Symmetric Spherical Designs on the sphere S2 with good geometric properties Last updated 27-Sep-2016

 Symmetric 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. Symmetric (antipodal) point sets contain both x and -x, so N is even. Symmetric point sets automatically integrate all odd degree spherical polynomials exactly. For symmetric spherical t-designs, t can always taken to be odd. 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 = 2, 4, ..., t-1, and t odd. 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-1 - a0 where a0 is the zeroth Legendre coefficient. These symmetric spherical designs have the number of points N = 2 ceil((t2+t+4)/4) except for t = 1 3, 5, 7, 11, 15 for which N = 2, 6, 12, 32, 70 and 120 respectively. For t = 1, 3, 5, 15 it is also possible to find symmetric spherical designs with N = 4, 8, 18, 122, in agreement with the formula above. These are not necessarily the smallest number of points for which a spherical t-design exists. For example Hardin and Sloane have a spherical 7-design with N = 24 points which is not symmetric.

 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 For symmetric point sets, N is even and xN/2+j = -xj for j = 1,...,N/2 Spherical t-designs up to degree 180, but no symmetry requirements, are available here. The file names have three components: point set, degree, number of points. Points last updated 31-Mar-2016 Table of values for symmetric spherical designs
 Degree t No. points N File SSQ At, N, ψ Separation Mesh norm Mesh ratio 1 2 ss001.00002 0.0e+00 0.0e+00 3.14159 1.57080 1.0000 3 6 ss003.00006 3.3e-31 3.5e-17 1.57080 0.95532 1.2163 5 12 ss005.00012 1.1e-29 6.9e-18 1.10715 0.65236 1.1784 7 32 ss007.00032 1.7e-28 2.1e-17 0.58626 0.44803 1.5284 9 48 ss009.00048 3.1e-26 4.2e-18 0.46115 0.38600 1.6741 11 70 ss011.00070 3.2e-27 -2.4e-17 0.37938 0.30172 1.5906 13 94 ss013.00094 4.6e-26 -8.4e-18 0.31456 0.26393 1.6781 15 120 ss015.00120 1.5e-26 7.9e-18 0.28998 0.23517 1.6220 17 156 ss017.00156 2.9e-26 8.7e-18 0.24567 0.20393 1.6602 19 192 ss019.00192 2.2e-25 -5.4e-18 0.22482 0.18990 1.6894 21 234 ss021.00234 1.4e-25 2.2e-17 0.20089 0.16892 1.6817 23 278 ss023.00278 2.5e-25 -3.3e-18 0.18218 0.15482 1.6997 25 328 ss025.00328 3.1e-25 -9.5e-18 0.17222 0.14206 1.6497 27 380 ss027.00380 7.6e-25 -4.6e-19 0.15666 0.13284 1.6960 29 438 ss029.00438 8.5e-25 5.8e-18 0.14480 0.12294 1.6981 31 498 ss031.00498 1.4e-24 -3.0e-17 0.13756 0.11515 1.6742 33 564 ss033.00564 1.7e-24 -4.0e-17 0.12915 0.10752 1.6650 35 632 ss035.00632 2.6e-24 2.1e-17 0.11969 0.10110 1.6895 37 706 ss037.00706 3.2e-24 3.8e-17 0.11654 0.09683 1.6618 39 782 ss039.00782 4.8e-24 2.6e-17 0.10823 0.09098 1.6812 41 864 ss041.00864 7.0e-24 9.7e-17 0.10253 0.08625 1.6826 43 948 ss043.00948 7.5e-24 -7.1e-18 0.09878 0.08245 1.6692 45 1038 ss045.01038 1.1e-23 2.5e-17 0.09357 0.07925 1.6939 47 