G-B rather than G-P --- not the first time I have failed to spot a special case, bah! So how do these "Gaussian curvature" angles relate to (say) dihedral angles at edges, or to face angles? WFL On 5/25/15, Dan Asimov <dasimov@earthlink.net> wrote:
For a convex closed surface K in space, there is a possibly-multivalued map
f: K -> S^2
to the unit sphere given by each point p of K is sent to the set of points of S^2 representing directions that are outward normal to the set of affine hyperplanes in space that intersect K in the point p (and possibly others), but that intersect no interior points of K.
By pulling back the area measure dA from S^2 via f, this defines a measure
mu = f*(dA) on K,
and the total measure
mu(K) = area(S^2) = 4pi.
When K is a convex polyhedron, this measure mu is concentrated at the vertices. At each vertex v, mu({v}) is equal to the solid angle subtended by the cone dual to the local cone of K around v. (I.e., dual to the cone that is the union of the faces of K containing v — the set of lines that are "perpendicular" at the points of a small simple closed curve about v to the faces containing v, interpolating in the obvious way across edges.)
This whole thing may also be thought of as the limit of the Gauss-Bonnet theorem applied to increasingly good approximations to K by smooth surfaces.
——Dan
On May 25, 2015, at 11:09 AM, James Propp <jamespropp@gmail.com> wrote:
How many of you already know (or can figure out) what four solid angles associated with a general tetrahedron add up to 4 pi?
I've been told that George Polya, when asked, replied with the correct answer instantly (it was part of his geometric toolkit), but that few of his contemporaries, when polled in an informal survey, were aware of this three-dimensional analogue of the familiar fact about the angles of a triangle summing to pi. I'm guessing that things haven't changed much in the intervening half-century.
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