Point picking (ie. uniform sampling) of the unit sphere (ie. a sphere centered at the origin and radius equals to 1)

sphere-picking-3d.py

import numpy

'''
Pick N points uniformly from the unit disc
This sampling algorithm does not use rejection sampling.
'''
def disc_uniform_pick(N):
	angle = (2 * numpy.pi) * numpy.random.random(N)
	out = numpy.stack([numpy.cos(angle), numpy.sin(angle)], axis = 1)
	out *= numpy.sqrt(numpy.random.random(N))[:,None]
	return out

'''
Pick N points uniformly from the unit sphere
This sampling algorithm does not use rejection sampling.
'''
def sphere_uniform_pick(N):
	U = disc_uniform_pick(N)
	U_norm_sqr = numpy.sum(U ** 2, axis = 1)
	out = numpy.concatenate([U, (1. - 2. * U_norm_sqr)[:,None]], axis = 1)
	out[:,:2] *= 2. * numpy.sqrt((1. - U_norm_sqr))[:,None]
	return out