The region examined here is the Euclidean ArrowHead. We define an arrowhead to be a Euclidean quadrilateral with one concave angle that is formed by joining two triangles with obtuse angles that have a pair of sides of equal length that are adjacent to the obtuse angle so that the concave angle of the resulting quadrilateral has a measure equal to the sum of the measures of the obtuse angles. The angle opposite the concave angle is the sum of the two non-obtuse angles adjacent to the sides that were joined together, and the remaining two angles are the remaining angles in each of the triangles. The sides of the arrowhead are then the sides of the triangles excluding the side they were joined on. Here we specify the two arrowheads we computed by their vertices. 1: [(1,0)(-.5,.5)(0,0)(-1,-1)] and 2: [(1,0)(-1,2)(0,0)(-3,-1)]. This was an important example for us because it allowed us to test our hypothesis on non-convex non-regular geometric regions of R^2.
Here are the counting graphs for the eigenvalue
spectrum, using the predicted values. The first approximately 120 eigenvalues
values were used here. The equation for the predicted counting function for the first arrowhead
is N'(t) = 3/16pi*t - (sqrt(13)/2+3sqrt(2)/2+sqrt(5))/4pi*sqrt(t) + c1 and for the second arrowhead
it is N'(t) = 3/8pi*t - (sqrt(5)+sqrt(10)+sqrt(17)+2sqrt(2))/4pi*sqrt(t) + c2. The constants c1 and c2
are equal to the standard constant for Euclidean polygons: Sum(theta/pi-pi/theta) over all internal angles.
Here are the difference graphs for the eigenvalue
spectrum, using the predicted values. The first approximately 120 values were used here.
Below is the Eigenvalue spectrum for the Euclidean ArrowHeads
under Dirichlet Boundary conditions. The calculated and predicted eigenvalue spectrum is presented here.
| ArrowHead 1 | ||||||||
|---|---|---|---|---|---|---|---|---|
| Refinement | ||||||||
| # | Initial | 1 | 2 | 3 | 4 | 5 | Predicted | |
| 1 | 55.96180846 | 50.83497449 | 49.43157514 | 49.02043552 | 48.8926266 | 48.85066001 | 48.83662491 | |
| 2 | 106.0452612 | 92.43863545 | 89.08206562 | 88.24510843 | 88.03419598 | 87.9807377 | 87.96285937 | |
| 3 | 144.0045197 | 122.0506502 | 116.5418327 | 115.0763878 | 114.6643481 | 114.5423192 | 114.5015084 | |
| 4 | 165.8944593 | 140.32979 | 133.8364041 | 132.2160236 | 131.8061793 | 131.7015468 | 131.6665539 | |
| 5 | 216.3849951 | 178.9826441 | 169.425115 | 167.0346615 | 166.427686 | 166.2716091 | 166.2194114 | |
| 6 | 259.0824869 | 208.3691861 | 195.3275532 | 192.0669528 | 191.2352822 | 191.0196294 | 190.9475075 | |
| 7 | 281.9951046 | 226.4080015 | 211.5255366 | 207.7824357 | 206.8273316 | 206.5806221 | 206.4981138 | |
| 8 | 332.9892709 | 261.2118632 | 241.7034924 | 236.7877168 | 235.5276876 | 235.1996553 | 235.0899497 | |
| 9 | 393.0960695 | 294.0697791 | 264.8679137 | 257.8508015 | 256.0897127 | 255.6376981 | 255.4865285 | |
