152ZFG Základy fyzikální geodézie - úloha3: Porovnání verzí

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==Zadání úlohy==
==Zadání úlohy==
Pomocí osmi identických bodů se zadanými prostorovými souřadnicemi v kartézských souřadnicových systémech S1 a S2 vypočítejte transformační klíč Helmertovy prostorové transformace a body transformujte ze systému S1 do systému S2. Vypočítejte aposteriorní charakteristiky přesnosti plynoucí ze zavedení podmínky MNČ.
Pro zadanou zeměpisnou šířku <math>\phi</math> vygenerujte rekurentními vzorci normované Legendreovy přidružené funkce <math>P_{nm}(\sin\phi)</math> a jejich 1. a 2. derivace, kde zvolte <math>n_{max}=30</math>. Úspešnost řešení otestujte dosazením do Legendreovy diferenciální rovnice a spočtete základní charakteristiky přesnosti (střední hodnotu odchylky a RMS). Dále pro zadaný stupeň a řád určete průběh Legendreovy přidružené funkce s měnící se zeměpisnou šířkou a totéž proveďte s Legendreovým polynomem příslušného stupně. Výsledky graficky zobrazte a diskutujte v závěru (průběh funkce, počet nulových bodů, proč je výhodné použít rekurentní vztahy oproti přímé definice ...).


==Numerické zadání úlohy==
==Numerické zadání úlohy==


Souřadnice transformovaných bodů v systému S2 jsou pro všechna zadání stejné:
{| class="border"
{| class="border"
|bod
|číslo zadání
|x [m]
|<math>\phi</math>
|y [m]
|n,m  
|z [m]
|číslo zadání [m]
|-
|<math>\phi</math>
| A || 0,000 || 0,000 || 0,000
|n,m
|-
| B || 100,000 || 0,000 || 0,000
|-
| C || 100,000 || 100,000 || 0,000
|-
| D || 0,000 || 100,000 || 0,000
|-
| E || 0,000 || 0,000 || 100,000
|-
| F || 100,000 || 0,000 || 100,000
|-
|-
| G || 100,000 || 100,000 || 100,000
| 1 || 0,0 || 0,0 || 23 || 0,0 || 0,0
|-
|-
| H || 0,000 || 100,000 || 100,000
|}
|}




Souřadnice transformovaných bodů v systému S1:
{| class="border"
|číslo zadání
|bod
|x [m]
|y [m]
|z [m]
|číslo zadání
|bod
|x [m]
|y [m]
|z [m]
|-
|rowspan="8"  | 1 || A || 93,170 || -295,133 || 48,646 ||rowspan="8"  | 51 || A || 30,977 || -52,407 || 155,159
|-
| B || -1,411 || -280,864 || 24,723 || B || 25,958 || 9,966 || 203,122
|-
| C || -22,338 || -261,377 || 119,082 || C || 31,794 || -37,672 || 265,678
|-
| D || 72,244 || -275,639 || 143,002 || D || 36,814 || -100,045 || 217,715
|-
| E || 111,543 || -199,539 || 32,973 || E || 109,452 || -44,876 || 153,571
|-
| F || 16,961 || -185,277 || 9,045 || F || 104,430 || 17,496 || 201,538
|-
| G || -3,961 || -165,780 || 103,413 || G || 110,259 || -30,144 || 264,102
|-
| H || 90,619 || -180,041 || 127,335 || H || 115,281 || -92,510 || 216,133
|-
|rowspan="8"  | 2 || A || -30,662 || -150,076 || -11,197 ||rowspan="8"  | 52 || A || -58,386 || 179,416 || -139,345
|-
| B || -65,948 || -64,049 || -107,415 || B || 6,015 || 194,213 || -161,630
|-
| C || -154,799 || -152,591 || -154,001 || C || -17,000 || 254,451 || -188,161
|-
| D || -119,508 || -238,628 || -57,778 || D || -81,406 || 239,658 || -165,875
|-
| E || -124,284 || -98,472 || 69,285 || E || -44,756 || 211,279 || -78,829
|-
| F || -159,575 || -12,443 || -26,942 || F || 19,643 || 226,073 || -101,114
|-
| G || -248,427 || -100,991 || -73,525 || G || -3,379 || 286,308 || -127,643
|-
| H || -213,138 || -187,020 || 22,695 || H || -67,779 || 271,515 || -105,356
|-
|rowspan="8"  | 3 || A || -58,640 || 90,084 || -100,064 ||rowspan="8"  | 53 || A || 15,080 || 334,609 || -140,155
|-
| B || 0,229 || 66,547 || -170,125 || B || -10,561 || 301,188 || -205,519
|-
| C || 8,938 || 157,707 || -193,421 || C || -82,489 || 298,753 || -176,064
|-
| D || -49,932 || 181,236 || -123,365 || D || -56,837 || 332,171 || -110,696
|-
| E || 14,759 || 98,137 || -41,104 || E || 0,375 || 404,779 || -170,263
|-
| F || 73,625 || 74,606 || -111,166 || F || -25,273 || 371,357 || -235,635
|-
| G || 82,330 || 165,765 || -134,457 || G || -97,197 || 368,929 || -206,162
|-
| H || 23,463 || 189,294 || -64,400 || H || -71,549 || 402,343 || -140,802
|-
|rowspan="8"  | 4 || A || -22,297 || 366,540 || 79,651 ||rowspan="8"  | 54 || A || -8,488 || -143,002 || -28,202
|-
| B || 58,657 || 324,191 || 109,487 || B || 23,660 || -248,992 || 37,987
|-
| C || 81,716 || 304,082 || 18,377 || C || 137,422 || -252,463 || -22,818
|-
| D || 0,769 || 346,426 || -11,462 || D || 105,272 || -146,469 || -89,012
|-
| E || 24,086 || 450,435 || 72,867 || E || 43,249 || -69,502 || 64,376
|-
| F || 105,043 || 408,096 || 102,706 || F || 75,397 || -175,489 || 130,568
|-
| G || 128,106 || 387,984 || 11,597 || G || 189,146 || -178,964 || 69,760
|-
| H || 47,145 || 430,327 || -18,239 || H || 156,998 || -72,964 || 3,571
