SovietRxiv

Auxiliary Tables for Solving the Principal Geodetic Problems on the Terrestrial Spheroid by the Method of B. F. Krutoy

A. G. Lesnyak
Submitted 1967 | SovietRxiv: ru-196701.09461 | Translated from Russian

Abstract Generated abstract

This paper presents auxiliary numerical tables intended to simplify computations in solving the fundamental geodetic problems on the terrestrial spheroid by B. F. Krutoy’s method, including direct resection. The tables use the argument defined by the sine of the azimuth of the rectified curve at its equatorial intersection and provide tabulated values and differences for several coefficients needed in the method. The author explains the interpolation procedure, mainly Newton interpolation with second-order differences and linear interpolation for selected tables, and gives a worked example for computing one coefficient. The contribution is a practical computational aid designed to reduce the labor and length of geodetic calculations while retaining the precision required by the tabulated quantities.

Full Text

Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 154 1967

Auxiliary Tables for Solving the Fundamental Geodetic Problems on the Terrestrial Spheroid by the Method of B. F. Krutoy

A. G. Lesnyak

(Presented in May 1966 by the mathematical section of the scientific-technical conference of the AVTF, dedicated to the 70th anniversary of TPI)

The use of the present tables makes it possible substantially to facilitate and shorten the computations in solving, by the method of B. F. Krutoy [1], the fundamental geodetic problems on the terrestrial spheroid, including direct resection. The argument for all tabular quantities is the quantity

\[ \nu=\sin A_{\mathrm{e}}, \]

where \(A_{\mathrm{e}}\) is the azimuth of the rectified curve \(\Gamma\) at the point of its intersection with the equator; evidently

\[ 0 \leq \nu \leq 1. \]

The tabular interval of variation of the argument is taken equal to 0.01; thus, the tables give values of the functions for 100 values of the argument.

Table 1 gives the values of the quantity \(D_2\) with nine decimal places, as well as the first differences \(\Delta_1\) and the second differences \(\Delta_2\). Interpolation is carried out by Newton’s formula with differences of the 2nd order:

\[ f(T_0+i\omega)=f(T_0)+i\Delta_1+\frac{i(i-1)}{2}\Delta_2, \]

where

\[ \begin{aligned} f(T_0) &\text{ is the tabular value of the function,}\\ f(T_0+i\omega) &\text{ is the required value of the function,}\\ \omega &\text{ is the tabular interval of variation of the argument.} \end{aligned} \]

Denote

\[ \frac{i(i-1)}{2}=A, \]

then

\[ f(T_0+i\omega)=f(T_0)+i\Delta_1+A\Delta_2. \]

Example: given \(\nu=0.343974\ 146\).
Find \(D_2\).

We find the tabular value \(D'_2\) for the argument \(\nu=0.34\)

\[ D'_2=-0.002\ 228\ 296. \]

\[ \Delta_1=+0.000017\ 334,\qquad i=0.3974, \]

\[ i\Delta_1=+0.000\ 006\ 888,\qquad \Delta_2=+0.000\ 000\ 503,\qquad i=0.40. \]

\[ A=-0.12,\qquad A\Delta_2=-0.000\ 000\ 060. \]