| 10 | 419.1344944 | 309.4346582 | 284.2082123 | 278.0222003 | 276.4834407 | 276.0989647 | 275.9703824 | |
| 11 | 460.2630093 | 339.8595161 | 308.9543154 | 301.4076033 | 299.514578 | 299.0331019 | 298.8720793 | |
| 12 | 523.7329759 | 384.1947675 | 336.367912 | 324.5074397 | 321.5490175 | 320.7982035 | 320.5471048 | |
| 13 | 564.6268218 | 394.5099812 | 351.4013224 | 341.4698258 | 339.0086312 | 338.3896866 | 338.1826898 | |
| 14 | 595.6801447 | 431.2579333 | 384.5493609 | 373.1128923 | 370.2642112 | 369.548955 | 369.3097482 | |
| 15 | 684.9166208 | 473.394784 | 413.8392435 | 399.0164393 | 395.3014861 | 394.3656645 | 394.0526928 | |
| 16 | 725.6998172 | 486.0763802 | 422.4654152 | 407.3117271 | 403.5293161 | 402.5559513 | 402.2304239 | |
| 17 | 806.7156631 | 509.217478 | 439.3833379 | 422.1017906 | 417.7124548 | 416.564684 | 416.180829 | |
| 18 | 917.7266689 | 538.9385833 | 469.2487608 | 452.4772482 | 448.3273948 | 447.2872462 | 446.939384 | |
| 19 | 0 | 578.6949962 | 497.7354006 | 477.8682028 | 472.9466512 | 471.7172082 | 471.3060392 | |
| 20 | 0 | 609.8254037 | 516.5410308 | 494.2127766 | 488.7291895 | 487.3641125 | 486.9075827 | |
| 21 | 0 | 634.809749 | 545.9347396 | 523.9315552 | 518.4836596 | 517.1246975 | 516.6702128 | |
| 22 | 0 | 671.033028 | 569.4035872 | 543.6274043 | 537.0933434 | 535.4182888 | 534.8580917 | |
| 23 | 0 | 685.1716047 | 579.7160966 | 552.9156704 | 546.2513413 | 544.5840692 | 544.0264748 | |
| 24 | 0 | 731.3594252 | 613.1953286 | 584.355826 | 577.2361435 | 575.4571755 | 574.862226 | |
| 25 | 0 | 752.9878177 | 626.4492549 | 595.0165694 | 587.1737129 | 585.1905756 | 584.5273447 | |
| 26 | 0 | 782.4201473 | 646.4936766 | 613.05913 | 604.7933604 | 602.7233644 | 602.0310849 | |
| 27 | 0 | 820.2333282 | 674.8949186 | 637.6881388 | 628.4353948 | 626.1253787 | 625.3528281 | |
| 28 | 0 | 843.2491717 | 697.7160474 | 662.2610348 | 653.5493228 | 651.379506 | 650.6538429 | |
| 29 | 0 | 876.7243622 | 718.6171546 | 679.7524361 | 670.2006714 | 667.8234869 | 667.0284728 | |
| 30 | 0 | 932.3635954 | 755.0357556 | 708.5417467 | 696.7101816 | 693.7196823 | 692.7195542 | |
| 31 | 0 | 952.2220545 | 777.8354098 | 733.1788306 | 721.6959578 | 718.7607667 | 717.7791356 | |
| 32 | 0 | 985.8609953 | 789.8804177 | 738.5815769 | 726.6979361 | 723.7955913 | 722.8249451 | |
| 33 | 0 | 1008.416415 | 805.130026 | 756.8499455 | 745.1828924 | 742.284544 | 741.3152344 | |
| 34 | 0 | 1038.404628 | 827.9963849 | 777.0019603 | 764.4238753 | 761.2699856 | 760.2152139 | |
| 35 | 0 | 1076.523456 | 865.5832771 | 811.8082908 | 798.4410196 | 795.086422 | 793.9645265 | |
| 36 | 0 | 1111.250518 | 891.7536426 | 831.3202218 | 815.8968297 | 812.0056796 | 810.7043421 | |
| 37 | 0 | 1164.564884 | 919.3704249 | 852.2641908 | 836.4234044 | 832.5228531 | 831.2183716 | |
| 38 | 0 | 1188.377442 | 929.0897029 | 863.0433316 | 846.5043199 | 842.2773362 | 840.8636842 | |
| 39 | 0 | 1191.529113 | 951.5964564 | 887.2414063 | 871.2936438 | 867.3041138 | 865.9698747 | |
