Efficient Spherical Designs on the sphere S2 with good geometric properties Last updated 05-Jun-2015

 Efficient Spherical Designs Unit sphere S2 in R3 has area |S2| = 4 π. The space Pt of polynomials of degree at most t on S2 has dimension dt = (t+1)2. Efficient spherical t-designs are sets of N ~ t2/2 points xj, j = 1,...,N on S2 such that equal weight cubature with these nodes is exact for all polynomials in Pt. There are several equivalent characterizations of a spherical t-design: See here for more details. The Weyl sums rl, k(XN)   :=   Σj=1,...,N Yl, k(xj) = 0 for k = 1,...,2l+1, and l = 1, 2, 3, ..., t. A potential function At, N, ψ(XN) = (1/N2) Σi Σj ψ(xiT xj) where ψ is a polynomial of degree t with strictly positive Legendre coefficients. The potential reported below uses ψ(z) = zt + zt-1 - a0 where a0 is the zeroth Legendre coefficient. These symmetric spherical designs have the number of points N = ceil((t+1)2/2)+1 except for t = 3, 5, 7, 9, 11.

 Geometrical Properties The geodesic distance between two points x and y on the unit sphere S2 is dist(x, y) = cos-1(xT y). The spherical cap with centre y and radius α is C(y, α) = {x on S2: dist(x, y) ≤ α } The minimum separation is mini ≠ j   dist(xi, xj) is twice the packing radius for identical caps with centres xj, j = 1,...,N. The mesh norm or covering radius is h = maxx in S2   minj=1,...,N   dist(x, xj). This is also the radius for covering the sphere by spherical caps with centers xj for j = 1,...,N. The mesh ratio is ρN = covering radius / packing radius   = 2 mesh norm / separation ≥ 1 A sequence if point sets XN is quasi-uniform if the mesh ratio if uniformly bounded. For a quasi-uniform sequence of point sets both the mesh norm and the covering radius have the optimal order N-1/2 There are many other measures related to the geometric quality of the point set: Riesz s-energy, in particular the Coulomb (s=1) and log (s=0) cases. Discrepancy, for example the spherical cap discrepancy or the Cui and Freeden discrepancy. Volume of the convex hull of the points. Properties of the Voronoi cells, such as area of largest and smallest Voronoi cells.

 Notes on point sets Caveat: All points are only numerical spherical designs within the limits of IEEE double precision. There are many approximate spherical designs with these numbers of points, but different geometric properties. For each point set, the text file has three items per row: the xj, yj, and zj Cartesian coordinates in [-1, 1] for the point xj = (xj, yj, zj) on S2. The equal cubature weights wj = |S2|/N for j = 1,...,N are not included in the files. The number of rows is equal to the number of points N. All points are on the unit sphere so have |xj|2 = xj2 + yj2 + zj2 = 1 for j = 1,...,N. All criteria and rotationally invariant, so the point sets are normalised with the first point at the north pole and the second point on the prime meridian Symmetric (antipodal) spherical t-designs up to degree 297 are available here. The file names have three components: point set, degree, number of points. Points last updated 29-Nov-2012