|-
|rowspan="8"  | 5 || A || -1,349 || -53,984 || 107,577 ||rowspan="8"  | 55 || A || -14,581 || 174,102 || 53,660
|-
| B || 12,440 || -86,892 || 148,631 || B || 28,563 || 70,321 || 134,653
|-
| C || 2,987 || -130,181 || 117,101 || C || -26,818 || -21,311 || 46,746
|-
| D || -10,801 || -97,276 || 76,046 || D || -69,968 || 82,470 || -34,241
|-
| E || 50,412 || -53,127 || 90,890 || E || 104,835 || 169,093 || -16,376
|-
| F || 64,199 || -86,029 || 131,941 || F || 147,989 || 65,326 || 64,612
|-
| G || 54,745 || -129,331 || 100,408 || G || 92,597 || -26,310 || -23,282
|-
| H || 40,963 || -96,415 || 59,353 || H || 49,448 || 77,466 || -104,272
|-
|rowspan="8"  | 6 || A || 20,503 || 393,620 || -122,535 ||rowspan="8"  | 56 || A || 52,342 || -91,495 || -67,506
|-
| B || -70,711 || 328,401 || -189,274 || B || 140,410 || -21,181 || -38,826
|-
| C || -110,464 || 271,128 || -78,977 || C || 115,863 || 46,740 || -129,970
|-
| D || -19,252 || 336,338 || -12,231 || D || 27,786 || -23,572 || -158,655
|-
| E || -63,906 || 491,048 || -102,367 || E || -19,532 || -28,522 || -1,227
|-
| F || -155,123 || 425,836 || -169,100 || F || 68,544 || 41,791 || 27,459
|-
| G || -194,875 || 368,565 || -58,798 || G || 43,995 || 109,719 || -63,687
|-
| H || -103,669 || 433,771 || 7,940 || H || -44,074 || 39,398 || -92,368
|-
|rowspan="8"  | 7 || A || 23,910 || 233,940 || -36,316 ||rowspan="8"  | 57 || A || 9,814 || 296,466 || 45,020
|-
| B || 47,640 || 328,518 || 31,422 || B || 28,940 || 250,611 || 2,156
|-
| C || 113,328 || 374,688 || -56,040 || C || -28,290 || 256,288 || -29,442
|-
| D || 89,611 || 280,104 || -123,782 || D || -47,417 || 302,135 || 13,430
|-
| E || -72,106 || 288,901 || -79,422 || E || 35,604 || 343,065 || 6,690
|-
| F || -48,378 || 383,477 || -11,686 || F || 54,720 || 297,210 || -36,181
|-
| G || 17,322 || 429,648 || -99,147 || G || -2,511 || 302,888 || -67,783
|-
| H || -6,403 || 335,061 || -166,890 || H || -21,628 || 348,742 || -24,913
|-
|rowspan="8"  | 8 || A || -20,027 || -70,648 || 74,791 ||rowspan="8"  | 58 || A || -3,202 || 109,027 || -276,757
|-
| B || -69,042 || -104,138 || -14,136 || B || 30,042 || 109,602 || -155,182
|-
| C || -1,308 || -46,278 || -73,262 || C || 68,834 || -9,904 || -165,220
|-
| D || 47,710 || -12,786 || 15,660 || D || 35,584 || -10,481 || -286,802
|-
| E || 46,612 || -154,089 || 69,484 || E || 112,022 || 149,085 || -308,451
|-
| F || -2,402 || -187,578 || -19,445 || F || 145,268 || 149,662 || -186,879
|-
| G || 65,327 || -129,720 || -78,576 || G || 184,049 || 30,157 || -196,919
|-
| H || 114,347 || -96,232 || 10,353 || H || 150,803 || 29,582 || -318,496
|-
|rowspan="8"  | 9 || A || 7,668 || 475,089 || 117,180 ||rowspan="8"  | 59 || A || -26,543 || -449,172 || 53,068
|-
| B || 55,432 || 448,741 || 105,023 || B || -94,065 || -405,727 || -12,443
|-
| C || 58,815 || 477,063 || 56,954 || C || -76,285 || -481,408 || -80,962
|-
| D || 11,051 || 503,407 || 69,119 || D || -8,756 || -524,847 || -15,459
|-
| E || 36,482 || 515,435 || 142,970 || E || -103,099 || -505,062 || 94,924
|-
| F || 84,247 || 489,084 || 130,818 || F || -170,629 || -461,617 || 29,420
|-
| G || 87,627 || 517,405 || 82,748 || G || -152,853 || -537,301 || -39,100
|-
| H || 39,864 || 543,748 || 94,911 || H || -85,317 || -580,741 || 26,406
|-
|rowspan="8"  | 10 || A || -10,754 || 207,875 || -47,920 ||rowspan="8"  | 60 || A || -64,418 || -97,966 || -83,457
|-
| B || -10,878 || 308,067 || -145,458 || B || -51,266 || -199,514 || 7,477
|-
| C || -7,130 || 210,560 || -245,610 || C || 62,152 || -242,037 || -56,400
|-
| D || -7,003 || 110,378 || -148,066 || D || 49,001 || -140,488 || -147,334
|-
| E || -150,523 || 205,165 || -50,514 || E || 11,196 || -16,528 || -3,437
|-
| F || -150,652 || 305,357 || -148,061 || F || 24,337 || -118,066 || 87,504
|-
| G || -146,903 || 207,851 || -248,211 || G || 137,753 || -160,591 || 23,619
|-
| H || -146,771 || 107,671 || -150,669 || H || 124,610 || -59,047 || -67,308
|-
|rowspan="8"  | 11 || A || 31,664 || -50,081 || 155,170 ||rowspan="8"  | 61 || A || 59,875 || 450,368 || 138,025
|-
| B || -52,182 || -140,593 || 125,299 || B || 108,055 || 323,175 || 152,809
|-
| C || -75,069 || -160,097 || 248,634 || C || 176,116 || 361,985 || 264,955
|-
| D || 8,775 || -69,583 || 278,506 || D || 127,936 || 489,180 || 250,175
|-
| E || -60,856 || 36,761 || 151,733 || E || -48,583 || 418,229 || 214,972
|-
| F || -144,707 || -53,750 || 121,860 || F || -0,408 || 291,039 || 229,751
|-
| G || -167,596 || -73,250 || 245,195 || G || 67,656 || 329,852 || 341,905