| 40 | 0 | 1235.419037 | 975.135884 | 904.8916848 | 887.7250252 | 883.4350572 | 882.000341 | |
| 41 | 0 | 1262.190693 | 989.8865237 | 921.3136776 | 904.5347473 | 900.3600432 | 898.9638753 | |
| 42 | 0 | 1303.548514 | 1011.139811 | 938.41953 | 920.4612942 | 915.9783471 | 914.4790919 | |
| 43 | 0 | 1347.918955 | 1057.021921 | 974.4742264 | 953.5656537 | 948.3570667 | 946.6151321 | |
| 44 | 0 | 1377.1793 | 1079.124391 | 993.549427 | 973.5276137 | 968.5782201 | 966.9229689 | |
| 45 | 0 | 1413.739969 | 1093.090314 | 1012.353009 | 992.4554775 | 987.494096 | 985.8348356 | |
| 46 | 0 | 1449.465258 | 1121.064781 | 1027.964342 | 1004.301026 | 998.3286269 | 996.3312467 | |
| 47 | 0 | 1468.524515 | 1144.394792 | 1052.111849 | 1028.681438 | 1022.775193 | 1020.799937 | |
| 48 | 0 | 1511.499198 | 1164.463899 | 1063.527257 | 1039.689709 | 1033.832736 | 1031.873958 | |
| 49 | 0 | 1533.856321 | 1193.747127 | 1096.674827 | 1072.236714 | 1066.102089 | 1064.050455 | |
| 50 | 0 | 1571.020081 | 1210.645579 | 1107.249368 | 1082.198144 | 1076.010773 | 1073.941499 | |
| 51 | 0 | 1591.071552 | 1234.771566 | 1131.688931 | 1105.833979 | 1099.201867 | 1096.983855 | |
| 52 | 0 | 1667.876899 | 1256.61019 | 1140.722415 | 1113.154987 | 1106.392714 | 1104.131173 | |
| 53 | 0 | 1683.02652 | 1280.717255 | 1167.904372 | 1139.379913 | 1132.189194 | 1129.784365 | |
| 54 | 0 | 1718.41099 | 1316.01888 | 1195.241414 | 1164.967862 | 1157.360531 | 1154.816372 | |
| 55 | 0 | 1771.106983 | 1351.614166 | 1209.184903 | 1175.615411 | 1167.465534 | 1164.739928 | |
| 56 | 0 | 1798.399504 | 1360.992047 | 1233.230009 | 1202.209191 | 1194.524826 | 1191.954904 | |
| 57 | 0 | 1808.900043 | 1379.856338 | 1253.286022 | 1222.017307 | 1214.182773 | 1211.562629 | |
| 58 | 0 | 1862.668963 | 1410.75109 | 1272.45847 | 1238.644561 | 1230.266669 | 1227.464807 | |
| 59 | 0 | 1872.15484 | 1434.152608 | 1296.986793 | 1261.824677 | 1253.006701 | 1250.057659 | |
| 60 | 0 | 1888.002071 | 1459.403966 | 1307.109261 | 1272.326813 | 1263.839038 | 1261.000428 | |
| 61 | 0 | 1948.785454 | 1489.546168 | 1341.600489 | 1305.52086 | 1296.46167 | 1293.431959 | |
| 62 | 0 | 1976.727831 | 1518.435241 | 1353.237494 | 1313.516918 | 1303.797786 | 1300.547367 | |
| 63 | 0 | 2013.02527 | 1522.092282 | 1368.227677 | 1329.33963 | 1319.599337 | 1316.341841 | |
| 64 | 0 | 2092.007153 | 1548.864288 | 1386.441582 | 1348.221253 | 1338.731979 | 1335.558432 | |
| 65 | 0 | 2121.285323 | 1584.5539 | 1421.173858 | 1380.892971 | 1370.840291 | 1367.478321 | |
| 66 | 0 | 2150.768183 | 1607.005559 | 1433.906992 | 1390.669585 | 1379.825054 | 1376.198261 | |
| 67 | 0 | 2192.633109 | 1633.928081 | 1457.253487 | 1413.873472 | 1403.127106 | 1399.533143 | |
| 68 | 0 | 2211.551531 | 1667.605913 | 1475.047316 | 1429.785327 | 1418.699704 | 1414.992282 | |
| 69 | 0 | 2258.340066 | 1692.551707 | 1499.763787 | 1451.004082 | 1438.947854 | 1434.915828 | |
| 70 | 0 | 2304.72489 | 1707.067816 | 1515.56676 | 1467.276375 | 1454.560511 | 1450.307879 | |
