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Solid angle subtended by a simplex
Posted:
Oct 6, 1993 10:08 PM


Someone recently asked on sci.math how to calculate the solid angle subtended by the regular nsimplex with respect to one of its vertices. A closed form solution would be nice.
I have reduced the question to an integral that I don't know how to evaluate in general.
QUESTION: Does anyone know a graceful way to make this calculation?
Dan Asimov Mail Stop T0451 NASA Ames Research Center Moffett Field, CA 940351000
asimov@nas.nasa.gov (415) 6044799
P.S. If the terminology is unfamiliar to you, here are some definitions:
Definition: A regular nsimplex in R^n is the the convex hull of a finite set F, where F is any set of n+1 distinct points in R^n such that all distances between two of them are equal.
Definition: Let p be a point of R^n and X any subset of R^n not containing p. Project the set X radially to the unit (n1)sphere centered at p, via
proj(x) = (x  p)/x  p. Then the solid angle subtended by X with respect to p is the (n1)dimensional volume of proj(X) (assuming this is welldefined).
P.P.S. By the way, for the tetrahedron I get 3*arccos(1/3)  pi.