 Degree t No. points N File SSQ At, N, ψ Separation Mesh norm Mesh ratio 1 3 sf001.00003 3.0e-32 3.7e-17 2.09440 1.57080 1.5000 2 6 sf002.00006 1.1e-31 3.5e-17 1.57080 0.95532 1.2163 3 8 sf003.00008 6.2e-30 2.7e-17 1.23096 0.95532 1.5521 4 14 sf004.00014 8.6e-30 -5.0e-18 0.86301 0.69133 1.6022 5 18 sf005.00018 5.2e-27 -3.6e-17 0.80386 0.57486 1.4303 6 26 sf006.00026 7.4e-29 9.8e-18 0.62274 0.49113 1.5773 7 32 sf007.00032 1.5e-28 5.8e-18 0.59531 0.43572 1.4639 8 42 sf008.00042 2.1e-28 -8.3e-18 0.48451 0.39580 1.6338 9 50 sf009.00050 7.0e-28 7.9e-18 0.45545 0.36076 1.5842 10 62 sf010.00062 8.1e-28 1.2e-17 0.39448 0.33078 1.6771 11 72 sf011.00072 2.1e-27 8.9e-18 0.37500 0.29886 1.5939 12 86 sf012.00086 2.9e-27 -8.4e-18 0.32411 0.27611 1.7038 13 98 sf013.00098 1.1e-26 -6.6e-18 0.30276 0.25671 1.6958 14 114 sf014.00114 1.2e-26 6.2e-18 0.28378 0.24021 1.6929 15 128 sf015.00128 2.8e-26 -1.2e-18 0.26439 0.22791 1.7240 16 146 sf016.00146 3.0e-26 4.8e-18 0.25678 0.21153 1.6475 17 163 sf017.00163 4.7e-26 1.9e-17 0.23332 0.20705 1.7748 18 182 sf018.00182 5.7e-26 -2.9e-18 0.22429 0.18801 1.6765 19 201 sf019.00201 7.7e-26 1.5e-17 0.20862 0.18427 1.7666 20 222 sf020.00222 1.3e-25 1.3e-17 0.21047 0.16966 1.6123 21 243 sf021.00243 1.4e-25 1.6e-17 0.19004 0.16773 1.7652 22 266 sf022.00266 1.8e-25 -5.2e-19 0.18867 0.15736 1.6681 23 289 sf023.00289 2.6e-25 -8.0e-18 0.17594 0.15540 1.7665 24 314 sf024.00314 3.2e-25 3.7e-18 0.17297 0.14512 1.6780 25 339 sf025.00339 4.3e-25 3.4e-18 0.16277 0.14066 1.7284 26 366 sf026.00366 4.3e-25 5.6e-18 0.15335 0.13332 1.7388 27 393 sf027.00393 4.9e-25 3.0e-18 0.14847 0.13049 1.7577 28 422 sf028.00422 5.8e-25 -1.8e-17 0.14900 0.12517 1.6801 29 451 sf029.00451 7.2e-25 -2.0e-17 0.14054 0.12142 1.7280 30 482 sf030.00482 8.8e-25 -3.5e-17 0.13808 0.11517 1.6681 31 513 sf031.00513 1.4e-24 -3.3e-17 0.13131 0.11318 1.7238 32 546 sf032.00546 1.7e-24 -5.0e-17 0.13152 0.10978 1.6694 33 579 sf033.00579 1.7e-24 -5.0e-17 0.12924 0.10541 1.6313 34 614 sf034.00614 2.2e-24 3.0e-17 0.12352 0.10303 1.6682 35 649 sf035.00649 2.4e-24 3.0e-17 0.11394 0.10047 1.7635 36 686 sf036.00686 3.4e-24 5.1e-17 0.11702 0.09695 1.6571 37 723 sf037.00723 3.1e-24 5.1e-17 0.11126 0.09620 1.7293 38 762 sf038.00762 4.0e-24 2.8e-17 0.10790 0.09247 1.7140 39 801 sf039.00801 4.1e-24 3.0e-17 0.10792 0.09327 1.7285 40 842 sf040.00842 5.0e-24 1.0e-16 0.10682 0.08754 1.6389 41 883 sf041.00883 5.5e-24 1.1e-16 0.09984 0.08581 1.7191 42 926 sf042.00926 6.8e-24 3.1e-19 0.10069 0.08286 