| 71 | 0 | 2326.777304 | 1727.848247 | 1520.519445 | 1473.237943 | 1462.164664 | 1458.46137 | |
| 72 | 0 | 2364.602703 | 1754.065072 | 1559.525754 | 1511.956298 | 1500.143132 | 1496.192393 | |
| 73 | 0 | 2412.752833 | 1780.53852 | 1580.386946 | 1531.215635 | 1518.949654 | 1514.847478 | |
| 74 | 0 | 2459.513404 | 1816.639484 | 1607.547329 | 1556.129259 | 1543.285565 | 1538.990181 | |
| 75 | 0 | 2471.267783 | 1835.772409 | 1619.234235 | 1565.908146 | 1551.82464 | 1547.11462 | |
| 76 | 0 | 0 | 1860.102029 | 1623.637461 | 1566.550118 | 1553.249832 | 1548.801748 | |
| 77 | 0 | 0 | 1888.358395 | 1656.077004 | 1598.898773 | 1584.739174 | 1580.003706 | |
| 78 | 0 | 0 | 1913.902198 | 1676.35422 | 1618.724292 | 1604.308401 | 1599.48722 | |
| 79 | 0 | 0 | 1940.561736 | 1703.47388 | 1643.967826 | 1628.855876 | 1623.801909 | |
| 80 | 0 | 0 | 1965.760865 | 1718.501632 | 1657.590832 | 1642.69028 | 1637.707012 | |
| 81 | 0 | 0 | 1985.255032 | 1741.213585 | 1679.719771 | 1664.526646 | 1659.445531 | |
| 82 | 0 | 0 | 2014.810899 | 1753.769767 | 1694.165884 | 1679.627937 | 1674.765936 | |
| 83 | 0 | 0 | 2061.525279 | 1792.770953 | 1724.148724 | 1707.170248 | 1701.492049 | |
| 84 | 0 | 0 | 2073.450417 | 1802.837873 | 1738.511658 | 1722.45054 | 1717.079137 | |
| 85 | 0 | 0 | 2106.079234 | 1829.789724 | 1758.304794 | 1740.649525 | 1734.744982 | |
| 86 | 0 | 0 | 2132.956792 | 1846.972734 | 1781.874593 | 1765.202881 | 1759.627274 | |
| 87 | 0 | 0 | 2140.07903 | 1853.10729 | 1782.876857 | 1765.934447 | 1760.268309 | |
| 88 | 0 | 0 | 2160.455292 | 1883.203071 | 1813.872153 | 1795.834557 | 1789.80215 | |
| 89 | 0 | 0 | 2213.52296 | 1904.144642 | 1826.19765 | 1807.329953 | 1801.019932 | |
| 90 | 0 | 0 | 2231.127646 | 1931.813161 | 1855.699145 | 1835.68437 | 1828.990725 | |
| 91 | 0 | 0 | 2259.914985 | 1937.506743 | 1859.93283 | 1842.122185 | 1836.165679 | |
| 92 | 0 | 0 | 2280.988626 | 1969.19405 | 1888.761539 | 1869.256522 | 1862.733358 | |
| 93 | 0 | 0 | 2300.824428 | 1973.62358 | 1897.423585 | 1878.488744 | 1872.156267 | |
| 94 | 0 | 0 | 2329.597367 | 2000.171039 | 1922.809258 | 1903.222516 | 1896.67202 | |
| 95 | 0 | 0 | 2367.378231 | 2017.61134 | 1936.348547 | 1916.997167 | 1910.525386 | |
| 96 | 0 | 0 | 2388.580362 | 2046.895976 | 1961.869756 | 1940.844188 | 1933.812499 | |
| 97 | 0 | 0 | 2397.206508 | 2058.933429 | 1977.742257 | 1957.630283 | 1950.904132 | |
| 98 | 0 | 0 | 2456.253718 | 2108.145319 | 2020.279498 | 1997.64293 | 1990.072466 | |
| 99 | 0 | 0 | 2471.494085 | 2116.785387 | 2028.045191 | 2006.152653 | 1998.831018 | |
| 100 | 0 | 0 | 2499.298762 | 2134.44957 | 2042.986025 | 2020.719408 | 2013.272667 | |
| 101 | 0 | 0 | 0 | 2157.940934 | 2065.663578 | 2042.327831 | 2034.523537 | |
| 102 | 0 | 0 | 0 | 2166.214827 | 2069.288708 | 2046.406024 | 2038.75325 | |
| 103 | 0 | 0 | 0 | 2191.958766 | 2099.438913 | 2075.448053 | 2067.424666 | |