Table 1

\(v\) \(D_2\) \(\Delta_1\) \(\Delta_2\) \(v\) \(D_2\) \(\Delta_1\) \(\Delta_2\)
0,01 −0,002 518 215 +756 0,32 −0,002 261 352 +16 327 +503
2 −0,002 517 458 1 251 +494 33 −0,002 245 025 16 830 503
3 −0,002 516 207 1 757 502 34 −0,002 228 196 17 333 503
4 −0,002 514 450 2 259 502 35 −0,002 210 862 17 836 503
5 −0,002 512 191 2 761 502 36 −0,002 193 026 18 340 503
6 −0,002 509 430 3 263 502 37 −0,002 174 687 18 843 503
7 −0,002 506 167 3 765 502 38 −0,002 155 844 19 346 503
8 −0,002 502 402 4 267 502 39 −0,002 136 498 19 850 504
9 −0,002 498 134 4 769 502 40 −0,002 116 648 20 354 504
10 −0,002 493 365 5 271 502 41 −0,002 096 294 20 857 504
11 −0,002 488 094 5 774 502 42 −0,002 075 437 21 361 504
12 −0,002 482 320 6 276 502 43 −0,002 054 076 21 865 504
13 −0,002 476 044 6 778 502 44 −0,002 032 211 22 369 504
14 −0,002 469 266 7 280 502 45 −0,002 009 842 22 873 504
15 −0,002 461 986 7 782 502 46 −0,001 986 969 23 377 504
16 −0,002 454 204 8 285 502 47 −0,001 963 592 23 881 504
17 −0,002 445 920 8 787 502 48 −0,001 939 711 24 386 504
18 −0,002 437 133 9 289 502 49 −0,001 915 325 24 890 504
19 −0,002 427 844 9 792 502 50 −0,001 890 435 25 394 504
20 −0,002 418 052 10 294 502 51 −0,001 865 040 25 899 505
21 −0,002 407 758 10 796 502 52 −0,001 839 141 26 404 505
22 −0,002 396 962 11 299 502 505
23 −0,002 385 663 11 801 503 53 −0,001 812 737 26 909 505
24 −0,002 373 861 12 304 503 54 −0,001 785 828 27 414 505
25 −0,002 361 557 12 807 503 55 −0,001 758 415 27 919 505
26 −0,002 348 750 13 309 503 56 −0,001 730 496 28 424 505
27 −0,002 335 441 13 812 503 57 −0,001 702 072 28 929 505
28 −0,002 321 629 14 315 503 58 −0,001 673 142 29 435 505
29 −0,002 307 314 14 818 503 59 −0,001 643 708 29 940 505
30 −0,002 292 496 15 321 503 60 −0,001 613 767 30 446 505
31 −0,002 277 176 15 824 503 61 −0,001 583 321 30 952 505
62 −0,001 552 370 506

Continuation of Table 1

\(v\) \(D_2\) \(\Delta_1\) \(\Delta_2\) \(v\) \(D_2\) \(\Delta_1\) \(\Delta_2\)
0,63 −0,001 520 912 +31 458 +505 0,85 −0,000 700 573 +42 623 +509
64 −0,001 488 948 31 964 506 86 −0,000 657 440 43 132 510
65 −0,001 456 479 32 470 506 87 −0,000 613 798 43 642 510
66 −0,001 423 503 32 976 506 88 −0,000 569 646 44 152 510
67 −0,001 390 020 33 482 506 89 −0,000 524 984 44 662 510
68 −0,001 356 031 33 989 507 90 −0,000 479 813 45 172 510
69 −0,001 321 536 34 496 507 91 −0,000 434 131 45 682 510
70 −0,001 286 533 35 003 507 92 −0,000 387 938 46 192 511
71 −0,001 251 023 35 510 507 93 −0,000 341 235 46 703 511
72 −0,001 215 007 36 017 507 94 −0,000 294 021 47 214 511
73 −0,001 178 483 36 524 507 95 −0,000 246 296 47 725 511
74 −0,001 141 451 37 031 508 96 −0,000 198 060 48 236 511
75 −0,001 103 912 37 539 508 97 −0,000 149 313 48 747 512
76 −0,001 065 856 38 047 508 98 −0,000 100 054 49 259 512
77 −0,001 027 311 38 554 508 99 −0,000 050 283 49 771
78 −0,000 988 249 39 062 508
79 −0,000 948 678 39 571 508
80 −0,000 908 599 40 079 508
81 −0,000 868 011 40 588 509
82 −0,000 826 915 41 096 509
83 −0,000 785 310 41 605 509
84 −0,000 743 196 42 114 509

