Efficient Spherical t-Designs on the Sphere S3 with Good Goemetric Properties Last updated 09-Oct-2017

 Notes on point sets Caveat: All calculated point sets are only numerical spherical designs within the limits of IEEE double precision. There are many approximate spherical designs with these nubmers of points, but different geometric properties. For each point set, the text file has four items per row: the cartesian coordinates of a point. The number of rows is equal to the number of points N. All points are on the unit sphere S3 in R4, so have ||x||2 = 1. All point sets are normalized by a rotation so that the first point is (1  0  0  0)T, the second point has its last two components zero and the third point has its last component zero. The file names have three components: point set name, degree t, number of points N. The equal cubature weight is wj = |S3|/N for j = 1,...,N, where |S3| = 1 (normalized surface measure) or |S3| = 2π2. A spherical t-design with N points has the equal weight cubature rule exact for all spherical polynomials of degree at most t.

 Degree t N Sym File At, N, 1 At, N, 2 At, N, 3 Geodesic Separation Mesh norm Mesh ratio 1 2 1 sdr001.00002 5.55e-17 0.00e+00 0.00e+00 3.14159 1.57080 1.0000 2 5 0 sdr002.00005 8.44e-17 4.44e-17 1.69e-16 1.82348 1.31812 1.4457 3 8 1 sdr003.00008 1.52e-17 -9.37e-17 -1.73e-17 1.57080 1.04720 1.3333 3 16 1 sdr003.00016 -3.09e-18 -9.66e-17 -1.44e-16 1.04720 1.04720 2.0000 5 24 1 sdr005.00024 -4.70e-17 1.26e-16 -6.84e-17 1.04720 0.78540 1.5000 7 48 1 sdr007.00048 -3.58e-18 -1.45e-16 -1.24e-16 0.78540 0.60856 1.5497 11 120 1 sdr011.00120 1.05e-17 -3.12e-17 -4.38e-16 0.62832 0.38814 1.2355 11 600 1 sdr011.00600 2.83e-17 -3.74e-17 -8.05e-17 0.27092 0.38814 2.8654

 Degree t N Sym File At, N, 1 At, N, 2 At, N, 3 Geodesic separation Mesh norm Mesh ratio 1 4 1 sdf001.00004 6.25e-17 0.00e+00 0.00e+00 1.57080 1.57080 2.0000 2 7 0 sdf002.00007 6.34e-17 5.61e-17 1.04e-16 1.35851 1.16830 1.7200 3 12 0 sdf003.00012 8.10e-18 -9.06e-17 -7.90e-17 1.23106 1.00162 1.6272 4 20 0 sdf004.00020 1.39e-19 5.56e-17 -2.41e-17 0.94143 0.83376 1.7713 5 32 0 sdf005.00032 -2.95e-17 1.27e-16 2.63e-17 0.78157 0.70408 1.8017 6 49 0 sdf006.00049 1.06e-17 1.04e-17 -9.16e-18 0.68833 0.60862 1.7684 7 70 0 sdf007.00070 2.14e-17 -1.16e-16 -1.81e-16 0.57651 0.54538 1.8920 8 97 0 sdf008.00097 -3.19e-17 1.60e-16 -1.49e-17 0.50279 0.48374 1.9242 9 130 0 sdf009.00130 -1.40e-17 7.05e-17 -5.94e-17 0.46879 0.44673 1.9059 10 171 0 sdf010.00171 1.22e-17 6.06e-17 -1.75e-16 0.44043 0.40818 1.8535 11 219 0 sdf011.00219 4.14e-18 -5.16e-17 -2.93e-16 0.38086 0.37483 1.9683 12 275 0 sdf012.00275 3.39e-17 -4.15e-17 -2.19e-16 0.34671 0.34090 1.9665 13 340 0 sdf013.00340 3.69e-17 5.60e-17 -3.90e-17 0.33278 0.32246 1.9380 14 415 0 sdf014.00415 4.87e-18 -1.98e-17 -1.49e-16 0.31115 0.29821 1.9168 15 501 0 sdf015.00501 2.26e-18 1.59e-16 -9.09e-18 0.29088 0.28979 1.9924 16 597 0 sdf016.00597 -1.47e-17 -7.58e-17 -1.08e-16 0.26727 0.26165 1.9579 17 705 0 sdf017.00705 -3.00e-17 7.87e-17 -4.29e-17 0.25355 0.25072 1.9777 18 825 0 sdf018.00825 1.18e-16 1.17e-16 1.08e-15 0.23858 0.23829 1.9976 19 959 0 sdf019.00959 3.70e-17 5.52e-18 -6.02e-17 0.22828 0.22666 1.9858 20 1106 0 sdf020.01106 1.93e-17 9.97e-17 1.18e-16 0.21633 0.21624 1.9992

 Degree t N Sym File At, N, 1 At, N, 2 At, N, 3 Separation Mesh norm Mesh ratio 1 4 1 sds001.00004 6.25e-17 0.00e+00 0.00e+00 1.57080 1.57080 2.0000 3 10 1 sds003.00010 -2.36e-17 -1.12e-16 -2.58e-16 1.31812 0.97760 1.4833 5 28 1 sds005.00028 -1.98e-17 1.30e-16 9.18e-17 0.83340 0.73031 1.7526 7 60 1 sds007.00060 5.74e-18 -1.28e-16 -1.12e-16 0.63236 0.56564 1.7890 9 114 1 sds009.00114 -2.67e-17 6.45e-17 -1.40e-16 0.48633 0.45479 1.8703 11 194 1 sds011.00194 1.73e-17 -4.72e-17 -2.20e-16 0.41258 0.38599 1.8711 13 308 1 sds013.00308 3.31e-17 5.47e-17 -1.13e-16 0.34535 0.32204 1.8650 15 458 1 sds015.00458 1.76e-18 1.56e-16 -2.74e-17 0.29137 0.28774 1.9751 17 650 1 sds017.00650 -2.94e-17 7.95e-17 -7.36e-17 0.26493 0.25837 1.9505 19 890 1 sds019.00890 3.46e-17 5.36e-18 -5.54e-17 0.23885 0.23797 1.9927 21 1184 1 sds021.01184 9.73e-18 1.35e-16 6.63e-17 0.21389 0.21131 1.9759 23 1538 1 sds023.01538 1.03e-17 -1.86e-16 -2.50e-16 0.19990 0.19515 1.9524 25 1954 1 sds025.01954 -1.13e-17 5.32e-17 -2.04e-17 0.17949 0.17776 1.9808 27 2440 1 sds027.02440 -1.05e-17 8.85e-17 -8.24e-18 0.16878 0.16625 1.9701 29 3000 1 sds029.03000 5.43e-16 8.09e-17 1.66e-14 0.15742 0.15450 1.9628 31 3642 1 sds031.03642 1.32e-16 -9.99e-17 5.06e-15 0.14612 0.14621 2.0013

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