1008               (1/96)*(LT(1,1,1) + FLT(1,1,1)) +
1009 \end_layout
1010
1011 \begin_layout Plain Layout
1012
1013               (1/64)*(R(1,1,1) + D(1,1,1) + FR(1,1,1) + FD(1,1,1)) +
1014 \end_layout
1015
1016 \begin_layout Plain Layout
1017
1018               (1/192)*(RT(1,1,1) + LD(1,1,1) + FRT(1,1,1) + FLD(1,1,1))
1019 \end_layout
1020
1021 \begin_layout Plain Layout
1022
1023 \end_layout
1024
1025 \begin_layout Plain Layout
1026
1027 sage: c0201 = (13/48)*(I(1,1,1) + F(1,1,1)) +
1028 \end_layout
1029
1030 \begin_layout Plain Layout
1031
1032               (17/192)*(R(1,1,1) + T(1,1,1) + FR(1,1,1) + FT(1,1,1)) +
1033 \end_layout
1034
1035 \begin_layout Plain Layout
1036
1037               (1/96)*(RT(1,1,1) + FRT(1,1,1)) +
1038 \end_layout
1039
1040 \begin_layout Plain Layout
1041
1042               (1/64)*(L(1,1,1) + D(1,1,1) + FL(1,1,1) + FD(1,1,1)) +
1043 \end_layout
1044
1045 \begin_layout Plain Layout
1046
1047               (1/192)*(RD(1,1,1) + LT(1,1,1) + FLT(1,1,1) + FRD(1,1,1))
1048 \end_layout
1049
1050 \begin_layout Plain Layout
1051
1052 \end_layout
1053
1054 \begin_layout Plain Layout
1055
1056 sage: c0300 = (13/48)*(I(1,1,1) + F(1,1,1)) +
1057 \end_layout
1058
1059 \begin_layout Plain Layout
1060
1061               (5/96)*(L(1,1,1) + R(1,1,1) + T(1,1,1) + D(1,1,1) +
1062 \end_layout
1063
1064 \begin_layout Plain Layout
1065
1066                       FL(1,1,1) + FR(1,1,1) + FT(1,1,1) + FD(1,1,1)) +
1067 \end_layout
1068
1069 \begin_layout Plain Layout
1070
1071               (1/192)*(RT(1,1,1) + RD(1,1,1) + LT(1,1,1) + LD(1,1,1) +
1072 \end_layout
1073
1074 \begin_layout Plain Layout
1075
1076                        FRT(1,1,1) + FRD(1,1,1) + FLT(1,1,1) + FLD(1,1,1))
1077 \end_layout
1078
1079 \begin_layout Plain Layout
1080
1081 \end_layout
1082
1083 \begin_layout Plain Layout
1084
1085 sage: c1020 = (1/4)*I(1,1,1) +
1086 \end_layout
1087
1088 \begin_layout Plain Layout
1089
1090               (1/6)*(F(1,1,1) + L(1,1,1) + T(1,1,1)) +
1091 \end_layout
1092
1093 \begin_layout Plain Layout
1094
1095               (1/12)*(LT(1,1,1) + FL(1,1,1) + FT(1,1,1))
1096 \end_layout
1097
1098 \begin_layout Plain Layout
1099
1100 \end_layout
1101
1102 \begin_layout Plain Layout
1103
1104 sage: c1002 = (1/4)*I(1,1,1) +
1105 \end_layout
1106
1107 \begin_layout Plain Layout
1108
1109               (1/6)*(F(1,1,1) + R(1,1,1) + T(1,1,1)) +
1110 \end_layout
1111
1112 \begin_layout Plain Layout
1113
1114               (1/12)*(RT(1,1,1) + FR(1,1,1) + FT(1,1,1))
1115 \end_layout
1116
1117 \begin_layout Plain Layout
1118
1119 \end_layout
1120
1121 \begin_layout Plain Layout
1122
1123 sage: c1011 = (1/3)*I(1,1,1) +
1124 \end_layout
1125
1126 \begin_layout Plain Layout
1127
1128               (5/24)*(F(1,1,1) + T(1,1,1)) +
1129 \end_layout
1130
1131 \begin_layout Plain Layout
1132
1133               (1/12)*FT(1,1,1) + (1/24)*(L(1,1,1) + R(1,1,1)) +
1134 \end_layout
1135
1136 \begin_layout Plain Layout
1137
1138               (1/48)*(LT(1,1,1) + RT(1,1,1) + FL(1,1,1) + FR(1,1,1))
1139 \end_layout
1140
1141 \begin_layout Plain Layout
1142
1143 \end_layout
1144
1145 \begin_layout Plain Layout
1146
1147 sage: c1110 = (1/3)*I(1,1,1) +
1148 \end_layout
1149
1150 \begin_layout Plain Layout
1151
1152               (5/24)*F(1,1,1) +
1153 \end_layout
1154
1155 \begin_layout Plain Layout