|-
| H || -83,752 || 17,258 || 275,064 || H || 19,475 || 457,043 || 327,126
|-
|rowspan="8"  | 12 || A || -31,701 || -485,743 || 57,217 ||rowspan="8"  | 62 || A || -41,024 || 66,815 || -130,056
|-
| B || 5,844 || -600,552 || 76,992 || B || -18,336 || 104,006 || -232,024
|-
| C || -20,037 || -629,030 || -39,214 || C || -19,825 || 208,263 || -194,326
|-
| D || -57,585 || -514,217 || -58,985 || D || -42,504 || 171,080 || -92,366
|-
| E || 81,893 || -454,272 || 24,212 || E || 67,502 || 60,479 || -108,230
|-
| F || 119,439 || -569,094 || 43,982 || F || 90,180 || 97,664 || -210,194
|-
| G || 93,561 || -597,560 || -72,223 || G || 88,697 || 201,923 || -172,495
|-
| H || 56,011 || -482,750 || -91,989 || H || 66,008 || 164,740 || -70,536
|-
|rowspan="8"  | 13 || A || -4,189 || -333,380 || 49,056 ||rowspan="8"  | 63 || A || -67,013 || -193,237 || -73,333
|-
| B || -55,985 || -366,730 || 75,963 || B || -1,556 || -96,768 || -147,320
|-
| C || -26,682 || -363,468 || 136,377 || C || -122,706 || -52,028 || -196,167
|-
| D || 25,102 || -330,118 || 109,463 || D || -188,163 || -148,493 || -122,181
|-
| E || -35,473 || -275,111 || 61,071 || E || -77,163 || -105,157 || 32,521
|-
| F || -87,257 || -308,459 || 87,990 || F || -11,703 || -8,693 || -41,464
|-
| G || -57,964 || -305,192 || 148,400 || G || -132,855 || 36,052 || -90,314
|-
| H || -6,168 || -271,852 || 121,488 || H || -198,312 || -60,423 || -16,332
|-
|rowspan="8"  | 14 || A || 88,609 || 12,300 || -33,202 ||rowspan="8"  | 64 || A || -31,542 || 143,668 || -143,998
|-
| B || 91,936 || 10,873 || 26,841 || B || 43,403 || 230,953 || -84,738
|-
| C || 96,089 || -49,117 || 25,201 || C || 135,993 || 141,681 || -70,361
|-
| D || 92,771 || -47,702 || -34,849 || D || 61,041 || 54,400 || -129,617
|-
| E || 148,531 || 16,538 || -36,427 || E || 19,035 || 177,739 || -258,154
|-
| F || 151,860 || 15,117 || 23,627 || F || 93,978 || 265,024 || -198,892
|-
| G || 156,018 || -44,875 || 21,979 || G || 186,571 || 175,750 || -184,503
|-
| H || 152,690 || -43,453 || -38,075 || H || 111,620 || 88,469 || -243,773
|-
|rowspan="8"  | 15 || A || 62,366 || -70,046 || -71,843 ||rowspan="8"  | 65 || A || 47,520 || 175,522 || 2,294
|-
| B || 62,519 || -17,971 || 27,704 || B || 99,131 || 108,969 || 9,227
|-
| C || 62,970 || -117,513 || 79,774 || C || 107,478 || 124,061 || 91,954
|-
| D || 62,813 || -169,587 || -19,771 || D || 55,870 || 190,620 || 85,014
|-
| E || 174,707 || -69,713 || -72,181 || E || -18,864 || 125,683 || 18,085
|-
| F || 174,861 || -17,641 || 27,364 || F || 32,738 || 59,132 || 25,015
|-
| G || 175,311 || -117,185 || 79,437 || G || 41,089 || 74,228 || 107,739
|-
| H || 175,152 || -169,256 || -20,106 || H || -10,523 || 140,780 || 100,805
|-
|rowspan="8"  | 16 || A || -72,703 || -75,016 || 102,975 ||rowspan="8"  | 66 || A || -53,662 || -192,983 || 45,141
|-
| B || -51,746 || -91,605 || -16,242 || B || -17,421 || -81,679 || 21,396
|-
| C || -39,883 || 29,111 || -30,955 || C || -32,369 || -101,732 || -95,398
|-
| D || -60,836 || 45,694 || 88,266 || D || -68,602 || -213,034 || -71,653
|-
| E || 47,073 || -84,070 || 125,289 || E || -166,488 || -154,580 || 52,981
|-
| F || 68,034 || -100,656 || 6,074 || F || -130,242 || -43,279 || 29,233
|-
| G || 79,896 || 20,048 || -8,640 || G || -145,191 || -63,324 || -87,552
|-
| H || 58,947 || 36,635 || 110,584 || H || -181,430 || -174,625 || -63,812
|-
|rowspan="8"  | 17 || A || -49,473 || -187,835 || -225,023 ||rowspan="8"  | 67 || A || 28,574 || -185,482 || -3,599
|-
| B || 11,721 || -96,543 || -173,052 || B || 112,033 || -165,763 || -24,837
|-
| C || 116,730 || -148,165 || -206,000 || C || 126,158 || -249,973 || -47,490
|-
| D || 55,542 || -239,454 || -257,974 || D || 42,708 || -269,694 || -26,245
|-
| E || -52,145 || -126,349 || -329,865 || E || 3,266 || -167,481 || -86,301
|-
| F || 9,042 || -35,066 || -277,895 || F || 86,727 || -147,756 || -107,547
|-
| G || 114,056 || -86,692 || -310,841 || G || 100,858 || -231,974 || -130,204
|-
| H || 52,868 || -177,982 || -362,815 || H || 17,398 || -251,697 || -108,962
|-
|rowspan="8"  | 18 || A || -3,291 || 58,392 || -67,764 ||rowspan="8"  | 68 || A || 29,360 || 117,807 || 150,925
|-
| B || -13,778 || 102,437 || -43,675 || B || 71,436 || 90,590 || 291,160
|-
| C || -5,168 || 128,262 || -87,137 || C || 69,409 || 236,665 || 320,118
|-
| D || 5,312 || 84,214 || -111,235 || D || 27,328 || 263,878 || 179,875
|-
| E || -52,753 || 53,542 || -80,440 || E || -113,483 || 107,712 || 191,829
|-
| F || -63,235 || 97,591 || -56,343 || F || -71,399 || 80,495 || 332,066