| 104 | 0 | 0 | 0 | 2199.907667 | 2101.92068 | 2078.954336 | 2071.273583 | |
| 105 | 0 | 0 | 0 | 2229.860956 | 2126.296245 | 2100.326803 | 2091.641708 | |
| 106 | 0 | 0 | 0 | 2250.341956 | 2148.595408 | 2123.773851 | 2115.472649 | |
| 107 | 0 | 0 | 0 | 2263.010528 | 2166.923379 | 2143.096746 | 2135.128282 | |
| 108 | 0 | 0 | 0 | 2282.081021 | 2186.00598 | 2162.163288 | 2154.189453 | |
| 109 | 0 | 0 | 0 | 2311.603879 | 2200.783658 | 2173.453558 | 2164.313411 | |
| 110 | 0 | 0 | 0 | 2333.348828 | 2229.242657 | 2203.683349 | 2195.135418 | |
| 111 | 0 | 0 | 0 | 2371.156639 | 2253.115073 | 2223.523026 | 2213.626405 | |
| 112 | 0 | 0 | 0 | 2389.547549 | 2274.802277 | 2246.126992 | 2236.536968 | |
| 113 | 0 | 0 | 0 | 2397.472026 | 2283.695077 | 2255.907763 | 2246.614707 | |
| 114 | 0 | 0 | 0 | 2408.662565 | 2297.615542 | 2270.077823 | 2260.868242 | |
| 115 | 0 | 0 | 0 | 2436.953434 | 2327.419296 | 2298.932766 | 2289.405869 | |
| 116 | 0 | 0 | 0 | 2461.625204 | 2339.334635 | 2310.580813 | 2300.964524 | |
| 117 | 0 | 0 | 0 | 2474.06077 | 2356.432139 | 2326.800538 | 2316.890689 | |
| 118 | 0 | 0 | 0 | 2506.548582 | 2372.76222 | 2340.454362 | 2329.649478 | |
| 119 | 0 | 0 | 0 | 0 | 2394.748667 | 2364.224664 | 2354.016364 | |
| 120 | 0 | 0 | 0 | 0 | 2425.69337 | 2392.51076 | 2381.413328 | |
| 121 | 0 | 0 | 0 | 0 | 2444.980894 | 2412.118408 | 2401.128038 | |
| 122 | 0 | 0 | 0 | 0 | 2454.783559 | 2424.272394 | 2414.068387 | |
| 123 | 0 | 0 | 0 | 0 | 2470.789585 | 2439.592237 | 2429.158747 | |
| 124 | 0 | 0 | 0 | 0 | 2481.464055 | 2448.141711 | 2436.997547 |
| ArrowHead 2 | ||||||||
|---|---|---|---|---|---|---|---|---|
| Refinement | ||||||||
| # | Initial | 1 | 2 | 3 | 4 | 5 | Predicted | |
| 1 | 36.0756097 | 31.26837052 | 30.03253219 | 29.70726463 | 29.61892082 | 29.59399577 | 29.58559544 | |
| 2 | 61.33471061 | 51.65856408 | 49.16989611 | 48.54431239 | 48.38407268 | 48.34229514 | 48.32821512 | |
| 3 | 93.24954246 | 75.75283099 | 70.78904363 | 69.55065708 | 69.2409026 | 69.16330478 | 69.13715248 | |
| 4 | 139.1268945 | 100.6074714 | 90.1116001 | 87.40657299 | 86.68033068 | 86.47724737 | 86.40880349 | |
| 5 | 168.5866582 | 110.7956882 | 98.40076661 | 95.48639917 | 94.75216083 | 94.56114264 | 94.496765 | |
| 6 | 199.8631931 | 139.5946379 | 122.4037142 | 118.2629904 | 117.2393268 | 116.9840518 | 116.8980181 | |
| 7 | 294.3799107 | 156.4970856 | 134.6443998 | 129.4048367 | 128.0930549 | 127.7590918 | 127.6465383 | |
| 8 | 914.5119745 | 180.1346201 | 154.8510588 | 148.5833079 | 147.0139211 | 146.6163261 | 146.4823271 | |
| 9 | 0 | 201.867957 | 171.1134093 | 163.5320119 | 161.6018759 | 161.0947751 | 160.9238702 | |
| 10 | 0 | 210.6418578 | 177.6042013 | 168.9144076 | 166.6171509 | 165.9858972 | 165.7731498 | |
| 11 | 0 | 235.092473 | 194.4746944 | 184.636136 | 182.2023303 | 181.5941831 | 181.3892231 | |
| 12 | 0 | 254.0045742 | 211.7501892 | 201.1838085 | 198.548432 | 197.8868206 | 197.663842 | |
