Find points on a 3D convex hull surface

Hi all,

I have a 3D convex hull and would like to populate it's surface with points. To be more specific, the convex hull gives the coordinates of the vertices of the smallest enclosing polygon, but I would like many more points, spread across the hull's surface.

Ultimately, I would like to transform these coordinates into spherical coordinates and generate a 'map' of the distance to the edge of the convexhull at every possible azimuth/elevation. A rough demonstration of this can be seen below. You will see that the step involving scatteredInterpolant would benefit from having many more points to work with, which is why I need more points falling on the convexhull surface...

Thanks for any help,

Rod.

% generate random data and find 3D convex hull
dat = [10 10 40].*rand(50,3);
K = convhulln(dat);

% convert these to spherical coordinates centred on hull centroid
cdat = dat-nanmean(dat(K,:));
[azi,ele,rad] = cart2sph(cdat(:,1),cdat(:,2),cdat(:,3));

% plot the result
figure
subplot(1,2,1)
scatter(azi,ele,20,rad)
daspect([1 1 1])
title('Actual points')

subplot(1,2,2)    
F = scatteredInterpolant([azi ele],rad,'natural');
[xe,ye] = meshgrid(-pi:0.1:pi,-pi/2:0.1:pi/2);
dmap = F(xe,ye);
imagesc(dmap);
axis xy
daspect([1 1 1])
title('Distance map')