|-
| G || -54,637 || 123,415 || -99,809 || G || -73,438 || 226,569 || 361,027
|-
| H || -44,151 || 79,368 || -123,899 || H || -115,522 || 253,776 || 220,780
|-
|rowspan="8"  | 19 || A || -26,093 || -7,702 || -58,331 ||rowspan="8"  | 69 || A || -29,898 || -316,569 || -27,757
|-
| B || -9,516 || 50,270 || -87,212 || B || -3,303 || -231,839 || -96,619
|-
| C || 1,414 || 77,141 || -26,977 || C || -34,166 || -158,019 || -17,718
|-
| D || -15,165 || 19,167 || 1,901 || D || -60,770 || -242,744 || 51,144
|-
| E || 37,754 || -27,356 || -61,147 || E || 74,829 || -316,334 || 12,989
|-
| F || 54,321 || 30,622 || -90,023 || F || 101,434 || -231,605 || -55,870
|-
| G || 65,251 || 57,496 || -29,791 || G || 70,567 || -157,781 || 23,033
|-
| H || 48,679 || -0,478 || -0,919 || H || 43,957 || -242,502 || 91,900
|-
|rowspan="8"  | 20 || A || -19,714 || 149,429 || 242,552 ||rowspan="8"  | 70 || A || -6,324 || 9,528 || -76,129
|-
| B || 1,831 || 90,967 || 295,510 || B || 41,615 || 60,008 || 18,026
|-
| C || -28,051 || 34,114 || 244,886 || C || 111,580 || 123,170 || -51,458
|-
| D || -49,595 || 92,582 || 191,928 || D || 63,629 || 72,686 || -145,608
|-
| E || 53,299 || 143,413 || 206,219 || E || -87,062 || 94,230 || -80,430
|-
| F || 74,838 || 84,941 || 259,177 || F || -39,121 || 144,713 || 13,730
|-
| G || 44,962 || 28,098 || 208,556 || G || 30,844 || 207,863 || -55,761
|-
| H || 23,420 || 86,563 || 155,596 || H || -17,099 || 157,393 || -149,912
|-
|rowspan="8"  | 21 || A || -1,496 || 50,358 || 30,936 ||rowspan="8"  | 71 || A || -17,985 || -78,194 || -28,835
|-
| B || 27,698 || -33,805 || 72,754 || B || -26,841 || -37,289 || -60,085
|-
| C || 21,239 || -79,295 || -14,278 || C || -31,192 || -69,479 || -100,980
|-
| D || -7,960 || 4,866 || -56,097 || D || -22,337 || -110,381 || -69,738
|-
| E || 92,257 || 73,432 || 11,916 || E || -69,271 || -82,530 || -19,975
|-
| F || 121,451 || -10,729 || 53,733 || F || -78,128 || -41,630 || -51,220
|-
| G || 114,997 || -56,216 || -33,290 || G || -82,480 || -73,809 || -92,122
|-
| H || 85,798 || 27,943 || -75,111 || H || -73,620 || -114,708 || -60,875
|-
|rowspan="8"  | 22 || A || 1,996 || 57,819 || 135,802 ||rowspan="8"  | 72 || A || 21,689 || -401,128 || 108,706
|-
| B || -21,399 || -15,302 || 170,653 || B || 5,663 || -475,434 || 192,354
|-
| C || -80,895 || -24,407 || 111,614 || C || 55,901 || -404,719 || 264,808
|-
| D || -57,495 || 48,707 || 76,761 || D || 71,928 || -330,405 || 181,171
|-
| E || 56,972 || 16,835 || 86,737 || E || -78,288 || -353,674 || 131,716
|-
| F || 33,570 || -56,290 || 121,594 || F || -94,312 || -427,992 || 215,354
|-
| G || -25,921 || -65,389 || 62,542 || G || -44,075 || -357,269 || 287,815
|-
| H || -2,525 || 7,732 || 27,688 || H || -28,043 || -282,955 || 204,169
|-
|rowspan="8"  | 23 || A || 1,756 || -73,171 || 74,601 ||rowspan="8"  | 73 || A || 16,964 || 375,085 || 247,244
|-
| B || 2,774 || -40,542 || 124,645 || B || -0,723 || 487,938 || 239,053
|-
| C || 5,957 || -90,542 || 157,181 || C || -103,043 || 468,448 || 191,430
|-
| D || 4,927 || -123,176 || 107,143 || D || -85,357 || 355,597 || 199,631
|-
| E || 61,402 || -71,074 || 72,013 || E || -31,354 || 375,058 || 351,084
|-
| F || 62,425 || -38,444 || 122,050 || F || -49,041 || 487,912 || 342,883
|-
| G || 65,603 || -88,446 || 154,593 || G || -151,362 || 468,421 || 295,263
|-
| H || 64,585 || -121,078 || 104,550 || H || -133,668 || 355,563 || 303,466
|-
|rowspan="8"  | 24 || A || -58,762 || -191,413 || -8,972 ||rowspan="8"  | 74 || A || 7,716 || 107,422 || -66,020
|-
| B || -48,805 || -172,763 || 51,033 || B || -1,935 || 147,772 || -22,712
|-
| C || -106,418 || -194,311 || 67,296 || C || -46,907 || 114,225 || -1,494
|-
| D || -116,380 || -212,962 || 7,278 || D || -37,248 || 73,877 || -44,805
|-
| E || -33,681 || -248,301 || 4,548 || E || 46,209 || 78,358 || -30,361
|-
| F || -23,723 || -229,650 || 64,556 || F || 36,550 || 118,711 || 12,944
|-
| G || -81,343 || -251,196 || 80,810 || G || -8,424 || 85,168 || 34,161
|-
| H || -91,300 || -269,851 || 20,797 || H || 1,234 || 44,823 || -9,156
|-
|rowspan="8"  | 25 || A || -14,816 || 312,924 || -88,682 ||rowspan="8"  | 75 || A || -30,726 || -336,403 || 19,400
|-
| B || -99,513 || 377,487 || -176,113 || B || -8,274 || -403,305 || 91,613
|-
| C || -66,758 || 286,964 || -274,700 || C || -106,287 || -425,369 || 101,646
|-
| D || 17,936 || 222,392 || -187,261 || D || -128,739 || -358,470 || 29,430
|-
| E || -118,459 || 231,536 || -48,393 || E || -21,595 || -408,737 || -50,451
|-
