Efficient Spherical Designs with Good Geometric Properties

Rob Womersley
School of Mathematics and Statistics
UNSW Sydney

Spherical t-designs on S² with N = t²/2 + t + O(1) points

• File for points sets for degrees t = 1, 2, 3, ..., 179, 180:   SF29-Nov-2012.zip (28.7 MB)
• See here for individual point sets and more information on spherical design criteria and their geometric properties.

Symmetric (antipodal) spherical t-designs on S² with N = t²/2 + t/2 + O(1) points

• File of points sets for degrees t = 1, 3, 5, ..., 325:   SS31-Mar-2016.zip (82.5 MB)
• See here for individual point sets and more information on symmetric spherical designs and their geometric properties.
• Recent updates (available from individual point sets page below and included in .zip file)
  • Added t = 287 on 19 May 2015
  • Added t = 289 on 23 May 2015
  • Added t = 297 on 4 June 2015
  • Added t = 279, 295, 301 on 15 June 2015. Point sets for t = 279, 295 currently have somewhat high mesh ratios.
  • Added t = 293 on 19 June 2015
  • Updated t = 279 with better mesh ratio on 24 June 2015
  • Added t = 291, 299 on 24 September 2015
  • Updated t = 89, 111, 117, 203, 213, 241, 279, 295, with better mesh ratios on 24 September 2015
  • Updated t = 111, 259, 279, 295 with better mesh ratios on 21 October 2015
  • Updated t = 299 with better mesh ratio on 28 October 2015
  • Updated t = 251, 281 with better mesh ratio on 29 November 2015
  • Added t = 303, 305, 309, 311 on 29 November 2015
  • Updated t = 299 with better mesh ratio on 26 January 2016
  • Added t = 313, 315, 317, 321 on 26 January 2016
  • Updated t = 315 with better mesh ratio on 18 February 2016
  • Added t = 319, 323 on 18 February 2016
  • Updated t = 299 with better mesh ratio on 27 March 2016
  • Added t = 325 on 31 March 2016
  • Point sets for t = 323, 325 have slightly high mesh ratios.

Spherical t-designs on S³ with N = t³/9 + O(t²) points

• S3SD     Last updated 22-Aug-2017.

Reference

• R. S. Womersley, Efficient Spherical Designs with Good Geometric Properties. In: Dick J., Kuo F., Wozniakowski H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer (2018) pp. 1243-1285
  https://doi.org/10.1007/978-3-319-72456-0_57

Reported at

• Workshop on Spherical Designs and Numerical Analysis, Shanghai Jiao Tong University, Shanghai, China, 21-27 April 2015. Slides: RSW_Shanghai15.pdf
• Optimal and Near Optimal Configurations on Lattices and Manifolds, MFO Workshop, Oberwolfach, Germany, 19-25 August 2012.
• Optimal Configurations on the Sphere and Other Manifolds, Department of Mathematics, Vanderbilt University, Nashville, USA, 17-20 May 2010.

Acknowledgement