I'm trying to calculate a measure of 'concentration' or 'clusteredness' for n points on a cylinder.
Say I start with a plane of width=12 (x) and height=10 (y) on which 10 points are situated. These points can only have cartesian coordinates which are integers. This plane is wrapped around a closed cylinder. For each of these points on the cylinder, the pairwise distance between points is calculated the shortest way around the cylinder based on the formula in this question:
What is the shortest distance between two 2D points on the surface of a cylinder?
The sum of all pairwise values between points is taken to get a measure distance between points. I would like to divide this value by the sum of pairwise distances of an imaginary set of 10 points which are maximally spread out on the cylinder, taking into account that their correspondent cartesian coordinates on the plane can also only take integer values. This would result in a measure which reflects the 'clusteredness' or 'concentration' of these points on the cylinder which is scaled between 0-1.
I have managed to get the pairwise distances between the actual points on the cylinder using the formula mentioned above, however, I can't seem to figure out what the configuration of the same number of points with maximum spread would be so that the summed pairwise distances for these points is maximal.
As I am a biologist and not excellent at math, any help would be greatly appreciated.