| F || -203,157 || 296,104 || -135,819 || F || 0,859 || -475,635 || 21,764
|-
| G || -170,389 || 205,581 || -234,405 || G || -97,154 || -497,698 || 31,789
|-
| H || -85,696 || 141,016 || -146,977 || H || -119,608 || -430,804 || -40,424
|-
|rowspan="8"  | 26 || A || 48,259 || -41,077 || -191,310 ||rowspan="8"  | 76 || A || -43,660 || 204,215 || -164,580
|-
| B || 15,108 || 18,228 || -225,919 || B || -137,467 || 266,163 || -67,163
|-
| C || 22,176 || -17,052 || -293,135 || C || -80,085 || 400,105 || -97,083
|-
| D || 55,328 || -76,361 || -258,529 || D || 13,731 || 338,149 || -194,495
|-
| E || -20,040 || -73,506 || -181,453 || E || -143,830 || 222,926 || -272,945
|-
| F || -53,188 || -14,207 || -216,064 || F || -237,633 || 284,876 || -175,536
|-
| G || -46,122 || -49,489 || -283,288 || G || -180,246 || 418,814 || -205,454
|-
| H || -12,970 || -108,788 || -248,679 || H || -86,438 || 356,866 || -302,868
|-
|rowspan="8"  | 27 || A || 72,657 || -16,129 || -62,558 ||rowspan="8"  | 77 || A || 64,212 || 120,027 || -257,009
|-
| B || 49,381 || -127,697 || -84,673 || B || 80,180 || 174,246 || -367,841
|-
| C || 62,283 || -152,717 || 27,952 || C || 18,757 || 80,811 || -422,392
|-
| D || 85,557 || -41,144 || 50,077 || D || 2,797 || 26,594 || -311,568
|-
| E || -40,354 || 3,996 || -45,133 || E || -42,800 || 181,741 || -242,230
|-
| F || -63,620 || -107,567 || -67,253 || F || -26,827 || 235,964 || -353,060
|-
| G || -50,715 || -132,594 || 45,378 || G || -88,243 || 142,527 || -407,615
|-
| H || -27,446 || -21,031 || 67,494 || H || -104,216 || 88,314 || -296,791
|-
|rowspan="8"  | 28 || A || 70,548 || -98,767 || -97,987 ||rowspan="8"  | 78 || A || 15,590 || 20,752 || 87,279
|-
| B || 20,523 || -114,915 || -70,311 || B || 10,979 || 72,163 || 141,186
|-
| C || 1,148 || -140,518 || -120,289 || C || -63,119 || 63,468 || 143,145
|-
| D || 51,172 || -124,376 || -147,964 || D || -58,513 || 12,056 || 89,234
|-
| E || 96,057 || -149,887 || -81,691 || E || 23,218 || -32,646 || 138,865
|-
| F || 46,030 || -166,024 || -54,022 || F || 18,607 || 18,766 || 192,767
|-
| G || 26,651 || -191,631 || -103,995 || G || -55,487 || 10,075 || 194,725
|-
| H || 76,676 || -175,485 || -131,667 || H || -50,881 || -41,335 || 140,821
|-
|rowspan="8"  | 29 || A || 19,154 || 157,384 || 153,406 ||rowspan="8"  | 79 || A || 7,879 || -100,695 || 129,989
|-
| B || 78,894 || 47,638 || 170,584 || B || -72,908 || -136,829 || 155,646
|-
| C || -30,186 || -13,998 || 156,108 || C || -32,830 || -173,666 || 229,988
|-
| D || -89,920 || 95,745 || 138,927 || D || 47,968 || -137,537 || 204,328
|-
| E || 40,151 || 149,384 || 29,293 || E || -11,004 || -24,351 || 178,007
|-
| F || 99,887 || 39,643 || 46,473 || F || -91,798 || -60,488 || 203,669
|-
| G || -9,189 || -22,001 || 31,991 || G || -51,717 || -97,333 || 278,004
|-
| H || -68,929 || 87,746 || 14,814 || H || 29,074 || -61,199 || 252,351
|-
|rowspan="8"  | 30 || A || -0,832 || 148,123 || -55,062 ||rowspan="8"  | 80 || A || -82,391 || 19,624 || 195,867
|-
| B || -3,177 || 83,383 || -82,664 || B || -126,755 || 13,600 || 235,381
|-
| C || 4,481 || 55,693 || -18,369 || C || -155,874 || -21,964 || 197,277
|-
| D || 6,819 || 120,437 || 9,226 || D || -111,512 || -15,951 || 157,758
|-
| E || -70,802 || 147,260 || -47,104 || E || -55,013 || -27,965 || 219,355
|-
| F || -73,142 || 82,515 || -74,709 || F || -99,374 || -33,982 || 258,868
|-
| G || -65,481 || 54,832 || -10,407 || G || -128,500 || -69,550 || 220,760
|-
| H || -63,142 || 119,582 || 17,182 || H || -84,143 || -63,530 || 181,251
|-
|rowspan="8"  | 31 || A || -62,187 || -290,838 || 74,947 ||rowspan="8"  | 81 || A || -10,413 || 205,562 || -99,978
|-
| B || -148,419 || -277,181 || 52,556 || B || -20,573 || 98,681 || -113,175
|-
| C || -122,249 || -237,171 || -23,850 || C || 75,377 || 83,682 || -65,540
|-
| D || -36,007 || -250,826 || -1,461 || D || 85,534 || 190,557 || -52,338
|-
| E || -63,819 || -370,443 || 32,697 || E || -59,316 || 198,329 || -3,776
|-
| F || -150,059 || -356,788 || 10,309 || F || -69,474 || 91,448 || -16,972
|-
| G || -123,886 || -316,776 || -66,108 || G || 26,476 || 76,444 || 30,672
|-
| H || -37,641 || -330,428 || -43,713 || H || 36,626 || 183,327 || 43,869
|-
|rowspan="8"  | 32 || A || 21,688 || 24,157 || 173,270 ||rowspan="8"  | 82 || A || -45,436 || 279,685 || -163,035
|-
| B || 35,864 || 10,746 || 240,885 || B || -84,684 || 320,165 || -149,056
|-
| C || 51,742 || 78,562 || 251,010 || C || -51,030 || 361,055 || -172,936
|-
| D || 37,573 || 91,973 || 183,389 || D || -11,771 || 320,577 || -186,922
|-
