Efficient Spherical t-Designs on the Sphere S3
with Good Goemetric Properties

Rob Womersley

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.

Values for known regular point sets on S3

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

Values for calculated point sets on S3 with no additional constraints

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

Values for calculated symmetric point sets on S3

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


Last updated 09-Oct-2017   |   Page contact