1130 ss047.01130 1.4e-23 7.9e-17 0.08886 0.07577 1.7053 49 1228 ss049.01228 1.7e-23 -3.5e-17 0.08564 0.07266 1.6971 51 1328 ss051.01328 2.1e-23 2.8e-17 0.08229 0.07058 1.7153 53 1434 ss053.01434 2.9e-23 8.9e-17 0.07986 0.06791 1.7007 55 1542 ss055.01542 3.4e-23 -1.1e-17 0.07721 0.06638 1.7196 57 1656 ss057.01656 4.2e-23 -7.2e-17 0.07418 0.06303 1.6992 59 1772 ss059.01772 5.1e-23 -1.9e-16 0.07220 0.06050 1.6757 61 1894 ss061.01894 6.5e-23 -6.1e-17 0.07007 0.05827 1.6632 63 2018 ss063.02018 7.9e-23 2.8e-17 0.06743 0.05680 1.6847 65 2148 ss065.02148 8.6e-23 3.0e-17 0.06644 0.05441 1.6379 67 2280 ss067.02280 9.9e-23 -1.0e-16 0.06336 0.05406 1.7063 69 2418 ss069.02418 1.1e-22 2.4e-16 0.06156 0.05211 1.6929 71 2558 ss071.02558 1.5e-22 7.1e-17 0.05959 0.05139 1.7247 73 2704 ss073.02704 1.9e-22 1.7e-16 0.05749 0.04946 1.7207 75 2852 ss075.02852 2.0e-22 1.5e-16 0.05687 0.04803 1.6890 77 3006 ss077.03006 2.4e-22 2.6e-16 0.05554 0.04625 1.6656 79 3162 ss079.03162 2.8e-22 2.9e-16 0.05385 0.04571 1.6978 81 3324 ss081.03324 3.6e-22 2.8e-16 0.05330 0.04457 1.6724 83 3488 ss083.03488 3.8e-22 2.8e-16 0.05057 0.04361 1.7247 85 3658 ss085.03658 4.0e-22 -1.4e-17 0.04992 0.04196 1.6809 87 3830 ss087.03830 5.0e-22 -1.8e-16 0.04898 0.04159 1.6982 89 4008 ss089.04008 5.5e-22 2.5e-16 0.04726 0.04045 1.7120 91 4188 ss091.04188 6.5e-22 -6.1e-17 0.04659 0.03955 1.6976 93 4374 ss093.04374 7.2e-22 -2.6e-16 0.04598 0.03886 1.6903 95 4562 ss095.04562 8.2e-22 8.8e-17 0.04488 0.03797 1.6919 97 4756 ss097.04756 9.4e-22 -1.3e-16 0.04388 0.03714 1.6928 99 4952 ss099.04952 1.0e-21 -1.6e-16 0.04250 0.03653 1.7191 101 5154 ss101.05154 1.1e-21 2.0e-16 0.04221 0.03533 1.6744 103 5358 ss103.05358 1.3e-21 -6.0e-17 0.04100 0.03495 1.7048 105 5568 ss105.05568 1.5e-21 -2.2e-16 0.04030 0.03417 1.6957 107 5780 ss107.05780 1.7e-21 1.9e-16 0.03971 0.03407 1.7162 109 5998 ss109.05998 2.0e-21 -4.1e-16 0.03859 0.03291 1.7058 111 6218 ss111.06218 2.1e-21 1.3e-16 0.03763 0.03236 1.7202 113 6444 ss113.06444 2.3e-21 1.1e-16 0.03767 0.03182 1.6893 115 6672 ss115.06672 2.8e-21 2.2e-16 0.03745 0.03163 1.6888 117 6906 ss117.06906 2.8e-21 -3.7e-16 0.03584 0.03094 1.7270 119 7142 ss119.07142 3.1e-21 -4.0e-16 0.03556 0.03034 1.7064 121 7384 ss121.07384 3.5e-21 -5.0e-16 0.03513 0.02972 1.6920 123 7628 ss123.07628 4.0e-21 1.7e-16 0.03486 0.02915 1.6723 125 7878 ss125.07878 4.3e-21 -3.1e-17 0.03393 0.02892 1.7050 127 8130 ss127.08130 4.7e-21 -1.1e-16 0.03354 0.02878 1.7160 129 8388 ss129.08388 4.9e-21 6.9e-16 0.03280 0.02772 1.6904 131 8648 ss131.08648 5.7e-21 2.8e-16 0.03203 0.02741 1.7119 133 8914 ss133.08914 6.3e-21 2.3e-16 0.03166 0.02687 1.6978 135 9182 ss135.09182 6.5e-21 4.3e-16 0.03177 0.02675 1.6838 137 9456 ss137.09456 7.5e-21 5.3e-16 0.03083 0.02614 1.6957 139 9732 ss139.09732 7.8e-21 2.0e-16 0.03028 0.02586 1.7082 141 10014 ss141.10014 8.7e-21 7.1e-17 0.03000 0.02539 