From: Dave Rusin
Date: Tue, 13 Oct 1998 10:07:14 -0500 (CDT)
To: py1ll@my-dejanews.com
Subject: Re: Solid Angles
>I have just derived an expression for the solid angle that the vertix of a
>right regular pyramid 'sees' its polygonal base. Is it already well known?
Yes; the solid angle in any polyhedron can be computed by decomposing it
into triangular pyramids, then adding the solid angles facing the bases
of each. These are perhaps more appropriately thought of as the areas of
the regions formed by intersecting the unit sphere with the pyramid.
That is, to find the solid angle at the vertex of a figure, scale the
figure to be large enough, draw the unit sphere centered at that vertex,
then consider the "triangular" region formed where the sphere and
pyramid intersect. That area is the solid angle. It's well known to be
equal to the amount by which the sum of the angles exceeds pi (180 degrees).
These angles are in turn equal to the dihedral angles between the planes
of the sides of the pyramid.
[deletia -- djr]