Table 2

$\gamma$ $D_4$ $\Delta_1$ $D_6$ $\gamma$ $D_4$ $\Delta_1$ $D_6$
0,01 +0,000 002 642 −2 −0,000 000 035 0,32 +0,000 002 131 −31 −0,000 000 025
2 641 3 34 33 100 31 25
3 638 4 34 34 069 32 24
4 634 5 34 35 037 33 24
5 630 6 34 36 004 33 23
6 624 7 34 37 +0,000 001 970 34 22
7 617 8 34 38 936 34 22
8 609 9 34 39 902 35 21
9 600 10 34 40 867 36 21
10 590 11 34 41 831 36 20
11 579 12 34 42 795 37 19
12 567 13 33 43 758 37 19
13 554 14 33 44 721 38 18
14 540 15 33 45 683 38 18
15 525 16 32 46 645 39 17
16 510 17 32 47 606 39 16
17 493 18 32 48 568 39 16
18 475 19 31 49 528 39 15
19 456 20 31 50 489 40 15
20 436 21 31 51 449 40 14
21 416 22 30 52 409 40 14
22 394 23 30 53 369 41 013
23 371 23 30 54 329 41 12
24 348 24 29 55 288 41 12
25 324 25 29 56 248 41 11
26 299 26 28 57 207 41 11
27 273 27 28 58 166 40 10
28 246 28 27 59 126 41 10
29 218 28 27 60 085 41 9
30 190 29 26 61 044 40 9
31 161 30 26 62 004 40 8

Continuation of Table 2

γ \(D_4\) \(\Delta_1\) \(D_6\) γ \(D_4\) \(\Delta_1\) \(D_6\)
0,63 +0,000 000 964 −40 −0,000 000 008 0,94 +0,000 000 036 −11
64 924 40 7 95 025 9
65 884 40 7 96 016 7
66 844 39 6 97 009 5
67 805 39 6 98 004 3
68 766 38 5 99 001
69 728 38 5
70 690 38 5
71 652 37 4
72 615 36 4
73 579 36 4
74 543 35 3
75 508 35 3
76 473 33 3
77 440 33 2
78 407 32 2
79 375 31 2
80 344 31 2
81 313 28 1
82 285 28 1
83 257 27 1
84 230 26 1
85 204 24 1
86 180 23 1
87 157 22 0
88 135 20
89 115 19
90 096 18
91 078 15
92 063 14
93 049 13

Table 3

\(v\) \(C_0\mu\) \(\Delta_1\) \(\Delta_2\) \(v\) \(C_0\mu\) \(\Delta_1\) \(\Delta_2\)
0,01 0,998 321 371 —502 0,43 0,998 014 804 —14 577
2 323 869 837 —335 44 000 226 14 913
3 323 031 1 172 45 0,997 985 314 15 248
4 321 859 1 507 46 970 065 15 584
5 320 352 1 842 47 954 482 15 919
6 318 509 2 177 48 938 562 16 655
7 316 332 2 512 49 922 308 16 591
8 313 820 2 847 50 905 717 16 926
9 310 973 3 182 51 888 791 17 262
10 307 791 3 517 52 871 530 17 597
11 304 274 3 852 53 853 933 17 933
12 300 421 4 187 54 836 000 18 268
13 296 234 4 522 55 817 731 18 604
14 291 713 4 858 56 799 127 18 940
15 286 855 5 192 57 780 187 19 276
16 281 663 5 527 58 760 911 19 612
17 276 136 5 862 59 741 300 19 947
18 270 274 6 197 60 721 352 20 283
19 264 077 6 532 61 701 069 20 619
20 257 545 6 867 62 680 450 20 955
21 250 678 7 202 63 659 495 21 291
22 243 475 7 537 64 638 204 21 627
23 235 938 7 872 65 616 577 21 963
24 228 065 8 207 66 594 614 22 299
25 219 858 8 543 67 572 315 22 635
26 211 315 8 878 68 549 680 22 971
27 202 438 9 213 69 526 709 23 308
28 193 224 9 548 335 70 503 401 23 644
29 183 676 9 883 336 71 476 757 23 980
30 173 793 10 218 72 455 777 24 316
31 163 575 10 554 73 431 461 24 652
32 153 021 10 889 74 406 809 24 989
33 142 132 11 224 75 381 820 25 325
34 130 908 11 559 76 356 495 25 652
35 119 349 11 895 77 330 833 25 998
36 107 454 12 230 78 304 835 26 334
37 095 225 12 565 79 278 501 26 671
38 082 660 12 900 80 251 830 27 008
39 069 759 13 236 81 224 822 27 344
40 056 523 13 571 82 197 478 27 681
41 042 952 13 906 83 169 797 28 018 —337
42 029 046 14 242 84 141 779 28 354