1.6458 43 969 sf043.00969 7.0e-24 -4.0e-19 0.09645 0.08188 1.6979 44 1014 sf044.01014 9.0e-24 2.8e-17 0.09798 0.08048 1.6427 45 1059 sf045.01059 9.4e-24 2.4e-17 0.09111 0.07872 1.7280 46 1106 sf046.01106 1.2e-23 7.6e-17 0.09486 0.07631 1.6088 47 1153 sf047.01153 1.3e-23 8.4e-17 0.08979 0.07508 1.6723 48 1202 sf048.01202 1.6e-23 -3.1e-17 0.08687 0.07479 1.7218 49 1251 sf049.01251 1.5e-23 -2.2e-17 0.08394 0.07251 1.7278 50 1302 sf050.01302 2.2e-23 3.4e-17 0.08578 0.07119 1.6598 51 1353 sf051.01353 2.0e-23 2.6e-17 0.08385 0.06944 1.6564 52 1406 sf052.01406 2.3e-23 1.0e-16 0.08088 0.06759 1.6713 53 1459 sf053.01459 2.8e-23 1.1e-16 0.07685 0.06742 1.7546 54 1514 sf054.01514 2.9e-23 -2.6e-17 0.07831 0.06559 1.6753 55 1569 sf055.01569 3.1e-23 -3.0e-17 0.07412 0.06498 1.7532 56 1626 sf056.01626 3.5e-23 -9.1e-17 0.07784 0.06291 1.6164 57 1683 sf057.01683 3.8e-23 -9.0e-17 0.07172 0.06309 1.7593 58 1742 sf058.01742 4.6e-23 -2.1e-16 0.07318 0.06145 1.6794 59 1801 sf059.01801 5.2e-23 -1.5e-16 0.07084 0.06077 1.7157 60 1862 sf060.01862 5.8e-23 -9.6e-17 0.07181 0.05921 1.6490 61 1923 sf061.01923 6.4e-23 -9.4e-17 0.06631 0.05840 1.7614 62 1986 sf062.01986 6.7e-23 2.6e-17 0.06992 0.05772 1.6510 63 2049 sf063.02049 7.1e-23 2.5e-17 0.06799 0.05637 1.6584 64 2114 sf064.02114 7.4e-23 3.3e-17 0.06624 0.05623 1.6977 65 2179 sf065.02179 8.9e-23 2.7e-17 0.06475 0.05520 1.7049 66 2246 sf066.02246 8.9e-23 -1.0e-16 0.06160 0.05368 1.7429 67 2313 sf067.02313 1.1e-22 -1.1e-16 0.06089 0.05336 1.7527 68 2382 sf068.02382 1.1e-22 2.5e-16 0.06205 0.05231 1.6863 69 2451 sf069.02451 1.2e-22 2.5e-16 0.05904 0.05162 1.7486 70 2522 sf070.02522 1.4e-22 1.0e-16 0.05946 0.05132 1.7262 71 2593 sf071.02593 1.5e-22 1.0e-16 0.05867 0.04960 1.6908 72 2666 sf072.02666 1.7e-22 1.7e-16 0.06035 0.04941 1.6375 73 2739 sf073.02739 1.6e-22 1.7e-16 0.05667 0.04882 1.7229 74 2814 sf074.02814 1.9e-22 1.8e-16 0.05818 0.04759 1.6358 75 2889 sf075.02889 2.0e-22 1.8e-16 0.05774 0.04748 1.6446 76 2966 sf076.02966 2.2e-22 2.8e-16 0.05473 0.04713 1.7224 77 3043 sf077.03043 2.3e-22 2.8e-16 0.05464 0.04590 1.6802 78 3122 sf078.03122 2.6e-22 3.2e-16 0.05540 0.04546 1.6413 79 3201 sf079.03201 2.7e-22 3.3e-16 0.05380 0.04492 1.6698 80 3282 sf080.03282 3.0e-22 2.6e-16 0.05253 0.04496 1.7115 81 3363 sf081.03363 3.4e-22 2.6e-16 0.05368 0.04358 1.6235 82 3446 sf082.03446 3.6e-22 3.1e-16 0.05322 0.04310 1.6197 83 3529 sf083.03529 3.6e-22 3.2e-16 0.05051 0.04257 1.6857 84 3614 sf084.03614 3.9e-22 -3.9e-17 0.05159 0.04240 1.6438 85 3699 sf085.03699 4.1e-22 -5.1e-17 0.04787 0.04183 1.7477 86 3786 sf086.03786 4.8e-22 -1.6e-16 0.04884 0.04266 1.7468 87 3873 sf087.03873 4.9e-22 -1.6e-16 0.04883 0.04138 1.6948 88 3962 sf088.03962 5.0e-22 2.5e-16 0.04934 0.04042 1.6382 89 4051 sf089.04051 5.4e-22 2.5e-16 0.04542 0.04033 1.7759 