conmap.m

This toolbox contains Matlab code for several graph and mesh partitioning methods, including geometr

function [xyzmap,xymap] = conmap(c,xyz)
% CONMAP : Conformal map for geometric partitioning.
%
% [xyzmap,xymap] = conmap(c,xyz):
% Input is points xyz on a unit sphere in d-space 
% (one point per row of matrix xyz),
% and a point c in d-space represented as a row vector.
%
% We compute the following conformal mapping:
%    First reflect the sphere so that c is on the last axis.
%    Second stereographically map the xyz points onto the d-1 - space.
%    Third scale the xy points so that c stretches to the origin.
%    Fourth stereog'y map the xy points back to the sphere.
%
% The first output argument is the new points on the sphere in d-space;
% the second output argument is the new points in the d-1 - space.
%
% See GEOPART.
%
% John Gilbert and Shanghua Teng, 1992-1993.
% Copyright (c) 1990-1996 by Xerox Corporation.  All rights reserved.
% HELP COPYRIGHT for complete copyright and licensing notice.


d = size(xyz,2);

% Compute the reflection and stretch.
[Q,r] = reflector(c);
alpha = sqrt((1+r)/(1-r));

% Reflect.
xyzref = xyz * Q;

% Points at the north pole will get NaN coords in the
% stereographic mapping, so handle them specially.
norths = find(abs(xyzref(:,d)-1) < sqrt(eps));
if length(norths)
    xyzn = xyzref(norths,:);
    xyzref(norths,:) = zeros(length(norths),d);
end;

xyref = stereodown(xyzref);

% Stretch.
xymap = xyref / alpha;
xyzmap = stereoup(xymap);

if length(norths)
    xyzmap(norths,:) = xyzn;
end;