Determining the exact uniform con?¬?guration of the orientation vectors is a hard mathematical
problem [5,6]. Although a continuous spherical uniform distribution is explicitly
de?¬?ned [7], there is unfortunately no single de?¬?nition of a corresponding discretized
uniform distribution. Researchers in different ?¬?elds such as geometry, climate modeling,
molecular structure, or electrostatics have studied the problem with their own de?¬?nitions,
each of which may lead to some different distribution [5,8]. However, although it may be
hard to obtain M vectors uniformly distributed on a sphere for our purposes, such vectors
need be computed only once and used repetitively for L points. Therefore, the computation
complexity of the discretization algorithms should not be regarded as an important factor
in our methodology.
An approximation algorithm by Rusin [9] is used in the numerical examples shown later
in the chapter. In this approximation method, a sphere is ?¬?rst cut by a series of evenly
spaced horizontal planes, each of which forms a constant-latitude circle on the sphere. On
each such circle, points are placed so that the arc distance between each pair of adjacent
points is the same. This distance is kept the same for all of the latitude circles. Thus, circles
closer to the pole have smaller radii and consequently a smaller number of points on them.
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