| 13 | 0 | 274.8575264 | 228.7801854 | 216.9150997 | 213.9184711 | 213.1615159 | 212.9064041 | |
| 14 | 0 | 287.8437264 | 236.8873621 | 223.9392196 | 220.6932566 | 219.8784244 | 219.6038066 | |
| 15 | 0 | 310.3251173 | 248.8851874 | 233.1124981 | 229.2427449 | 228.2807895 | 227.9565878 | |
| 16 | 0 | 327.261254 | 266.0456829 | 249.0723731 | 244.747578 | 243.6572791 | 243.2898226 | |
| 17 | 0 | 342.2954346 | 276.3171896 | 259.3106496 | 255.0342594 | 253.9358641 | 253.5656789 | |
| 18 | 0 | 353.6564572 | 284.7723147 | 267.2425036 | 262.904345 | 261.8122187 | 261.4441464 | |
| 19 | 0 | 372.7116622 | 297.2783056 | 277.1575427 | 272.1640681 | 270.917814 | 270.4977969 | |
| 20 | 0 | 388.714337 | 313.1372659 | 292.7293558 | 287.5939882 | 286.2864567 | 285.8457878 | |
| 21 | 0 | 417.7808182 | 327.8294079 | 303.2306045 | 297.1426799 | 295.6263981 | 295.1153753 | |
| 22 | 0 | 434.9428987 | 338.4552364 | 313.8067248 | 307.7729708 | 306.2691773 | 305.7623634 | |
| 23 | 0 | 447.5921559 | 354.5617258 | 324.8876137 | 317.6207063 | 315.8135474 | 315.2044921 | |
| 24 | 0 | 465.0239863 | 356.0655659 | 332.6457441 | 326.9226974 | 325.4954128 | 325.0143842 | |
| 25 | 0 | 491.7880833 | 385.3015409 | 351.4804989 | 342.9440856 | 340.8062583 | 340.08576 | |
| 26 | 0 | 501.1653363 | 396.2364754 | 363.5955084 | 355.3567007 | 353.2953398 | 352.6006125 | |
| 27 | 0 | 515.5753807 | 402.8209184 | 372.3227613 | 364.6972031 | 362.7629665 | 362.1110831 | |
| 28 | 0 | 534.1272617 | 416.2121209 | 378.7435987 | 369.4809879 | 367.1657862 | 366.3855085 | |
| 29 | 0 | 543.9490914 | 432.9486581 | 389.939819 | 379.2729201 | 376.6069093 | 375.7084006 | |
| 30 | 0 | 573.6844036 | 445.3540642 | 409.821706 | 399.0377788 | 396.1240677 | 395.1420782 | |
| 31 | 0 | 604.2314959 | 462.5439412 | 413.3599984 | 402.6462748 | 400.1635513 | 399.3268148 | |
| 32 | 0 | 627.7983646 | 472.3424087 | 423.1551717 | 411.6457712 | 408.8029136 | 407.8448033 | |
| 33 | 0 | 630.8277084 | 486.4015624 | 437.1725626 | 424.6994151 | 421.5641957 | 420.5075527 | |
| 34 | 0 | 654.9263128 | 489.1828482 | 444.5009904 | 432.5857816 | 429.560752 | 428.5412455 | |
| 35 | 0 | 692.3966298 | 510.0599768 | 457.9083402 | 446.2712655 | 443.4324879 | 442.4757528 | |
| 36 | 0 | 702.9603475 | 529.90857 | 474.2977477 | 459.9294861 | 456.3502755 | 455.1439969 | |
| 37 | 0 | 717.344351 | 540.4515759 | 482.359205 | 466.7788036 | 462.878541 | 461.5640603 | |
| 38 | 0 | 728.9196405 | 551.1273272 | 493.6218478 | 480.768572 | 477.5397387 | 476.4515456 | |
| 39 | 0 | 758.4824987 | 562.0803034 | 503.8873695 | 488.6540927 | 484.5443557 | 483.1592772 | |
| 40 | 0 | 782.4818616 | 580.2125159 | 508.4051836 | 492.4905826 | 488.9073626 | 487.6997328 | |
| 41 | 0 | 791.5910221 | 590.0492372 | 534.1669716 | 516.4255422 | 511.8651091 | 510.3281354 | |
| 42 | 0 | 822.7145821 | 610.4987172 | 539.7729596 | 524.7272374 | 520.5989724 | 519.2076495 | |
| 43 | 0 | 831.364694 | 629.8266073 | 548.6472245 | 529.5217031 | 525.2401478 | 523.7971625 | |