1156
1157               (1/8)*(L(1,1,1) + T(1,1,1)) +
1158 \end_layout
1159
1160 \begin_layout Plain Layout
1161
1162               (5/96)*(FL(1,1,1) + FT(1,1,1)) +
1163 \end_layout
1164
1165 \begin_layout Plain Layout
1166
1167               (1/48)*(D(1,1,1) + R(1,1,1) + LT(1,1,1)) +
1168 \end_layout
1169
1170 \begin_layout Plain Layout
1171
1172               (1/96)*(FD(1,1,1) + LD(1,1,1) + RT(1,1,1) + FR(1,1,1))
1173 \end_layout
1174
1175 \begin_layout Plain Layout
1176
1177 \end_layout
1178
1179 \begin_layout Plain Layout
1180
1181 sage: c1101 = (1/3)*I(1,1,1) +
1182 \end_layout
1183
1184 \begin_layout Plain Layout
1185
1186               (5/24)*F(1,1,1) +
1187 \end_layout
1188
1189 \begin_layout Plain Layout
1190
1191               (1/8)*(R(1,1,1) + T(1,1,1)) +
1192 \end_layout
1193
1194 \begin_layout Plain Layout
1195
1196               (5/96)*(FR(1,1,1) + FT(1,1,1)) +
1197 \end_layout
1198
1199 \begin_layout Plain Layout
1200
1201               (1/48)*(D(1,1,1) + L(1,1,1) + RT(1,1,1)) +
1202 \end_layout
1203
1204 \begin_layout Plain Layout
1205
1206               (1/96)*(FD(1,1,1) + LT(1,1,1) + RD(1,1,1) + FL(1,1,1))
1207 \end_layout
1208
1209 \begin_layout Plain Layout
1210
1211 \end_layout
1212
1213 \begin_layout Plain Layout
1214
1215 sage: c1200 = (1/3)*I(1,1,1) +
1216 \end_layout
1217
1218 \begin_layout Plain Layout
1219
1220               (5/24)*F(1,1,1) +
1221 \end_layout
1222
1223 \begin_layout Plain Layout
1224
1225               (7/96)*(L(1,1,1) + R(1,1,1) + T(1,1,1) + D(1,1,1)) +
1226 \end_layout
1227
1228 \begin_layout Plain Layout
1229
1230               (1/32)*(FL(1,1,1) + FR(1,1,1) + FT(1,1,1) + FD(1,1,1)) +
1231 \end_layout
1232
1233 \begin_layout Plain Layout
1234
1235               (1/96)*(RT(1,1,1) + RD(1,1,1) + LT(1,1,1) + LD(1,1,1))
1236 \end_layout
1237
1238 \begin_layout Plain Layout
1239
1240 \end_layout
1241
1242 \begin_layout Plain Layout
1243
1244 sage: c2010 = (3/8)*I(1,1,1) +
1245 \end_layout
1246
1247 \begin_layout Plain Layout
1248
1249               (7/48)*(F(1,1,1) + T(1,1,1) + L(1,1,1)) +
1250 \end_layout
1251
1252 \begin_layout Plain Layout
1253
1254               (1/48)*(R(1,1,1) + D(1,1,1) + B(1,1,1) + LT(1,1,1) + FL(1,1,1)
1255  + FT(1,1,1)) +
1256 \end_layout
1257
1258 \begin_layout Plain Layout
1259
1260               (1/96)*(RT(1,1,1) + BT(1,1,1) + FR(1,1,1) + FD(1,1,1) + LD(1,1,1)
1261  + BL(1,1,1))
1262 \end_layout
1263
1264 \begin_layout Plain Layout
1265
1266 \end_layout
1267
1268 \begin_layout Plain Layout
1269
1270 sage: c2001 = (3/8)*I(1,1,1) +
1271 \end_layout
1272
1273 \begin_layout Plain Layout
1274
1275               (7/48)*(F(1,1,1) + T(1,1,1) + R(1,1,1)) +
1276 \end_layout
1277
1278 \begin_layout Plain Layout
1279
1280               (1/48)*(L(1,1,1) + D(1,1,1) + B(1,1,1) + RT(1,1,1) + FR(1,1,1)
1281  + FT(1,1,1)) +
1282 \end_layout
1283
1284 \begin_layout Plain Layout
1285
1286               (1/96)*(LT(1,1,1) + BT(1,1,1) + FL(1,1,1) + FD(1,1,1) + RD(1,1,1)
1287  + BR(1,1,1))
1288 \end_layout
1289
1290 \begin_layout Plain Layout
1291
1292 \end_layout
1293
1294 \begin_layout Plain Layout
1295
1296 sage: c2100 = (3/8)*I(1,1,1) +
1297 \end_layout
1298
1299 \begin_layout Plain Layout
1300
1301               (1/12)*(T(1,1,1) + R(1,1,1) + L(1,1,1) + D(1,1,1)) +
1302 \end_layout
1303
1304 \begin_layout Plain Layout
1305