Continuation of Table 3

$\gamma$ $C_0\mu$ $\Delta_1$ $\Delta_2$ $\gamma$ $C_0\mu$ $\Delta_1$ $\Delta_2$
0,85 0,997 113 425 −28 691 0,93 0,996 874 462 −31 387
86 084 734 29 028 94 843 075 31 724
87 055 706 29 365 95 811 351 32 061
88 026 341 29 702 96 779 289 32 399
89 0,996 996 639 30 039 97 746 891 32 736
90 966 601 30 376 98 714 155 33 073
91 936 225 30 713 99 681 081
92 905 512 31 050

Table 4

$\gamma$ $C_0$ $\Delta_1$ $\Delta_2$ $\gamma$ $C_0$ $\Delta_1$ $\Delta_2$
0,01 1,005 051 269 −1 515 0,43 1,004 118 019 −43 939
2 49 754 2 525 −1 010 44 074 080 44 949
3 47 229 3 535 45 029 131 45 960
4 43 694 4 545 46 1,003 983 171 46 970
5 39 150 5 555 47 936 201 47 981
6 33 595 6 565 48 888 220 48 991
7 27 030 7 575 49 839 229 50 002 −1 011
8 19 456 8 584 50 789 227 51 012
9 10 871 9 594 51 738 214 52 023
10 01 277 10 604 52 686 191 53 034
11 1,004 990 673 11 614 53 633 158 54 044
12 979 058 12 624 54 579 113 55 055
13 966 434 13 634 55 524 058 56 066
14 952 800 14 644 56 467 993 57 076
15 938 155 15 654 57 410 916 58 087
16 922 501 16 664 58 352 829 59 098
17 905 837 17 674 59 293 731 60 109
18 888 163 18 684 60 233 622 61 120
19 869 478 19 694 61 172 503 62 130
20 849 784 20 704 62 110 372 63 141
21 829 030 21 714 63 047 231 64 152
22 807 365 22 724 64 1,002 983 079 65 163
23 784 641 23 734 65 917 916 66 174
24 760 906 24 744 66 851 741 67 185
25 736 162 25 755 67 784 556 68 196
26 710 407 26 765 68 716 360 69 207
27 683 642 27 775 69 647 152 70 219
28 655 867 28 785 70 576 933 71 230
29 627 082 29 795 71 505 704 72 241
30 597 237 30 805 72 433 463 73 252
31 566 482 31 816 73 360 210 74 264
32 534 666 32 826 74 285 947 75 275
33 501 841 33 836 75 210 672 76 286
34 468 005 34 846 76 134 386 77 298
35 433 159 35 856 77 057 088 78 309
36 397 302 36 867 78 1,001 978 779 79 320
37 360 436 37 877 79 899 458 80 332
38 322 559 38 887 80 819 126 81 344
39 283 672 39 898 81 737 783 82 355
40 243 774 40 908 82 655 428 83 367
41 202 866 41 918 83 572 061 84 378
42 160 948 42 929 84 487 682 85 390 −1 012