1.6926 143 10298 ss143.10298 9.7e-21 2.2e-16 0.03018 0.02554 1.6924 145 10588 ss145.10588 1.0e-20 -6.3e-16 0.02889 0.02479 1.7164 147 10880 ss147.10880 1.1e-20 6.5e-16 0.02856 0.02462 1.7242 149 11178 ss149.11178 1.2e-20 -3.3e-16 0.02868 0.02406 1.6775 151 11478 ss151.11478 1.3e-20 1.3e-16 0.02833 0.02392 1.6882 153 11784 ss153.11784 1.4e-20 -7.8e-16 0.02743 0.02352 1.7147 155 12092 ss155.12092 1.5e-20 -4.8e-16 0.02753 0.02336 1.6970 157 12406 ss157.12406 1.6e-20 4.3e-16 0.02693 0.02315 1.7195 159 12722 ss159.12722 1.8e-20 8.6e-17 0.02669 0.02286 1.7128 161 13044 ss161.13044 1.9e-20 -7.8e-16 0.02673 0.02255 1.6870 163 13368 ss163.13368 2.0e-20 1.8e-16 0.02623 0.02217 1.6905 165 13698 ss165.13698 2.2e-20 -6.3e-16 0.02570 0.02166 1.6863 167 14030 ss167.14030 2.3e-20 -9.0e-16 0.02553 0.02156 1.6892 169 14368 ss169.14368 2.6e-20 -1.8e-16 0.02534 0.02137 1.6868 171 14708 ss171.14708 2.7e-20 -1.3e-16 0.02475 0.02103 1.6991 173 15054 ss173.15054 3.0e-20 3.9e-16 0.02460 0.02079 1.6903 175 15402 ss175.15402 3.2e-20 1.9e-16 0.02473 0.02040 1.6503 177 15756 ss177.15756 3.4e-20 3.5e-17 0.02424 0.02047 1.6895 179 16112 ss179.16112 3.6e-20 -2.0e-16 0.02363 0.02023 1.7116 181 16474 ss181.16474 3.9e-20 -7.4e-16 0.02335 0.01975 1.6912 183 16838 ss183.16838 4.0e-20 1.2e-16 0.02401 0.01975 1.6458 185 17208 ss185.17208 4.3e-20 -2.8e-17 0.02242 0.01935 1.7257 187 17580 ss187.17580 3.4e-20 -1.1e-16 0.02323 0.01921 1.6540 189 17958 ss189.17958 5.1e-20 -1.8e-16 0.02219 0.01898 1.7103 191 18338 ss191.18338 5.2e-20 2.6e-16 0.02210 0.01883 1.7042 193 18724 ss193.18724 5.7e-20 1.2e-15 0.02229 0.01867 1.6756 195 19112 ss195.19112 6.0e-20 -7.0e-16 0.02192 0.01870 1.7058 197 19506 ss197.19506 6.3e-20 6.4e-16 0.02184 0.01817 1.6639 199 19902 ss199.19902 6.8e-20 -6.3e-16 0.02168 0.01789 1.6506 201 20304 ss201.20304 7.3e-20 1.2e-15 0.02095 0.01781 1.7002 203 20708 ss203.20708 7.6e-20 5.0e-16 0.02093 0.01778 1.6988 205 21118 ss205.21118 8.0e-20 -6.3e-17 0.02063 0.01756 1.7022 207 21530 ss207.21530 8.5e-20 2.5e-16 0.02023 0.01744 1.7239 209 21948 ss209.21948 9.1e-20 -1.1e-16 0.02014 0.01720 1.7075 211 22368 ss211.22368 9.9e-20 9.1e-16 0.02013 0.01713 1.7018 213 22794 ss213.22794 1.0e-19 -5.1e-16 0.01955 0.01675 1.7138 215 23222 ss215.23222 1.1e-19 -3.4e-16 0.02005 0.01679 1.6741 217 23656 ss217.23656 1.1e-19 9.8e-17 0.01965 0.01658 1.6879 219 24092 ss219.24092 1.2e-19 2.3e-17 0.01943 0.01658 1.7066 221 24534 ss221.24534 1.3e-19 -1.4e-15 0.01937 0.01627 1.6792 223 24978 ss223.24978 1.4e-19 -8.1e-17 0.01917 0.01638 1.7090 225 25428 ss225.25428 1.4e-19 -4.9e-16 0.01881 0.01601 1.7017 227 25880 ss227.25880 1.5e-19 1.5e-16 0.01845 0.01584 1.7166 229 26338 ss229.26338 1.6e-19 -1.0e-15 0.01828 0.01565 1.7119 231 26798 ss231.26798 1.7e-19 3.6e-17 0.01812 0.01575 1.7379 233 27264 ss233.27264 1.8e-19 -6.2e-16 0.01849 0.01551 1.6773 235 27732 ss235.27732 