| 44 | 0 | 849.7154903 | 635.2800721 | 560.7855239 | 541.5172068 | 536.6263711 | 534.9780439 | |
| 45 | 0 | 864.8712626 | 647.8121067 | 577.4588661 | 554.4574596 | 548.7562248 | 546.834774 | |
| 46 | 0 | 894.7063933 | 672.0590309 | 583.8287702 | 564.1133381 | 558.2158038 | 556.2281954 | |
| 47 | 0 | 949.555214 | 680.2993051 | 589.4379702 | 569.3744118 | 565.1447025 | 563.7191905 | |
| 48 | 0 | 981.017771 | 699.2329403 | 607.5856031 | 583.5735153 | 577.6434169 | 575.6448336 | |
| 49 | 0 | 1002.498928 | 706.341199 | 620.8995002 | 594.0357511 | 587.3650566 | 585.1168749 | |
| 50 | 0 | 1028.509414 | 722.6892595 | 625.9054977 | 604.3358967 | 598.7619123 | 596.8833478 | |
| 51 | 0 | 1100.53191 | 743.419766 | 635.8800023 | 614.7121948 | 609.6390531 | 607.9292844 | |
| 52 | 0 | 1149.79695 | 750.2626671 | 652.3052528 | 624.0321562 | 617.0296765 | 614.6696753 | |
| 53 | 0 | 1168.160444 | 766.4491024 | 665.7622812 | 640.4032838 | 633.968995 | 631.8004877 | |
| 54 | 0 | 0 | 784.5364341 | 678.8027011 | 651.6024115 | 644.5253517 | 642.1402153 | |
| 55 | 0 | 0 | 802.8836651 | 685.9181241 | 659.7361266 | 653.2070848 | 651.0066434 | |
| 56 | 0 | 0 | 816.2829948 | 698.7153696 | 669.4130632 | 661.3909523 | 658.6873113 | |
| 57 | 0 | 0 | 821.6855292 | 703.8292508 | 670.3872244 | 663.2757163 | 660.8789701 | |
| 58 | 0 | 0 | 841.2949995 | 724.5706646 | 694.5537428 | 685.7480748 | 682.7803566 | |
| 59 | 0 | 0 | 861.4283361 | 730.027502 | 700.0485068 | 693.3506963 | 691.0933758 | |
| 60 | 0 | 0 | 867.8343808 | 745.3624946 | 714.4350885 | 706.978759 | 704.4657998 | |
| 61 | 0 | 0 | 887.5943771 | 753.6713314 | 718.9707504 | 710.3303239 | 707.418296 | |
| 62 | 0 | 0 | 898.5147822 | 768.694816 | 731.6075254 | 722.3865627 | 719.2788803 | |
| 63 | 0 | 0 | 926.0355704 | 779.9417883 | 748.0851728 | 737.8490768 | 734.3992706 | |
| 64 | 0 | 0 | 935.0305168 | 789.7397427 | 749.6438324 | 740.869572 | 737.912439 | |
| 65 | 0 | 0 | 951.689647 | 793.1300389 | 756.8745671 | 748.8525019 | 746.1488763 | |
| 66 | 0 | 0 | 959.6887429 | 812.7278821 | 775.4061769 | 765.3596538 | 761.9737381 | |
| 67 | 0 | 0 | 970.8208557 | 821.4085873 | 778.7893337 | 768.7575815 | 765.3766439 | |
| 68 | 0 | 0 | 996.1855986 | 834.1478703 | 798.885255 | 787.6705614 | 783.8909446 | |
| 69 | 0 | 0 | 1009.670244 | 846.3817266 | 802.4558671 | 794.0852789 | 791.2641928 | |
| 70 | 0 | 0 | 1031.123472 | 857.4168973 | 813.2978265 | 801.9352982 | 798.1058577 | |
| 71 | 0 | 0 | 1039.345917 | 860.8355877 | 815.9338348 | 805.0114499 | 801.3303481 | |
| 72 | 0 | 0 | 1046.157386 | 878.0601452 | 828.6827507 | 816.784796 | 812.7749041 | |
| 73 | 0 | 0 | 1074.362797 | 880.9789128 | 841.3717167 | 830.8119631 | 827.2530766 | |
| 74 | 0 | 0 | 1077.467667 | 900.4209896 | 857.3349132 | 845.2369694 | 841.1596765 | |
| 75 | 0 | 0 | 1098.508727 | 912.3993356 | 859.9336426 | 848.7775938 | 845.0177416 | |
| 76 | 0 | 0 | 1119.573276 | 930.9675824 | 878.3913864 | 865.0829669 | 860.597715 | |