1306               (1/64)*(FT(1,1,1) + FR(1,1,1) + FL(1,1,1) + FD(1,1,1)) +
1307 \end_layout
1308
1309 \begin_layout Plain Layout
1310
1311               (7/48)*F(1,1,1) +
1312 \end_layout
1313
1314 \begin_layout Plain Layout
1315
1316               (1/48)*B(1,1,1) +
1317 \end_layout
1318
1319 \begin_layout Plain Layout
1320
1321               (1/96)*(RT(1,1,1) + LD(1,1,1) + LT(1,1,1) + RD(1,1,1)) +
1322 \end_layout
1323
1324 \begin_layout Plain Layout
1325
1326               (1/192)*(BT(1,1,1) + BR(1,1,1) + BL(1,1,1) + BD(1,1,1))
1327 \end_layout
1328
1329 \begin_layout Plain Layout
1330
1331 \end_layout
1332
1333 \begin_layout Plain Layout
1334
1335 sage: c3000 = (3/8)*I(1,1,1) +
1336 \end_layout
1337
1338 \begin_layout Plain Layout
1339
1340               (1/12)*(T(1,1,1) + F(1,1,1) + L(1,1,1) + R(1,1,1) + D(1,1,1)
1341  + B(1,1,1)) +
1342 \end_layout
1343
1344 \begin_layout Plain Layout
1345
1346               (1/96)*(LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + RT(1,1,1) + BT(1,1,1)
1347  + FR(1,1,1)  +
1348 \end_layout
1349
1350 \begin_layout Plain Layout
1351
1352                       FD(1,1,1) + LD(1,1,1) + BD(1,1,1) + BR(1,1,1) + RD(1,1,1)
1353  + BL(1,1,1))
1354 \end_layout
1355
1356 \end_inset
1357
1358
1359 \end_layout
1360
1361 \begin_layout Example*
1362 We can see what the constant values are now:
1363 \end_layout
1364
1365 \begin_layout Example*
1366 \begin_inset listings
1367 inline false
1368 status open
1369
1370 \begin_layout Plain Layout
1371
1372 sage: c0030
1373 \end_layout
1374
1375 \begin_layout Plain Layout
1376
1377 17/8
1378 \end_layout
1379
1380 \begin_layout Plain Layout
1381
1382 \end_layout
1383
1384 \begin_layout Plain Layout
1385
1386 sage: c0003
1387 \end_layout
1388
1389 \begin_layout Plain Layout
1390
1391 27/8
1392 \end_layout
1393
1394 \begin_layout Plain Layout
1395
1396 \end_layout
1397
1398 \begin_layout Plain Layout
1399
1400 sage: c0021
1401 \end_layout
1402
1403 \begin_layout Plain Layout
1404
1405 61/24
1406 \end_layout
1407
1408 \begin_layout Plain Layout
1409
1410 \end_layout
1411
1412 \begin_layout Plain Layout
1413
1414 sage: c0012
1415 \end_layout
1416
1417 \begin_layout Plain Layout
1418
1419 71/24
1420 \end_layout
1421
1422 \begin_layout Plain Layout
1423
1424 \end_layout
1425
1426 \begin_layout Plain Layout
1427
1428 sage: c0120
1429 \end_layout
1430
1431 \begin_layout Plain Layout
1432
1433 55/24
1434 \end_layout
1435
1436 \begin_layout Plain Layout
1437
1438 \end_layout
1439
1440 \begin_layout Plain Layout
1441
1442 sage: c0102
1443 \end_layout
1444
1445 \begin_layout Plain Layout
1446
1447 73/24
1448 \end_layout
1449
1450 \begin_layout Plain Layout
1451
1452 \end_layout
1453
1454 \begin_layout Plain Layout
1455
1456 sage: c0111
1457 \end_layout
1458
1459 \begin_layout Plain Layout
1460
1461 8/3
1462 \end_layout
1463
1464 \begin_layout Plain Layout
1465
1466 \end_layout
1467
1468 \begin_layout Plain Layout
1469
1470 sage: c0210
1471 \end_layout
1472
1473 \begin_layout Plain Layout
1474
1475 29/12
1476 \end_layout
1477
1478 \begin_layout Plain Layout
1479
1480 \end_layout
1481
1482 \begin_layout Plain Layout
1483
1484 sage: c0201
1485 \end_layout
1486
1487 \begin_layout Plain Layout
1488
1489 11/4
1490 \end_layout
1491
1492 \begin_layout Plain Layout
1493
1494 \end_layout
1495
1496 \begin_layout Plain Layout