1.9e-19 -9.6e-16 0.01819 0.01532 1.6850 237 28206 ss237.28206 2.0e-19 1.1e-15 0.01782 0.01542 1.7307 239 28682 ss239.28682 2.0e-19 -1.5e-15 0.01761 0.01518 1.7235 241 29164 ss241.29164 2.1e-19 -1.5e-15 0.01803 0.01532 1.6989 243 29648 ss243.29648 2.3e-19 8.7e-16 0.01712 0.01482 1.7311 245 30138 ss245.30138 2.3e-19 8.6e-16 0.01743 0.01457 1.6717 247 30630 ss247.30630 2.5e-19 6.9e-16 0.01725 0.01460 1.6922 249 31128 ss249.31128 2.6e-19 -5.1e-16 0.01677 0.01458 1.7389 251 31628 ss251.31628 2.7e-19 -1.1e-15 0.01672 0.01433 1.7147 253 32134 ss253.32134 2.9e-19 9.8e-16 0.01703 0.01412 1.6579 255 32642 ss255.32642 3.0e-19 -2.5e-16 0.01650 0.01419 1.7198 257 33156 ss257.33156 3.2e-19 -1.6e-15 0.01659 0.01413 1.7026 259 33672 ss259.33672 3.4e-19 -5.0e-16 0.01633 0.01420 1.7395 261 34194 ss261.34194 3.5e-19 -9.0e-17 0.01650 0.01394 1.6895 263 34718 ss263.34718 3.6e-19 7.3e-16 0.01600 0.01360 1.7011 265 35248 ss265.35248 3.8e-19 9.8e-16 0.01615 0.01376 1.7038 267 35780 ss267.35780 4.1e-19 1.7e-16 0.01540 0.01337 1.7356 269 36318 ss269.36318 4.2e-19 -9.0e-16 0.01601 0.01344 1.6786 271 36858 ss271.36858 4.2e-19 1.3e-15 0.01568 0.01370 1.7476 273 37404 ss273.37404 4.6e-19 -1.7e-15 0.01520 0.01318 1.7337 275 37952 ss275.37952 4.8e-19 1.6e-15 0.01516 0.01318 1.7395 277 38506 ss277.38506 5.0e-19 -4.1e-17 0.01527 0.01298 1.7000 279 39062 ss279.39062 5.1e-19 -9.5e-16 0.01515 0.01323 1.7466 281 39624 ss281.39624 5.4e-19 -2.9e-16 0.01505 0.01295 1.7209 283 40188 ss283.40188 5.7e-19 1.7e-15 0.01495 0.01263 1.6901 285 40758 ss285.40758 5.8e-19 -8.5e-16 0.01478 0.01266 1.7129 287 41330 ss287.41330 6.2e-19 6.9e-16 0.01487 0.01265 1.7013 289 41908 ss289.41908 6.3e-19 -1.1e-15 0.01480 0.01265 1.7090 291 42488 ss291.42488 6.8e-19 -4.0e-17 0.01489 0.01236 1.6603 293 43074 ss293.43074 7.0e-19 9.5e-16 0.01419 0.01231 1.7350 295 43662 ss295.43662 7.3e-19 1.7e-15 0.01428 0.01244 1.7419 297 44256 ss297.44256 7.4e-19 -9.0e-17 0.01427 0.01211 1.6981 299 44852 ss299.44852 7.9e-19 -8.3e-16 0.01407 0.01212 1.7223 301 45454 ss301.45454 8.2e-19 5.8e-16 0.01405 0.01195 1.7018 303 46058 ss303.46058 8.5e-19 3.0e-16 0.01395 0.01184 1.6970 305 46668 ss305.46668 8.6e-19 5.8e-16 0.01401 0.01177 1.6792 307 47280 ss307.47280 9.2e-19 -1.0e-15 0.01390 0.01172 1.6859 309 47898 ss309.47898 9.4e-19 -1.8e-15 0.01365 0.01179 1.7269 311 48518 ss311.48518 9.9e-19 5.9e-16 0.01347 0.01160 1.7226 313 49144 ss313.49144 1.1e-18 1.6e-15 0.01387 0.01150 1.6593 315 49772 ss315.49772 1.1e-18 -3.8e-17 0.01330 0.01144 1.7202 317 50406 ss317.50406 1.1e-18 -1.2e-15 0.01343 0.01137 1.6942 319 51042 ss319.51042 1.1e-18 7.8e-16 0.01343 0.01123 1.6730 321 51684 ss321.51684 1.2e-18 -2.0e-15 0.01325 0.01118 1.6879 323 52328 ss323.52328 1.2e-18 -3.2e-16 0.01314 0.01154 1.7562 325 52978 ss325.52978 1.3e-18 1.2e-15 0.01243 0.01101 1.7715

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