| E || -45,394 || 37,383 || 189,943 || E || -71,915 || 271,648 || -214,121
|-
| F || -31,224 || 23,972 || 257,570 || F || -111,174 || 312,136 || -200,137
|-
| G || -15,339 || 91,783 || 267,694 || G || -77,514 || 353,016 || -224,014
|-
| H || -29,509 || 105,189 || 200,071 || H || -38,255 || 312,536 || -237,997
|-
|rowspan="8"  | 33 || A || 44,183 || -183,082 || -7,037 ||rowspan="8"  | 83 || A || -25,582 || -156,980 || 151,171
|-
| B || 38,428 || -219,288 || 83,764 || B || 36,128 || -53,249 || 236,515
|-
| C || 29,366 || -128,530 || 119,376 || C || 35,072 || -146,795 || 350,972
|-
| D || 35,115 || -92,329 || 28,585 || D || -26,636 || -250,524 || 265,628
|-
| E || -53,143 || -189,398 || -15,719 || E || 108,740 || -205,366 || 112,860
|-
| F || -58,891 || -225,601 || 75,075 || F || 170,451 || -101,635 || 198,212
|-
| G || -67,959 || -134,848 || 110,694 || G || 169,393 || -195,183 || 312,666
|-
| H || -62,213 || -98,642 || 19,902 || H || 107,685 || -298,907 || 227,318
|-
|rowspan="8"  | 34 || A || 39,453 || 117,439 || -16,522 ||rowspan="8"  | 84 || A || -46,635 || 166,791 || 13,631
|-
| B || 33,907 || 39,298 || 86,838 || B || -71,644 || 60,654 || 67,095
|-
| C || 42,312 || -64,157 || 9,066 || C || 46,747 || 43,133 || 87,695
|-
| D || 47,865 || 13,995 || -94,293 || D || 71,758 || 149,260 || 34,235
|-
| E || 168,764 || 120,807 || -7,024 || E || -56,922 || 223,156 || 120,711
|-
| F || 163,206 || 42,664 || 96,330 || F || -81,942 || 117,013 || 174,178
|-
| G || 171,612 || -60,794 || 18,564 || G || 36,453 || 99,495 || 194,776
|-
| H || 177,166 || 17,359 || -84,797 || H || 61,469 || 205,633 || 141,314
|-
|rowspan="8"  | 35 || A || 41,180 || 17,938 || -167,878 ||rowspan="8"  | 85 || A || -51,182 || -51,542 || 18,228
|-
| B || 37,801 || 67,734 || -245,735 || B || -118,414 || -85,354 || 130,674
|-
| C || 106,456 || 121,247 || -214,492 || C || -103,162 || -216,350 || 100,410
|-
| D || 109,831 || 71,451 || -136,634 || D || -35,929 || -182,537 || -12,038
|-
| E || 103,051 || -38,713 || -206,801 || E || 65,247 || -53,904 || 87,134
|-
| F || 99,674 || 11,090 || -284,655 || F || -1,994 || -87,717 || 199,581
|-
| G || 168,327 || 64,599 || -253,411 || G || 13,265 || -218,713 || 169,308
|-
| H || 171,707 || 14,805 || -175,566 || H || 80,501 || -184,899 || 56,863
|-
|rowspan="8"  | 36 || A || 18,551 || -42,342 || 65,222 ||rowspan="8"  | 86 || A || -10,928 || 153,829 || -86,720
|-
| B || 22,249 || -14,504 || -47,899 || B || -17,656 || 30,629 || -10,708
|-
| C || 81,529 || -112,378 || -70,056 || C || 125,085 || 37,646 || 13,298
|-
| D || 77,831 || -140,224 || 43,063 || D || 131,820 || 160,845 || -62,715
|-
| E || -81,736 || -99,169 || 47,947 || E || -35,020 || 229,818 || 34,307
|-
| F || -78,037 || -71,326 || -65,177 || F || -41,751 || 106,619 || 110,312
|-
| G || -18,748 || -169,207 || -87,320 || G || 100,994 || 113,639 || 134,323
|-
| H || -22,459 || -197,050 || 25,792 || H || 107,727 || 236,836 || 58,305
|-
|rowspan="8"  | 37 || A || 21,819 || 222,345 || 35,209 ||rowspan="8"  | 87 || A || 29,723 || -86,649 || -35,592
|-
| B || 64,934 || 299,487 || 73,760 || B || 17,622 || -41,898 || 100,238
|-
| C || 62,447 || 257,521 || 160,516 || C || -74,921 || 59,585 || 58,562
|-
| D || 19,334 || 180,377 || 121,968 || D || -62,830 || 14,829 || -77,269
|-
| E || 108,020 || 182,552 || 18,432 || E || -79,309 || -177,746 || -15,287
|-
| F || 151,129 || 259,692 || 56,978 || F || -91,403 || -132,992 || 120,547
|-
| G || 148,641 || 217,725 || 143,745 || G || -183,957 || -31,512 || 78,877
|-
| H || 105,528 || 140,591 || 105,197 || H || -171,861 || -76,268 || -56,958
|-
|rowspan="8"  | 38 || A || -16,265 || -55,427 || -71,646 ||rowspan="8"  | 88 || A || 50,033 || 278,002 || -139,931
|-
| B || -25,671 || -131,166 || -43,586 || B || 50,039 || 216,948 || -236,550
|-
| C || -106,270 || -124,213 || -51,827 || C || 50,034 || 313,571 || -297,607
|-
| D || -96,861 || -48,463 || -79,890 || D || 50,022 || 374,625 || -200,989
|-
| E || -10,987 || -84,193 || -147,527 || E || 164,336 || 278,018 || -139,927
|-
| F || -20,397 || -159,943 || -119,457 || F || 164,339 || 216,961 || -236,546
|-
| G || -100,991 || -152,981 || -127,706 || G || 164,335 || 313,582 || -297,603
|-
| H || -91,590 || -77,240 || -155,770 || H || 164,319 || 374,643 || -200,984
|-
|rowspan="8"  | 39 || A || -53,192 || 129,839 || 73,707 ||rowspan="8"  | 89 || A || 3,685 || 475,694 || -68,483
|-
| B || -23,735 || 106,391 || 30,249 || B || 50,250 || 450,269 || -84,841
|-
| C || 20,100 || 142,101 || 40,698 || C || 80,158 || 484,614 || -53,084
|-
