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Suppose one has a distribution of $N$ points on the sphere. Is there an agreed upon metric for the difference of this distribution and $N$ equidistant points on the sphere? To me entropy seems like the right concept, but I'm not sure how to go about defining it.

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there exists a great variety of metrics for the uniformity of points on a (hyper)sphere; the reference list of this 2010 paper will point you to them. For a comparison of the different metrics:

  • A. Figueiredo, Comparison of tests of uniformity defined on the hypersphere, Statistics and Probability Letters, 77, 329 (2007).
  • P.J. Diggle, N.I. Fisher, and A.J. Lee, A comparison of tests of uniformity for spherical data, Australian Journal of Statistics, 27, 53 (1985).
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    $\begingroup$ In other words, there is no unique concept of "N equidistant points on the sphere," except for a few very symmetric cases where the metrics agree. $\endgroup$
    – user25199
    Sep 5, 2013 at 11:35

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