1497
1498 sage: c0300
1499 \end_layout
1500
1501 \begin_layout Plain Layout
1502
1503 5/2
1504 \end_layout
1505
1506 \begin_layout Plain Layout
1507
1508 \end_layout
1509
1510 \begin_layout Plain Layout
1511
1512 sage: c1020
1513 \end_layout
1514
1515 \begin_layout Plain Layout
1516
1517 8/3
1518 \end_layout
1519
1520 \begin_layout Plain Layout
1521
1522 \end_layout
1523
1524 \begin_layout Plain Layout
1525
1526 sage: c1002
1527 \end_layout
1528
1529 \begin_layout Plain Layout
1530
1531 23/6
1532 \end_layout
1533
1534 \begin_layout Plain Layout
1535
1536 \end_layout
1537
1538 \begin_layout Plain Layout
1539
1540 sage: c1011
1541 \end_layout
1542
1543 \begin_layout Plain Layout
1544
1545 13/4
1546 \end_layout
1547
1548 \begin_layout Plain Layout
1549
1550 \end_layout
1551
1552 \begin_layout Plain Layout
1553
1554 sage: c1110
1555 \end_layout
1556
1557 \begin_layout Plain Layout
1558
1559 23/8
1560 \end_layout
1561
1562 \begin_layout Plain Layout
1563
1564 \end_layout
1565
1566 \begin_layout Plain Layout
1567
1568 sage: c1101
1569 \end_layout
1570
1571 \begin_layout Plain Layout
1572
1573 27/8
1574 \end_layout
1575
1576 \begin_layout Plain Layout
1577
1578 \end_layout
1579
1580 \begin_layout Plain Layout
1581
1582 sage: c1200
1583 \end_layout
1584
1585 \begin_layout Plain Layout
1586
1587 3
1588 \end_layout
1589
1590 \begin_layout Plain Layout
1591
1592 \end_layout
1593
1594 \begin_layout Plain Layout
1595
1596 sage: c2010
1597 \end_layout
1598
1599 \begin_layout Plain Layout
1600
1601 10/3
1602 \end_layout
1603
1604 \begin_layout Plain Layout
1605
1606 \end_layout
1607
1608 \begin_layout Plain Layout
1609
1610 sage: c2001
1611 \end_layout
1612
1613 \begin_layout Plain Layout
1614
1615 4
1616 \end_layout
1617
1618 \begin_layout Plain Layout
1619
1620 \end_layout
1621
1622 \begin_layout Plain Layout
1623
1624 sage: c2100
1625 \end_layout
1626
1627 \begin_layout Plain Layout
1628
1629 7/2
1630 \end_layout
1631
1632 \begin_layout Plain Layout
1633
1634 \end_layout
1635
1636 \begin_layout Plain Layout
1637
1638 sage: c3000
1639 \end_layout
1640
1641 \begin_layout Plain Layout
1642
1643 4
1644 \end_layout
1645
1646 \end_inset
1647
1648
1649 \end_layout
1650
1651 \begin_layout Example*
1652 Now that we have the coefficients, we'll choose a particular tetrahedron
1653  and compute the polynomial over it.
1654  If we look at the
1655 \begin_inset Quotes eld
1656 \end_inset
1657
1658 top
1659 \begin_inset Quotes erd
1660 \end_inset
1661
1662  face of the cube (in the positive
1663 \begin_inset Formula $z$
1664 \end_inset
1665
1666  direction), there are only four tetrahedra to choose from.
1667  We'll be consider the
1668 \begin_inset Quotes eld
1669 \end_inset
1670
1671 right
1672 \begin_inset Quotes erd
1673 \end_inset
1674
1675  tetrahedron; that is, the one with vertices,
1676 \end_layout
1677
1678 \begin_layout Example*
1679 \begin_inset Formula \begin{eqnarray*}
1680 v_{0}=\left(0.5,1.5,1.5\right) &  & \mbox{at the front-right of the cube}\\
1681 v_{1}=\left(1.5,1.5,1.5\right) &  & \mbox{at the back-right of the cube}\\
1682 v_{2}=\left(1,1,1.5\right) &  & \mbox{at the center of the top face of the cube}\\
1683 v_{3}=\left(1,1,1\right) &  & \mbox{at the center of the cube}\end{eqnarray*}
1684
1685 \end_inset
1686
1687
1688 \end_layout
1689
1690 \end_body
1691 \end_document