| D || -9,356 || 165,552 || 84,151 || D || 33,588 || 510,041 || -36,721
|-
| E || -30,466 || 91,359 || 109,873 || E || -0,736 || 440,246 || -25,970
|-
| F || -1,001 || 67,914 || 66,420 || F || 45,829 || 414,815 || -42,337
|-
| G || 42,831 || 103,618 || 76,868 || G || 75,737 || 449,169 || -10,583
|-
| H || 13,370 || 127,067 || 120,318 || H || 29,166 || 474,598 || 5,778
|-
|rowspan="8"  | 40 || A || -47,254 || -39,090 || -47,292 ||rowspan="8"  | 90 || A || 40,806 || -248,916 || 46,046
|-
| B || -44,379 || -81,587 || -14,591 || B || 64,657 || -294,153 || 16,219
|-
| C || -95,649 || -74,176 || -0,454 || C || 71,528 || -259,296 || -31,137
|-
| D || -98,528 || -31,681 || -33,159 || D || 47,679 || -214,070 || -1,306
|-
| E || -62,953 || -71,075 || -87,471 || E || 94,554 || -233,299 || 65,333
|-
| F || -60,073 || -113,573 || -54,767 || F || 118,401 || -278,535 || 35,513
|-
| G || -111,345 || -106,159 || -40,627 || G || 125,272 || -243,676 || -11,850
|-
| H || -114,228 || -63,675 || -73,338 || H || 101,419 || -198,451 || 17,980
|-
|rowspan="8"  | 41 || A || -74,761 || 30,164 || 183,471 ||rowspan="8"  | 91 || A || -38,767 || 105,181 || -70,416
|-
| B || -70,723 || 95,811 || 316,035 || B || -15,246 || 102,931 || -130,225
|-
| C || -70,850 || 228,428 || 250,361 || C || -30,973 || 40,700 || -134,058
|-
| D || -74,898 || 162,771 || 117,799 || D || -54,488 || 42,950 || -74,264
|-
| E || -222,685 || 31,843 || 187,150 || E || -96,502 || 121,202 || -93,723
|-
| F || -218,651 || 97,487 || 319,713 || F || -72,998 || 118,953 || -153,533
|-
| G || -218,781 || 230,096 || 254,042 || G || -88,711 || 56,727 || -157,367
|-
| H || -222,817 || 164,453 || 121,479 || H || -112,223 || 58,978 || -97,562
|-
|rowspan="8"  | 42 || A || -41,878 || -226,942 || -3,086 ||rowspan="8"  | 92 || A || -6,294 || 255,663 || -231,651
|-
| B || -38,826 || -304,573 || -25,689 || B || 38,261 || 154,536 || -249,454
|-
| C || -107,693 || -295,222 || -67,118 || C || 116,077 || 175,124 || -171,665
|-
| D || -110,749 || -217,588 || -44,521 || D || 71,528 || 276,260 || -153,860
|-
| E || 0,483 || -206,134 || -68,822 || E || -73,289 || 212,332 || -153,138
|-
| F || 3,540 || -283,767 || -91,425 || F || -28,737 || 111,199 || -170,950
|-
| G || -65,337 || -274,417 || -132,855 || G || 49,081 || 131,789 || -93,168
|-
| H || -68,390 || -196,782 || -110,245 || H || 4,523 || 232,920 || -75,365
|-
|rowspan="8"  | 43 || A || -7,819 || -55,712 || 12,582 ||rowspan="8"  | 93 || A || -75,723 || 244,498 || -135,815
|-
| B || -60,330 || -80,620 || 3,655 || B || -82,559 || 229,007 || -46,701
|-
| C || -74,640 || -37,204 || -33,324 || C || -85,390 || 139,720 || -62,434
|-
| D || -22,133 || -12,297 || -24,402 || D || -78,555 || 155,200 || -151,551
|-
| E || 14,446 || -86,572 || -32,251 || E || 14,681 || 240,532 || -129,582
|-
| F || -38,074 || -111,488 || -41,174 || F || 7,850 || 225,046 || -40,465
|-
| G || -52,388 || -68,062 || -78,169 || G || 5,021 || 135,755 || -56,195
|-
| H || 0,130 || -43,150 || -69,240 || H || 11,847 || 151,234 || -145,314
|-
|rowspan="8"  | 44 || A || 12,124 || 29,493 || -61,344 ||rowspan="8"  | 94 || A || 26,421 || -45,661 || 265,356
|-
| B || 6,872 || 5,332 || 8,602 || B || 2,414 || 39,092 || 221,112
|-
| C || 80,039 || 14,134 || 17,146 || C || 59,763 || 15,365 || 144,530
|-
| D || 85,293 || 38,298 || -52,807 || D || 83,771 || -69,402 || 188,771
|-
| E || 1,041 || 99,080 || -38,137 || E || -50,083 || -90,051 || 221,820
|-
| F || -4,216 || 74,920 || 31,812 || F || -74,083 || -5,292 || 177,589
|-
| G || 68,953 || 83,727 || 40,347 || G || -16,738 || -29,019 || 100,998
|-
| H || 74,216 || 107,887 || -29,595 || H || 7,266 || -113,787 || 145,242
|-
|rowspan="8"  | 45 || A || -14,290 || 305,473 || -38,546 ||rowspan="8"  | 95 || A || 59,456 || 169,514 || -55,338
|-
| B || -33,289 || 244,676 || -26,483 || B || 72,883 || 88,087 || -110,613
|-
| C || -88,751 || 266,977 || -1,417 || C || 27,474 || 132,458 || -187,016
|-
| D || -69,763 || 327,772 || -13,482 || D || 14,051 || 213,898 || -131,738
|-
| E || -41,951 || 302,495 || -97,100 || E || 146,782 || 205,112 || -86,567
|-
| F || -60,942 || 241,699 || -85,042 || F || 160,205 || 123,674 || -141,848
|-
| G || -116,416 || 264,001 || -59,971 || G || 114,796 || 168,052 || -218,246
|-
| H || -97,415 || 324,801 || -72,038 || H || 101,372 || 249,496 || -162,967
|-
|rowspan="8"  | 46 || A || 42,815 || 56,809 || 65,624 ||rowspan="8"  | 96 || A || 87,799 || 9,898 || 157,693
|-
| B || 59,197 || 180,782 || 131,551 || B || 168,746 || 47,893 || 60,760
|-