90 4142 sf090.04142 6.0e-22 -9.9e-17 0.04895 0.03939 1.6093 91 4233 sf091.04233 6.2e-22 -1.1e-16 0.04729 0.03933 1.6634 92 4326 sf092.04326 6.7e-22 -2.2e-16 0.04813 0.03885 1.6147 93 4419 sf093.04419 7.1e-22 -2.1e-16 0.04666 0.03817 1.6358 94 4514 sf094.04514 7.5e-22 4.3e-17 0.04635 0.03826 1.6508 95 4609 sf095.04609 7.9e-22 3.5e-17 0.04625 0.03774 1.6320 96 4706 sf096.04706 9.0e-22 -1.7e-16 0.04577 0.03689 1.6119 97 4803 sf097.04803 8.8e-22 -1.8e-16 0.04292 0.03675 1.7122 98 4902 sf098.04902 1.0e-21 -1.6e-16 0.04526 0.03623 1.6013 99 5001 sf099.05001 1.0e-21 -1.6e-16 0.04235 0.03698 1.7461 100 5102 sf100.05102 1.1e-21 1.4e-16 0.04320 0.03569 1.6522 101 5203 sf101.05203 1.2e-21 1.5e-16 0.04334 0.03513 1.6210 102 5306 sf102.05306 1.3e-21 -1.1e-16 0.04241 0.03505 1.6529 103 5409 sf103.05409 1.3e-21 -1.2e-16 0.04281 0.03446 1.6100 104 5514 sf104.05514 1.5e-21 -2.7e-16 0.04237 0.03427 1.6177 105 5619 sf105.05619 1.5e-21 -2.8e-16 0.03948 0.03451 1.7484 106 5726 sf106.05726 1.6e-21 1.9e-16 0.04100 0.03350 1.6341 107 5833 sf107.05833 1.6e-21 1.9e-16 0.03927 0.03372 1.7172 108 5942 sf108.05942 1.8e-21 -4.3e-16 0.03848 0.03399 1.7664 109 6051 sf109.06051 1.9e-21 -4.3e-16 0.03805 0.03327 1.7488 110 6162 sf110.06162 2.0e-21 1.9e-16 0.03956 0.03236 1.6360 111 6273 sf111.06273 2.1e-21 1.9e-16 0.03763 0.03207 1.7043 112 6386 sf112.06386 2.2e-21 6.0e-17 0.03793 0.03222 1.6989 113 6499 sf113.06499 2.3e-21 6.1e-17 0.03729 0.03149 1.6892 114 6614 sf114.06614 2.8e-21 2.2e-16 0.03810 0.03122 1.6386 115 6729 sf115.06729 2.5e-21 2.2e-16 0.03669 0.03096 1.6878 116 6846 sf116.06846 2.8e-21 -4.3e-16 0.03682 0.03081 1.6736 117 6963 sf117.06963 2.8e-21 -4.3e-16 0.03554 0.03035 1.7081 118 7082 sf118.07082 3.0e-21 -4.5e-16 0.03602 0.03063 1.7006 119 7201 sf119.07201 3.3e-21 -4.5e-16 0.03582 0.03078 1.7187 120 7322 sf120.07322 3.4e-21 -5.2e-16 0.03483 0.02966 1.7031 121 7443 sf121.07443 3.3e-21 -5.2e-16 0.03378 0.02950 1.7464 122 7566 sf122.07566 3.6e-21 1.8e-16 0.03558 0.02948 1.6568 123 7689 sf123.07689 4.0e-21 1.8e-16 0.03521 0.02922 1.6596 124 7814 sf124.07814 4.1e-21 -9.1e-17 0.03478 0.02980 1.7140 125 7939 sf125.07939 4.2e-21 -8.7e-17 0.03375 0.02877 1.7049 126 8066 sf126.08066 4.4e-21 -8.7e-17 0.03319 0.02831 1.7061 127 8193 sf127.08193 4.5e-21 -7.6e-17 0.03207 0.02828 1.7635 128 8322 sf128.08322 4.9e-21 6.9e-16 0.03299 0.02834 1.7183 129 8451 sf129.08451 5.1e-21 7.1e-16 0.03415 0.02786 1.6318 130 8582 sf130.08582 5.4e-21 2.8e-16 0.03215 0.02777 1.7272 131 8713 sf131.08713 5.6e-21 2.8e-16 0.03255 0.02762 1.6970 132 8846 sf132.08846 5.8e-21 2.8e-16 0.03240 0.02781 1.7166 133 8979 sf133.08979 6.1e-21 2.8e-16 0.03346 0.02689 1.6075 134 9114 sf134.09114 6.4e-21 4.5e-16 0.03205 0.02671 1.6666 135 9249 sf135.09249 7.0e-21 4.5e-16 0.03088 0.02698 1.7475 136 9386 sf136.09386 