Get random orientations in 3D from a uniform orientation distribution.

vecs = sphrand(N)
vecs = sphrand(N,nOctants)
[phi,theta] = sphrand(...)
[x,y,z] = sphrand(...)

sphrand does in a random way what sphgrid does more systematically. It calculates N points on the unit sphere. In the limit of these points are evenly distributed, since the underlying distribution is a uniform one.

If nOctants is specified, it determines the number of octants over which to generate the points. It can be 1 (one octant), 2 (two octants, part of upper hemisphere with y>0), 4 (upper hemisphere) or 8 (whole sphere).

The result containing the set of points on the sphere can be in one of three forms:

• vecs, a 3xN array of cartesian vectors of unit length pointing from the origin to the sphere surface.
• x, y, and z, the three cartesian coordinates of the unit vectors.
• phi and theta, the polar angles for the points on the sphere. phi denotes the counterclockwise angle between the x axis and the projection of a vector onto the xy plane (azimuth), theta is the angle between a vector and the z axis (colatitude, elevation complement). Both theta and phi are in radians.

The command

[phi,theta] = sphrand(10000,8);

generates 10000 points randomly distributed over the whole sphere. The plot

plot(phi*180/pi,theta*180/pi,'.');
set(gca,'YDir','reverse'); axis tight
xlabel('phi'); ylabel('theta');

shows that in a theta vs. phi projection there is reduced point density around the north and south poles.

ang2vec, vec2ang, sphgrid, sphtri