| C || 42,456 || 248,403 || 8,546 || C || 118,896 || 169,835 || 66,937
|-
| D || 26,080 || 124,431 || -57,384 || D || 37,963 || 131,834 || 163,874
|-
| E || -96,594 || 63,254 || 88,136 || E || 179,213 || 42,745 || 246,898
|-
| F || -80,215 || 187,223 || 154,067 || F || 260,161 || 80,738 || 149,963
|-
| G || -96,957 || 254,856 || 31,063 || G || 210,308 || 202,685 || 156,145
|-
| H || -113,336 || 130,881 || -34,871 || H || 129,369 || 164,682 || 253,079
|-
|rowspan="8"  | 47 || A || 16,659 || 216,633 || -245,483 ||rowspan="8"  | 97 || A || 15,320 || 50,805 || 120,047
|-
| B || 69,269 || 215,380 || -216,984 || B || 16,360 || 122,350 || 229,948
|-
| C || 78,347 || 272,789 || -231,217 || C || 13,706 || 12,471 || 301,494
|-
| D || 25,740 || 274,043 || -259,713 || D || 12,665 || -59,065 || 191,589
|-
| E || -10,379 || 233,470 || -194,822 || E || 146,436 || 48,013 || 120,621
|-
| F || 42,217 || 232,216 || -166,326 || F || 147,467 || 119,556 || 230,527
|-
| G || 51,294 || 289,627 || -180,557 || G || 144,812 || 9,675 || 302,069
|-
| H || -1,311 || 290,873 || -209,051 || H || 143,779 || -61,856 || 192,165
|-
|rowspan="8"  | 48 || A || -85,888 || 301,572 || -21,383 ||rowspan="8"  | 98 || A || 24,696 || 289,570 || -67,006
|-
| B || -104,608 || 351,179 || -120,655 || B || -33,685 || 352,622 || -151,554
|-
| C || -200,846 || 394,044 || -81,085 || C || 67,752 || 359,704 || -216,316
|-
| D || -182,129 || 344,439 || 18,178 || D || 126,129 || 296,647 || -131,759
|-
| E || -30,635 || 393,030 || 13,903 || E || -4,212 || 187,067 || -123,498
|-
| F || -49,346 || 442,638 || -85,362 || F || -62,592 || 250,121 || -208,043
|-
| G || -145,584 || 485,510 || -45,798 || G || 38,848 || 257,198 || -272,797
|-
| H || -126,869 || 435,905 || 53,478 || H || 97,217 || 194,146 || -188,253
|-
|rowspan="8"  | 49 || A || 29,200 || -214,223 || 271,379 ||rowspan="8"  | 99 || A || -21,738 || -63,055 || 1,821
|-
| B || -20,130 || -197,446 || 186,408 || B || 54,881 || 4,990 || -19,130
|-
| C || 22,774 || -107,751 || 179,213 || C || 24,411 || 8,511 || -119,116
|-
| D || 72,106 || -124,541 || 264,192 || D || -52,202 || -59,530 || -98,175
|-
| E || 104,453 || -254,365 || 219,770 || E || -86,078 || 16,289 || 24,217
|-
| F || 55,119 || -237,577 || 134,791 || F || -9,475 || 84,337 || 3,277
|-
| G || 98,023 || -147,884 || 127,604 || G || -39,942 || 87,862 || -96,718
|-
| H || 147,360 || -164,667 || 212,581 || H || -116,549 || 19,814 || -75,767
|-
|rowspan="8"  | 50 || A || -46,150 || -37,165 || -63,817 ||rowspan="8" | 100 || A || -13,094 || -292,595 || -36,143
|-
| B || -113,887 || -30,922 || -190,379 || B || 45,011 || -174,946 || -68,398
|-
| C || -8,846 || 52,106 || -242,495 || C || 152,991 || -207,928 || 5,840
|-
| D || 58,887 || 45,878 || -115,939 || D || 94,883 || -325,573 || 38,088
|-
| E || 24,724 || -154,257 || -107,523 || E || 43,665 || -350,296 || -144,34
|-
| F || -43,010 || -148,016 || -234,076 || F || 101,779 || -232,648 || -176,594
|-
| G || 62,026 || -64,979 || -286,205 || G || 209,752 || -265,623 || -102,354
|-
| H || 129,768 || -71,222 || -159,642 || H || 151,649 || -383,269 || -70,100
|-
|rowspan="8" | 101 || A || -23,015 || -35,636 || -89,787 ||rowspan="8" | 102 || A || 74,487 || 131,213 || -147,528
|-
| B || 67,309 || -11,943 || -172,643 || B || 43,481 || 203,541 || -267,277
|-
| C || 129,377 || -113,094 || -133,901 || C || 28,144 || 79,865 || -338,002
|-
| D || 39,057 || -136,789 || -51,048 || D || 59,158 || 7,545 || -218,252
|-
| E || -82,804 || -104,861 || -174,752 || E || -64,564 || 128,730 || -113,025
|-
| F || 7,520 || -81,177 || -257,608 || F || -95,576 || 201,049 || -232,773
|-
| G || 69,593 || -182,324 || -218,864 || G || -110,905 || 77,375 || -303,500
|-
| H || -20,727 || -206,010 || -136,009 || H || -79,891 || 5,0560 || -183,744
|}
[[Kategorie:Výuka]]
[[Kategorie:Výuka]]

Verze z 9. 2. 2010, 21:30

Zadání úlohy

Pro zadanou zeměpisnou šířku vygenerujte rekurentními vzorci normované Legendreovy přidružené funkce a jejich 1. a 2. derivace, kde zvolte . Úspešnost řešení otestujte dosazením do Legendreovy diferenciální rovnice a spočtete základní charakteristiky přesnosti (střední hodnotu odchylky a RMS). Dále pro zadaný stupeň a řád určete průběh Legendreovy přidružené funkce s měnící se zeměpisnou šířkou a totéž proveďte s Legendreovým polynomem příslušného stupně. Výsledky graficky zobrazte a diskutujte v závěru (průběh funkce, počet nulových bodů, proč je výhodné použít rekurentní vztahy oproti přímé definice ...).

Numerické zadání úlohy

číslo zadání n,m číslo zadání [m] n,m
1 0,0 0,0 23 0,0 0,0