Wikipedia:Reference desk/Archives/Mathematics/2020 October 13

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October 13

Quality of point distribution on a sphere

I'm looking for a function, ideally one calculable in less than O(n*n), that I can use to measure how evenly spread a set of points are over the surface of a sphere.

Does anybody have any pointers for where to start looking? — Preceding unsigned comment added by 2A01:E34:EF5E:4640:8089:C4C5:1392:7CE8 (talk) 18:43, 13 October 2020 (UTC)

A large set of points drawn independently from a uniform distribution over the surface will have some clusters and some "bald" spots; is my assumption correct that this should have a lower quality score than when the points repel each other and try to maintain a maximal social distance? (So that the set of vertices of a spherical regular dodecahedron has max quality for a set of 20 points?) One possible measure could be based on the potential energy of the configuration, using some force field (e.g., a repellent force of magnitude $\cos {\tfrac {1}{2}}\varphi$ between two points at an angular distance $\varphi$). This measure can be calculated in $O(n^{2})$ steps (in the RAM model). Lower energy corresponds to a more even spread.  --Lambiam 19:58, 13 October 2020 (UTC)

Thankyou for your detailed answer. I had considered a potential-based approach, but it occurred to me that I could construct the Voronoi diagram and look at the area of cells in O(nlogn). This seems like overkill though so I was wondering if there weren't a well-known accepted metric.

You might use the measures from the Thompson problem or the Tammes problem. See also [1]. It also occurs to me to find the largest empty circle, but I don't know how efficiently that can be calculated on the sphere. --Amble (talk) 20:44, 13 October 2020 (UTC)

The measure from the Thompson problem uses the force field model but with a different force, and Tammes nearly fits the model. Using a step function as the potential, where the cutoff point for the step is adjustable, will, with a bit of tweaking, get you the Tammes version. Any number of potential functions could be used, so it's not a matter of finding a measure but which measure is best suited to the problem at hand. --RDBury (talk) 21:24, 13 October 2020 (UTC)

Some examples are given in Diggle and Fisher (1983): [2]. As RDBury points out, there are lots of options, which may be more or less suited to the problem. I pointed to the Thompson and Tammes problems because they have existing literature and you can probably find some sample code. --Amble (talk) 21:37, 13 October 2020 (UTC)

The area on the sphere between two parallel planes is proportional to the distance between those planes (a theorem of Archimedes; this is used in more than one algo for sphere arrangements). So perhaps we can reduce the problem to a simpler one: how evenly are the coordinates in x, in y and in z distributed? You could do this by squaring the difference between a coordinate and the corresponding (2k-2n+1)/2n, where k is the coordinate's rank. —Tamfang (talk) 01:46, 20 October 2020 (UTC)

If two points are close to each other on the sphere, each pair of their coordinates will also be close, so nicely separated coordinates imply separated points. But some very even sphere distributions (e.g. the dodecahedral vertices) may have very uneven coordinate distributions, so the coordinate test may discard good point distributions or score them much lower than much worse distributions.  --Lambiam 09:12, 20 October 2020 (UTC)

Quadratic formula division

Can you divide numbers in the numerator of a simplified quadratic formula by a number in the denominator?

For example, if I had 2 ± sqrt(31) all over -3, could the negative sign in the denominator move up to the top? And could I divide the 2 by the -3 to get -2/3 ± sqrt31?

Thanks, User:Heyoostorm^_talk! 22:36, 13 October 2020 (UTC)

Not quite. You need to distribute the -1/3 over both elements of the sum, so you'd get -2/3 ± (sqrt31)/3 = -2/3 ± sqrt(31/9).

In certain contexts where the + vs the - is tracked, it would be more appropriate to write -2/3 ∓ sqrt(31/9), but that is not an issue here. Where it applies is in equations such as sin(±π/2) = ±1. Dividing both halves of that equation by -3 gives you (-1/3)sin(±π/2) = ∓1/3. See Plus–minus sign#In mathematics for more details.

But if all you are doing is "moving the negative sign up to the top" with a quadratic formula, then no problem. But you should keep in mind that is because (-1)±x = ∓x, which in that context is the same thing as ±x. -- ToE 23:58, 13 October 2020 (UTC)

Thanks, that helps. User:Heyoostorm^_talk! 14:30, 14 October 2020 (UTC)