kavraiskii.txt * * * * * * * * * * * * * * * * * * * * * * * = === = === = === = === = === = === = === = === = === = === = === = * * * * * * * * * * * * * * * * * * * * * * * http://members.lycos.co.uk/stereotomography/ http://stereotomography.tripod.com/ http://georef.cos.com/cgi-bin/getRec?un=1985-032079 http://www.brebeuf.qc.ca/fortin/categories/liens-geologie.htm ---------------------------------------------- The Kavraiskii's "constant" ( 0.30035..... ) ---------------------------------------------- Source : Stereotomography by Philippe G. Dor and André F. Labossière ( Received July 14, 1983; accepted after revision June 8, 1984 ) Engineering Geology, December 1984, 20(4): 311-324 [ ISSN 0013-7952 ], Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands 0013-7952/84/$03.00 (c) 1984 Elsevier Science Publishers B.V. (c) 1984 Philippe G. Dor & André F. Labossière [ u.s.(c) TXu-162-403, 84-06-05 ] ( page 314 ) « ... Stereograms ~~~~~~~~~~~ In a stereographic projection it is possible to represent a plane as a cyclographic trace of a great circle, and a line as a point. Two types of nets are commonly used for structural or geomechanical studies: the Schmidt equal area net and the Wulff equal angle net. However, many other types of stereograms are yet possible as the orthographic net, the Kavraiskii net ( Duncan, 1981, see below [*] ), and others where their order is n>1 (convenient for discontinuity attitudes of low dip), and which we can reproduce using the following equation for their relative radius r . # (n) ______________________ [ where : # = theta ] ~~~~~~~~~~~~~~~~~~~~~~ / \ | # # | | cos - - sin - | | 2 2 | D \ / r = --- --------------------- n=~0.30035... # 2 n (Kavraiskii) (n) / \ | # # | | cos - + sin - | | 2 2 | \ / where D = diameter of the stereogram, # = plunge (or dip), orthographic (O) n=-1, Schmidt (S) n=0, Kavraiskii (K) n=~0.30035, Wulff (W) n=1. Thus, despite the large variety of choices in stereograms, we resorted to the Schmidt net and to the stereonet (Wulff net), and obtained some clearly interesting results. ... » [*] Duncan, A.C., 1981. A review of Cartesian coordinate construction from a sphere, for generation of two dimensional geological net projections. Comput. Geosci., 7(4): 367-385 * * * * * * * * * * * * * * * * * * * * * * * = === = === = === = === = === = === = === = === = === = === = === = * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * The Kavraiskii's "constant" * * * * * * * * * * * * * * * * * * * 0.30035 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * K-A-V-R-A-I-S-K-I-I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * La "constante" de Kavraiskii * * * * * * * * * * * * * * * * * * * * * * * * * * * 0,30035 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / \ | # # | | cos - - sin - | | 2 2 | / \ D \ / D | # | r = --- ------------------------ = --- | 1 - --------- | # 2 K 2 | 90 deg. | (K) / \ (#) \ / | # # | | cos - + sin - | | 2 2 | \ / ___________________ [ où : # = théta ] ~~~~~~~~~~~~~~~~~~~ +----+-------------------------+ +----+-------------------------+ | | * * | | | # | K * * # | K | | | (#) * * | (#) | | | * * | | +----+-------------------------+ +----+-------------------------+ *----*-------------------------* +----+-------------------------+ | | * * | | | 0 | { 0,273 239 5 } * * 50 | 0,297 562 415 313 817 | | | * * | | | 1 | 0,273 965 079 806 458 * * 51 | 0,297 835 137 706 546 | | | * * | | | 2 | 0,274 679 044 968 856 * * 52 | 0,298 100 483 128 396 | | | * * | | | 3 | 0,275 381 591 722 495 * * 53 | 0,298 358 489 549 358 | | | * * | | | 4 | 0,276 072 867 321 610 * * 54 | 0,298 609 193 734 683 | | | * * | | | 5 | 0,276 753 014 919 336 * * 55 | 0,298 852 631 263 784 | | | * * | | | 6 | 0,277 422 173 705 871 * * 56 | 0,299 088 836 548 422 | | | * * | | | 7 | 0,278 080 479 040 248 * * 57 | 0,299 317 842 850 149 | | | * * | | | 8 | 0,278 728 062 576 125 * * 58 | 0,299 539 682 297 072 | | | * * | | | 9 | 0,279 365 052 381 890 * * 59 | 0,299 754 385 899 923 | | | * * | | +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ | | * * | | | 10 | 0,279 991 573 055 421 * * 60 | 0,299 961 983 567 480 | | | * * | | | 11 | 0,280 607 745 833 733 * * 61 | 0,300 162 504 121 346 | | | * * | | | 12 | 0,281 213 688 697 858 * * 62 | 0,300 355 975 310 098 | | | * * | | | 13 | 0,281 809 516 473 123 * * 63 | 0,300 542 423 822 844 | | | * * | | | 14 | 0,282 395 340 925 131 * * 64 | 0,300 721 875 302 176 | | | * * | | | 15 | 0,282 971 270 851 632 * * 65 | 0,300 894 354 356 552 | | | * * | | | 16 | 0,283 537 412 170 497 * * 66 | 0,301 059 884 572 120 | | | * * | | | 17 | 0,284 093 868 003 998 * * 67 | 0,301 218 488 523 985 | | | * * | | | 18 | 0,284 640 738 759 584 * * 68 | 0,301 370 187 786 946 | | | * * | | | 19 | 0,285 178 122 207 281 * * 69 | 0,301 515 002 945 722 | | | * * | | +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ | | * * | | | 20 | 0,285 706 113 553 939 * * 70 | 0,301 652 953 604 637 | | | * * | | | 21 | 0,286 224 805 514 452 * * 71 | 0,301 784 058 396 826 | | | * * | | | 22 | 0,286 734 288 380 073 * * 72 | 0,301 908 334 992 949 | | | * * | | | 23 | 0,287 234 650 083 997 * * 73 | 0,302 025 800 109 415 | | | * * | | | 24 | 0,287 725 976 264 324 * * 74 | 0,302 136 469 516 132 | | | * * | | | 25 | 0,288 208 350 324 504 * * 75 | 0,302 240 358 043 804 | | | * * | | | 26 | 0,288 681 853 491 427 * * 76 | 0,302 337 479 590 757 | | | * * | | | 27 | 0,289 146 564 871 200 * * 77 | 0,302 427 847 129 331 | | | * * | | | 28 | 0,289 602 561 502 774 * * 78 | 0,302 511 472 711 809 | | | * * | | | 29 | 0,290 049 918 409 474 * * 79 | 0,302 588 367 475 939 | | | * * | | +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ | | * * | | | 30 | 0,290 488 708 648 546 * * 80 | 0,302 658 541 649 990 | | | * * | | | 31 | 0,290 919 003 358 799 * * 81 | 0,302 722 004 557 413 | | | * * | | | 32 | 0,291 340 871 806 427 * * 82 | 0,302 778 764 621 073 | | | * * | | | 33 | 0,291 754 381 429 072 * * 83 | 0,302 828 829 367 047 | | | * * | | | 34 | 0,292 159 597 878 233 * * 84 | 0,302 872 205 428 039 | | | * * | | | 35 | 0,292 556 585 060 064 * * 85 | 0,302 908 898 546 367 | | | * * | | | 36 | 0,292 945 405 174 620 * * 86 | 0,302 938 913 576 562 | | | * * | | | 37 | 0,293 326 118 753 645 * * 87 | 0,302 962 254 487 519 | | | * * | | | 38 | 0,293 698 784 696 945 * * 88 | 0,302 978 924 364 346 | | | * * | | | 39 | 0,294 063 460 307 377 * * 89 | 0,302 988 925 409 643 | | | * * | | +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ +----+-------------------------+ | | * * | | | 40 | 0,294 420 201 324 552 * * 90 | { 0,302 992 2 } | | | * * | | | 41 | 0,294 769 061 957 271 * *----*-------------------------* | | * | 42 | 0,295 110 094 914 748 * +----+-------------------------+ | | * * | | | 43 | 0,295 443 351 436 679 * * # | K | | | * * | (#) | | 44 | 0,295 768 881 322 179 * * | | | | * +----+-------------------------+ | 45 | 0,296 086 732 957 639 * | | * | 46 | 0,296 396 953 343 553 * | | * | 47 | 0,296 699 588 120 325 * | | * | 48 | 0,296 994 681 593 124 * | | * | 49 | 0,297 282 276 755 784 * | | * +----+-------------------------+ +----+-------------------------+ | | * | # | K * ___________________ | | (#) * [ où : # = théta ] | | * ~~~~~~~~~~~~~~~~~~~ +----+-------------------------+ moyenne de K [réel] = 0,293 393 808 416 893 ( logarithmique ) ( entre 1 et 89 degrés ) moyenne de K [réel] = 0,293 522 467 964 471 ( arithmétique ) ( entre 1 et 89 degrés ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * +-----------------+--------------------------+ | | | | # | K | | | (#) | | (en degrés) | | | | | +-----------------+--------------------------+ | | | | 0,1 | 0,273 312 623 295 896 | | 0,01 | 0,273 246 857 864 159 | | 0,001 | 0,273 240 276 083 093 | | 0,0001 | 0,273 239 617 794 197 | | 0,00001 | 0,273 239 550 840 614 | | 0,000001 | 0,273 239 532 492 291 | | 0,0000001 | ~ 0,273 239 544 409 333 | | | | | | [ valeur approchée ] | | | [ pour 0 degré ] | | | { 0,273 239 5 } <= | | | | +-----------------+--------------------------+ +-----------------+--------------------------+ | | | | # | K | | | (#) | | (en degrés) | | | | | +-----------------+--------------------------+ | | | | 89,9 | 0,302 992 225 609 648 | | 89,99 | 0,302 992 258 608 862 | | 89,999 | 0,302 992 258 932 316 | | 89,9999 | 0,302 992 258 973 093 | | 89,99999 | 0,302 992 260 229 236 | | 89,999999 | ~ 0,302 992 242 473 131 | | 89,9999999 | ~ 0,302 992 120 962 939 | | | | | | [ valeur approchée ] | | | [ pour 90 degrés ] | | | { 0,302 992 2 } <= | | | | +-----------------+--------------------------+ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * où le domaine est : 0,2732395 < K < 0,3029922 (#) pour : 0 degré < # < 90 degrés ************************************************************************* * * * 0,031 313 351 208 171 * * K [estimé] = 0,262 739 117 600 074 ( # ) * * * * coefficient de corrélation linéaire = * * 96,645 740 762 266 037 % * * * ************************************************************************* ln A = - 1,336 593 687 399 358 = a A = 0,262 739 117 600 074 B = 0,031 313 351 208 171 = b ln K = ln A + B (ln #) B K [est.] = A ( # ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ----------------------------------------- Some greek letters converted in Unicode ~~~~ ~~~~~ ~~~~~~~ ~~~~~~~~~ ~~ ~~~~~~~ alpha with tonos = & # 9 4 0 ; gamma = & # 9 4 7 ; delta = & # 9 4 8 ; epsilon = & # 9 4 9 ; eta with tonos = & # 9 4 2 ; [ # = ] theta = & # 9 5 2 ; mu = & # 9 5 6 ; omicron = & # 9 5 9 ; omicron with tonos = & # 9 7 2 ; rho = & # 9 6 1 ; sigma = & # 9 6 3 ; final sigma = & # 9 6 2 ; tau = & # 9 6 4 ; phi = & # 9 6 6 ; omega = & # 9 6 9 ; -----------------------------------------