How to quantify the spread/separation of points on a circle

Hi, this may be a little vague

I have a set of points in the unit circle and I want to quantify their separation, for example, they will get a low score if they are all at the same location, and they will get a high score if they are very separated. For example, if you imagine how a set of charges would arrange themselves on a circular conducting plate, that should maximise their separation- this state would get the highest "separation" score.

Does anyone know of any algorithms or approaches I can use to do this, in the most mathematically respectable way? I really am a novice to the concept.

Thanks for any feedback, Mike

Answer 1

[Matt J]

If your points are the columns of a matrix A, you can quantify as

 id=interdists(A);
 score = sum(id(:));

where the interdist() function below generates a matrix of distances between all the points.

    function Graph=interdists(A,B)
    %Finds the graph of distances between point coordinates
    %
    % (1) Graph=interdists(A,B)
    %
    % in:
    % 
    % A:  matrix whose columns are coordinates of points, for example 
    %          [[x1;y1;z1], [x2;y2;z2] ,..., [xM;yM;zM]]
    %     but the columns may be points in a space of any dimension, not just 3D.
    % 
    % B:  A second matrix whose columns are coordinates of points in the same
    %     Euclidean space. Default B=A.
    % 
    % 
    % out:
    % 
    % Graph: The MxN  matrix of separation distances in l2 norm between the coordinates. 
    %        Namely, Graph(i,j) will be the distance between A(:,i) and B(:,j).
    %
    %
    % (2) interdists(A,'noself') is the same as interdists(A), except the output 
    %     diagonals will be NaN instead of zero. Hence, for example, operations
    %     like min(interdists(A,'noself')) will ignore self-distances.
    %

    noself=false;
    if nargin<2  
        B=A; 
    elseif ischar(B)&&strcmpi(B,'noself')
        noself=true;
        B=A;
    end

    N=size(A,1);
    B=reshape(B,N,1,[]);

    tmp=bsxfun(@minus, A, B);
    Graph=sqrt(dot(tmp,tmp,1));

    Graph=squeeze(Graph);

    if noself
        n=length(Graph);
       Graph(linspace(1,n^2,n))=nan; 
    end