2009/7/5 Ian Mallett <geometr...@gmail.com>: > @Stéfan: I thought of the first method. Let's hear the second approach.
Please see the attached example. I start off by drawing random azimuth and elevation angles, as well as a radii: N = 1000 max_radius = 5 az = np.random.uniform(low=0, high=np.pi * 2, size=N) el = np.random.uniform(low=0, high=np.pi, size=N) r = np.random.uniform(size=N) You can imagine your volume consisting of a large number of concentric spherical surfaces (almost like those Russian nested dolls). We'd like to have all of those surfaces equally densely packed, but their surfaces increase in area by the third power with radius. To counter this effect we do r = r ** (1/3.) Now, imagine the elevation contours (like latitude on the earth) for one of those spherical surfaces. If we choose them equally spaced, we'll have a much higher concentration of points near the north and south poles. Instead, we choose them according to el = np.arccos(1 - 2*el) so that we have more contours close to the equator (where the contours are longer and need more points). >From a statistical point of view, the derivation is done using transformation of random variables: http://en.wikipedia.org/wiki/Probability_integral_transform http://en.wikipedia.org/wiki/Inverse_transform_sampling Regards Stéfan
random_sphere.py
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