Continuation of Table 4

$\nu$ $C_0$ $\Delta_1$ $\Delta_2$ $\nu$ $C_0$ $\Delta_1$ $\Delta_2$
0,85 1,001 404 292 −86 402 0,93 1,000 682 742 −94 498
86 315 890 87 414 94 588 244 95 510
87 228 476 88 426 95 492 734 96 522
88 140 050 89 438 96 396 212 97 534
89 050 613 90 950 97 298 678 98 547
90 1,000 960 163 91 462 98 200 131 99 559
91 868 702 92 474 99 100 572
92 776 228 93 486

Table 5

\(\nu\) \(\dfrac{e^2}{2}\nu\mu\) \(\Delta_1\) \(\nu\) \(\dfrac{e^2}{2}\nu\mu\) \(\Delta_1\)
0,01 0,000 033 243 +33 243 0,32 0,001 064 144 +33 278
2 066 486 243 33 097 422 281
3 099 730 243 34 130 703 283
4 132 973 244 35 163 986 285
5 166 217 244 36 197 271 287
6 199 461 245 37 230 558 291
7 232 706 245 38 263 849 292
8 265 950 246 39 297 141 295
9 299 196 246 40 330 436 298
10 332 442 247 41 363 734 301
11 365 689 247 42 397 035 304
12 398 936 249 43 430 339 306
13 432 185 249 44 463 645 309
14 465 434 250 45 496 954 313
15 498 684 251 46 530 267 315
16 531 935 252 47 563 582 318
17 565 187 254 48 596 900 322
18 598 441 254 49 630 222 325
19 631 695 256 50 663 547 328
20 664 951 257 51 696 876 332
21 698 208 259 52 730 208 335
22 731 467 260 53 763 543 339
23 764 727 261 54 795 882 342
24 797 988 263 55 830 224 346
25 831 251 265 56 863 570 350
26 864 516 267 57 896 920 354
27 897 783 268 58 930 274 358
28 931 051 270 59 963 632 361
29 964 321 273 60 996 993 366
30 997 594 274 61 0,002 030 359 370
31 0,001 030 868 276 62 063 729 374

Continuation of Table 5

\(\gamma\) \(\dfrac{e^2}{2}\,\psi\mu\) \(\Delta_1\) \(\gamma\) \(\dfrac{e^2}{2}\,\psi\mu\) \(\Delta_1\)
0,63 0,002 097 103 +33 378 0,94 0,003 134 133 +33 543
64 130 481 382 95 167 676 550
65 163 863 387 96 201 226 557
66 197 250 391 97 234 783 563
67 230 611 396 98 268 346 569
68 264 037 400 99 301 915
69 297 437 405
70 330 842 410
71 364 252 414
72 397 666 420
73 431 086 424
74 464 510 429
75 497 939 435
76 531 374 439
77 564 813 444
78 598 257 450
79 631 707 455
80 665 162 461
81 698 623 466
82 732 089 472
83 765 561 477
84 799 038 483
85 832 521 488
86 866 009 495
87 899 504 500
88 933 004 506
89 966 510 512
90 0,003 000 022 519
91 033 541 524
92 067 065 531
93 100 596 537

Table 6

$\nu$ $\dfrac{3}{16} e^2 k^2 \nu\mu$ $\Delta_1$ $\nu$ $\dfrac{3}{16} e^2 k^2 \nu\mu$ $\Delta_1$
0,01 0,000 000 083 +84 0,32 0,000 002 399 +57
2 167 83 33 456 56
3 250 83 34 512 54
4 333 83 35 566 52
5 416 83 36 618 50
6 499 82 37 668 49
7 581 82 38 717 46
8 663 82 39 763 45
9 745 81 40 808 43
10 826 81 41 851 40
11 907 80 42 891 39
12 987 80 43 930 36
13 0,000 001 067 79 44 966 35
14 146 78 45 0,000 003 001 32
15 224 77 46 033 29
16 301 77 47 062 28
17 378 76 48 090 24
18 454 75 49 114 23
19 529 74 50 137 20
20 603 73 51 157 17
21 676 72 52 174 15
22 748 71 53 189 12
23 819 70 54 201 10
24 888 69 55 211 7
25 957 67 56 218 3
26 0,000 002 024 66 57 221 1
27 090 65 58 222 −1
28 155 63 59 221 5
29 218 62 60 216 8
30 280 60 61 208 11
31 340 59 62 197 14