| 77 | 0 | 0 | 1139.503782 | 935.9648712 | 884.9813259 | 871.2367749 | 866.6045365 | |
| 78 | 0 | 0 | 1148.435864 | 943.0333368 | 894.4080868 | 882.3589616 | 878.2981216 | |
| 79 | 0 | 0 | 1155.535931 | 953.6614051 | 898.278616 | 884.3877163 | 879.7061549 | |
| 80 | 0 | 0 | 1177.633123 | 959.6566789 | 906.7289863 | 893.7583873 | 889.3869888 | |
| 81 | 0 | 0 | 1187.177348 | 972.906572 | 919.4010992 | 907.2666867 | 903.177103 | |
| 82 | 0 | 0 | 1198.081253 | 989.1070141 | 931.9582338 | 917.8716781 | 913.124176 | |
| 83 | 0 | 0 | 1224.643524 | 997.857536 | 947.1014091 | 930.9129857 | 925.4571044 | |
| 84 | 0 | 0 | 1234.346008 | 1013.082241 | 950.1839187 | 937.3395315 | 933.0106695 | |
| 85 | 0 | 0 | 0 | 1023.646002 | 963.6283924 | 948.6708406 | 943.6297923 | |
| 86 | 0 | 0 | 0 | 1037.146389 | 974.2541818 | 959.7086137 | 954.8064135 | |
| 87 | 0 | 0 | 0 | 1041.37192 | 981.4841471 | 966.0867301 | 960.8974367 | |
| 88 | 0 | 0 | 0 | 1051.786911 | 988.9491386 | 973.5013242 | 968.2950457 | |
| 89 | 0 | 0 | 0 | 1064.077719 | 1003.710517 | 987.3208775 | 981.7971817 | |
| 90 | 0 | 0 | 0 | 1070.46542 | 1012.448838 | 994.3826602 | 988.2939315 | |
| 91 | 0 | 0 | 0 | 1085.964736 | 1013.121761 | 1000.397186 | 996.1087038 | |
| 92 | 0 | 0 | 0 | 1100.546073 | 1035.62405 | 1017.412961 | 1011.275394 | |
| 93 | 0 | 0 | 0 | 1112.672907 | 1043.693208 | 1027.420714 | 1021.936499 | |
| 94 | 0 | 0 | 0 | 1126.895813 | 1049.082597 | 1030.627954 | 1024.408303 | |
| 95 | 0 | 0 | 0 | 1128.169259 | 1066.611027 | 1047.850152 | 1041.527293 | |
| 96 | 0 | 0 | 0 | 1142.353912 | 1069.347012 | 1053.410527 | 1048.039555 | |
| 97 | 0 | 0 | 0 | 1147.812295 | 1076.346729 | 1059.222794 | 1053.451624 | |
| 98 | 0 | 0 | 0 | 1166.482761 | 1085.175556 | 1065.789416 | 1059.255828 | |
| 99 | 0 | 0 | 0 | 1176.033919 | 1103.917572 | 1083.220208 | 1076.244708 | |
| 100 | 0 | 0 | 0 | 1190.667647 | 1113.779236 | 1096.809386 | 1091.090146 | |
| 101 | 0 | 0 | 0 | 1206.973164 | 1128.670609 | 1107.098495 | 1099.828182 | |
| 102 | 0 | 0 | 0 | 1213.328805 | 1135.010155 | 1113.353783 | 1106.055074 | |
| 103 | 0 | 0 | 0 | 1222.726161 | 1136.968043 | 1118.128756 | 1111.779471 | |
| 104 | 0 | 0 | 0 | 1233.931297 | 1144.458834 | 1123.388212 | 1116.286915 | |
| 105 | 0 | 0 | 0 | 1246.368722 | 1164.341861 | 1142.673169 | 1135.370308 | |
| 106 | 0 | 0 | 0 | 0 | 1177.741164 | 1156.299688 | 1149.073404 | |
| 107 | 0 | 0 | 0 | 0 | 1183.014573 | 1158.307268 | 1149.980323 | |
| 108 | 0 | 0 | 0 | 0 | 1185.754478 | 1163.349464 | 1155.798444 | |
| 109 | 0 | 0 | 0 | 0 | 1201.366124 | 1178.878065 | 1171.299057 | |
| 110 | 0 | 0 | 0 | 0 | 1208.634442 | 1187.136597 | 1179.891316 | |
| 111 | 0 | 0 | 0 | 0 | 1226.938395 | 1202.876639 | 1194.767258 | |
| 112 | 0 | 0 | 0 | 0 | 1231.111855 | 1204.879724 | 1196.038875 | |
| 113 | 0 | 0 | 0 | 0 | 1243.303267 | 1218.594579 | 1210.267167 | |
| 114 | 0 | 0 | 0 | 0 | 1245.283911 | 1223.850286 | 1216.626648 |