Symmetric Spherical Designs on the sphere S2
with good geometric properties

Rob Womersley

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.
Residual SSQ

Spherical design potential

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.
Minimum angle between points

Mesh norm (covering radius)

Mesh ratio (covering radius/packing radius)

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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