7.3e-21 5.4e-16 0.03062 0.02606 1.7021 137 9523 sf137.09523 7.8e-21 5.5e-16 0.03230 0.02597 1.6082 138 9662 sf138.09662 8.1e-21 2.5e-16 0.03005 0.02672 1.7783 139 9801 sf139.09801 8.6e-21 2.5e-16 0.03034 0.02712 1.7876 140 9942 sf140.09942 9.3e-21 7.5e-17 0.02981 0.02634 1.7672 141 10083 sf141.10083 8.9e-21 7.1e-17 0.02988 0.02570 1.7207 142 10226 sf142.10226 9.4e-21 2.2e-16 0.02995 0.02548 1.7016 143 10369 sf143.10369 9.4e-21 2.1e-16 0.02925 0.02498 1.7082 144 10514 sf144.10514 9.9e-21 -6.5e-16 0.02854 0.02529 1.7725 145 10659 sf145.10659 1.0e-20 -6.5e-16 0.02831 0.02504 1.7691 146 10806 sf146.10806 1.1e-20 6.6e-16 0.02851 0.02502 1.7551 147 10953 sf147.10953 1.1e-20 6.6e-16 0.02866 0.02512 1.7530 148 11102 sf148.11102 1.2e-20 -3.8e-16 0.02818 0.02414 1.7129 149 11251 sf149.11251 1.1e-20 -3.8e-16 0.02764 0.02447 1.7705 150 11402 sf150.11402 1.3e-20 1.4e-16 0.02793 0.02433 1.7422 151 11553 sf151.11553 1.4e-20 1.3e-16 0.02752 0.02394 1.7400 152 11706 sf152.11706 1.3e-20 -8.5e-16 0.02743 0.02379 1.7348 153 11859 sf153.11859 1.4e-20 -8.4e-16 0.02681 0.02372 1.7699 154 12014 sf154.12014 1.5e-20 -4.5e-16 0.02706 0.02349 1.7356 155 12169 sf155.12169 1.6e-20 -4.5e-16 0.02683 0.02404 1.7917 156 12326 sf156.12326 1.6e-20 0.0e+00 0.02690 0.02335 1.7365 157 12483 sf157.12483 1.7e-20 0.0e+00 0.02694 0.02384 1.7704 158 12642 sf158.12642 1.7e-20 0.0e+00 0.02620 0.02272 1.7344 159 12801 sf159.12801 1.7e-20 0.0e+00 0.02622 0.02272 1.7329 160 12962 sf160.12962 1.9e-20 0.0e+00 0.02645 0.02299 1.7381 161 13123 sf161.13123 1.9e-20 -8.0e-16 0.02560 0.02245 1.7537 162 13286 sf162.13286 2.1e-20 1.2e-16 0.02564 0.02276 1.7752 163 13449 sf163.13449 2.2e-20 1.1e-16 0.02569 0.02215 1.7238 164 13614 sf164.13614 2.1e-20 -5.7e-16 0.02568 0.02232 1.7386 165 13779 sf165.13779 2.2e-20 -5.7e-16 0.02509 0.02250 1.7937 166 13946 sf166.13946 2.3e-20 -9.6e-16 0.02508 0.02173 1.7323 167 14113 sf167.14113 2.3e-20 -9.5e-16 0.02517 0.02207 1.7537 168 14282 sf168.14282 2.5e-20 -1.8e-16 0.02480 0.02144 1.7293 169 14451 sf169.14451 2.6e-20 -1.8e-16 0.02488 0.02247 1.8064 170 14622 sf170.14622 2.7e-20 -1.3e-16 0.02631 0.02100 1.5960 171 14793 sf171.14793 2.8e-20 -1.3e-16 0.02609 0.02090 1.6021 172 14966 sf172.14966 2.9e-20 3.5e-16 0.02422 0.02115 1.7468 173 15139 sf173.15139 3.0e-20 3.4e-16 0.02429 0.02115 1.7419 174 15314 sf174.15314 3.0e-20 1.4e-16 0.02445 0.02107 1.7236 175 15489 sf175.15489 3.3e-20 1.3e-16 0.02407 0.02076 1.7249 176 15666 sf176.15666 3.5e-20 1.2e-16 0.02377 0.02057 1.7309 177 15843 sf177.15843 3.4e-20 1.3e-16 0.02377 0.02119 1.7829 178 16022 sf178.16022 3.6e-20 -2.3e-16 0.02328 0.02021 1.7367 179 16201 sf179.16201 3.6e-20 -2.3e-16 0.02385 0.02016 1.6905 180 16382 sf180.16382 3.7e-20 -7.8e-16 0.02356 0.02022 1.7164

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