Evenly distributed points on a sphere

[Mike_In_Plano]

I have a practical application which relies upon the generation of evenly distributed points on a sphere.

When I first considered this problem, I learned that some number of polyhedrons (Platonic polyhedrons) had each vertex lying evenly spaced from the others within the surface of a sphere. I also found that there was a finite number of these shapes and I needed far more points on my sphere (10's of thousands)

Next I learned that there was an accepted methodology to distribute points in a roughly even fashion through a numeric technique whereby each point is treated as having a repulsion to his neighbors and the system of points is adjusted until the net repulsion reaches a minimum.

This latter technique seems valid enough given that one knows the critical number of points to introduce to ensure that the distribution is even (i.e. the distances between points is consistent over all cases.) However, how does one go about finding N, such that all points may be evenly spaced?

 

[mfb]

If all points should have the same distances to other points, you are limited to archimedean solids and platonic solids. They allow up to 120 points, but the distribution is not really uniform.

This thread might be interesting.

 

[jim mcnamara]

Two commonly used algorithms to generate n equidistributed points on the surface of a sphere:

http://www.cmu.edu/biolphys/deserno/pdf/sphere_equi.pdf

The consequences of which one to use are yours to decide.

 

[Mike_In_Plano]

Thank you so much, MFB, and Jim. I especially like the link to the paper. Short and sweet :)

- Mike

 

[blue_raver22]

Thank you for sharing your practical application and your research on evenly distributed points on a sphere. This is a common problem in various fields of science and mathematics, and there are several approaches that can be taken to achieve an even distribution of points on a sphere.

One approach is to use a geometric construction known as the "Fibonacci sphere." This method involves placing points on a sphere in a spiral pattern that follows the Fibonacci sequence, resulting in evenly distributed points. This technique has been used in computer graphics and visualization to create realistic and evenly spaced particle systems.

Another approach is to use a numerical algorithm, similar to the one you mentioned, that minimizes the energy of the system by adjusting the positions of points until they are evenly distributed. This method is commonly used in molecular dynamics simulations to generate evenly spaced atoms on a spherical surface.

In terms of finding the optimal number of points, it will depend on the specific application and the desired level of accuracy. In some cases, a smaller number of points may be sufficient, while in others, a larger number may be necessary. This can be determined through experimentation and testing.

Overall, the generation of evenly distributed points on a sphere is an important problem with many practical applications. I encourage you to continue exploring different methods and techniques to find the best solution for your specific application.