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Topic: Solid angle subtended by a simplex
Replies: 0

 Daniel A. Asimov Posts: 35 Registered: 12/6/04
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 n-simplex 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 T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799

P.S. If the terminology is unfamiliar to you, here are some definitions:

Definition:
A regular n-simplex 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 (n-1)-sphere centered at p, via

proj(x) = (x - p)/||x - p||.

Then the solid angle subtended by X with respect to p is the
(n-1)-dimensional volume of proj(X) (assuming this is well-defined).

P.P.S. By the way, for the tetrahedron I get 3*arccos(1/3) - pi.