Continuation of Table 6

$\gamma$ $\dfrac{3}{16} e^2 k^2 \gamma\mu$ $\Delta_1$ $\gamma$ $\dfrac{3}{16} e^2 k^2 \gamma\mu$ $\Delta_1$
0,63 0,000 003 183 −17 0,94 0,000 000 921 −141
64 166 20 95 780 146
65 146 24 96 634 151
66 122 27 97 483 156
67 095 30 98 327 161
68 065 34 99 166
69 031 37
70 0,000 002 994 41
71 953 45
72 908 48
73 860 51
74 809 56
75 753 59
76 694 63
77 631 67
78 564 70
79 494 75
80 419 79
81 340 83
82 257 87
83 170 92
84 078 96
85 0,000 001 982 99
86 883 105
87 778 108
88 670 114
89 556 118
90 438 122
91 316 127
92 189 131
93 058 137

Table 7

$\nu$ $\dfrac{5}{16} e^{2} k^{4}\psi_{\mu}$ $\nu$ $\dfrac{5}{16} e^{2} k^{4}\psi_{\mu}$
0,01 0,000 000 001 0,50 0,000 000 026
2 2 51 26
3 3 52 26
4 4 53 26
5 5 54 25
6 6 55 25
7 6 56 25
8 7 57 24
9 8 58 24
10 9 59 24
11 10 60 23
12 11 61 22
13 12 62 22
14 12 63 22
15 13 64 21
16 14 65 20
17 15 66 20
18 16 67 19
19 16 68 18
20 17 69 18
21 18 70 17
22 19 71 16
23 19 72 16
24 20 73 15
25 20 74 14
26 21 75 14
27 22 76 13
28 22 77 12
29 23 78 11
30 23 79 10
31 24 80 10
32 24 81 9
33 24 82 8
34 25 83 8
35 25 84 7
36 25 85 6
37 26 86 6
38 26 87 5
39 26 88 4
40 26 89 4
41 26 90 3
42 27 91 2
43 27 92 2
44 27 93 2
45 27 94 1
46 27 95 1
47 27 96 1
48 27 97 0
49 26 98 0
50 26 99 0

Finally we have:

\[ D=-0.002\ 221\ 368. \]

Table 2 contains the values of the quantities $D_{4}$ and $D_{6}$, and also $\Delta_{1}$ for $D_{4}$. There is no need to take the quantity $D_{8}$ into account, since it does not exceed one unit in the tenth decimal place.

Table 3 contains the values of the quantity $C_{0}\psi$, and also $\Delta_{1}$ and $\Delta_{2}$.

Table 4 contains the values of the quantity $C_{0}$, and also $\Delta_{1}$ and $\Delta_{2}$ for $C_{0}$.

Table 5 contains the values of the quantity $\dfrac{e^{2}}{2}\psi$, and also $\Delta_{1}$.

Table 6 contains the values of the quantity \(\dfrac{3}{16} e^{2} k^{2}\psi\), as well as \(\Delta_{1}\).

Table 7 gives the values of the quantity \(\dfrac{5}{16} e^{2} k^{4}\psi\).

Interpolation in Tables 2, 3, and 4 with second-order differences is carried out in the same way as in Table 1. In Tables 5 and 6, only linear interpolation is carried out. When using Table 7, the need for interpolation disappears altogether.

The table of the coefficients \(A\) is in Barlow’s tables, published by “Mir,” 1965.

LITERATURE

  1. B. F. Krutoy, General methods for solving the fundamental computational problems on the terrestrial spheroid. Tomsk, Proceedings of TPI, vol. 131, 1965.

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[v1] 1967

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Auxiliary Tables for Solving the Principal Geodetic Problems on the Terrestrial Spheroid by the Method of B. F. Krutoy