sphere distribution problems

A list of pages concerning the frequent question “How can I arrange N points evenly on a sphere?”

Overviews
- Dave Rusin
- list of papers by E B Saff
- Lienhard Wimmer (cites many papers)
- Sphere Packing and Kissing Numbers at the Geometry Junkyard, David Eppstein's ever-growing collection of links
- Optimal Configurations on the Sphere and Other Manifolds, a conference in May 2010
- Rob Womersley
- comp.graphics.algorithms FAQ (largely rewritten by me in September 2007)
- Hardin, Michaels, Saff (2016)
Applications
- “A Disco Ball in Space” (used Golden Section helix to place mirrors)
- “spherepoint” plugin for Animation:Master
Exact optimization. Several of these can be considered special cases of the problem "minimize or maximize the sum of pairwise distances raised to an exponent"; for these the exponent is shown.
- Overviews
  - Yanmu Zhou (1996)
  - Applet by Cris Cecka et al. at Syracuse U.
- Tammes or Fejes Tóth Problem: packing: maximize the least separation (exponent -∞)
- Thomson problem / Fekete points: electrostatic repulsion (exponent -1)
  - Programs: Leech, Bulatov (1996)
  - BASIC code by Rusin (1995)
  - UBASIC code by Rusin (1995)
  - Solutions: Hardin, Sloane & Smith (1997)
  - Descriptions of solutions: Kevin Brown
  - Martin Trump's Pretty Polyhedra
  - Bob Allanson's Java models
  - Results of Ann Davis, Scott Malloy et al., using the algorithm by Michael Neubauer, Mark Schilling, William Watkins & Joel Zeitlin, CSUN, 1998
  - Java simulation by Scott Malloy
  - Liz Dolan and Jorge Moré (2001) use the Thomson Problem to test optimization software
  - “Asymptotics for minimal discrete energy on the sphere” by Kuijlaars & Saff
  - “Particles On a Sphere: A Specimen Multidisciplinary Problem” (1996) by Timothy Rolfe
  - Simon Tatham's results and Python code
  - Symen Hovinga searches for repulsion figures in higher dimensions
  - table of optima by David Wales and Sikida Ulker
  - Paul Bourke's code
- Whyte's problem: maximize product of distances ("exponent" 0 - sum of logarithms)
- Maximize sum of distances (exponent +1)
  - Jentje Goslinga
- Covering: minimize the greatest Voronoi radius
  - citation of Fejes-Tóth results
  - Solutions: Hardin, Sloane & Smith
- Equal area
  - Recursive Zonal Equal Area Sphere Partitioning Toolbox
  - Max Tegmark: icosahedron
- Maximize volume of convex hull
  - Solutions: Hardin, Sloane & Smith
  - Images by Hugo Pfoertner
- Maximize inradius of convex hull
  - Marek Teichmann specifies that the inscribed sphere be concentric with the original sphere; that makes the problem equivalent to covering. But is the condition redundant? Is the maximum inscribed sphere ever not concentric?
- Minimize volume of tangent polyhedron
- t-designs
  - Hardin & Sloane
- Illumination by light sources at infinity
  - Hugo Pfoertner seems to have this field to himself. He tackles three forms of the problem:
    - minimize the maximum luminance (analogous to packing)
    - maximize the minimum luminance (analogous to covering)
    - minimize the difference between minimum and maximum
Algorithmic approximations
- Saff & Kuijlaars (NB: this page's code confuses phi with theta); original paper (PDF)
- Golden Angle spiral is a better packing
- Rusin's disco ball is better still, though not as pretty or as versatile
- Comparison of these three approximations
- Adrian Rossiter's Antiprism, a set of small Python programs
- Martin Roberts's improvement on the Golden Angle spiral

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this page created 2002 Mar 31; last addition 2024 Feb 19