On 7/12/07, Bill Gosper <gosper@alum.mit.edu> wrote:
Circa Xmas '02 I sent Subj: more solid angles In my version of Macsyma's Geofuncs package is now (c1) solid_angle(a,b,c) 2 (cos(c) + cos(b) + cos(a) + 1) (d1) acos(-------------------------------------- - 1) (cos(a) + 1) (cos(b) + 1) (cos(c) + 1)
where a, b, and c are the ordinary angles at the trihedral vertex. Further simplified to
cos(c) + cos(b) + cos(a) + 1 (d1) 2 acos ---------------------------- . a b c 4 cos(-) cos(-) cos(-) 2 2 2
It seems too nice to be new, but an hour or so of (low bandwidth) Googling failed to turn it up. Here are some playings with it.
Whenever I think I might have discovered a new formula of this type, it turns out to be there already in Ian Todhunter's "Spherical Trigonometry".
... Is there a polyhedral analog of n-gon angle sum = (n-2) pi, presumably involving vertex angles, dihedrals, and maybe vertex solid angles? Disappointingly, for the regular tetrahedron, (c683) (%pi/3,solid_angle(%%,%%,%%))
5 (d683) 2 acos(---------) 3 sqrt(3)
(c684) dfloat(4*%)
(d684) 2.20514239373012d0
while for the tetrahedral corner of a cube, (c692) (%pi/2,solid_angle(%%,%%,%%)+3*solid_angle(%pi/4,%pi/4,%pi/3))
2 3 ------- + - sqrt(2) 2 %pi (d692) 6 acos(-------------------) + --- 2 %pi 2 2 sqrt(3) cos (---) 8
(c693) dfloat(%)
(d693) 2.59030705515727d0 I.e., the solid angles of the vertices don't even sum to a constant.
Which gives me yet another excuse to inflict some determinant geometry on everybody ... Although the notion of spherical measure is intuitively appealing (and additive in an obvious fashion) there's reason to suppose that it's not the most useful way to quantify a multidimensional angle. Given 3 planes in 3-space (or indeed n primes in n-space) with equations \sum_j a_ij x_j = 0 for i = 1,...,n, normalised so that \sum_j (a_ij)^2 = 0 for i = 1,...,n, consider the determinant S = |a_ij|. [I believe Coxeter calls this the "n-dimensional cosine" in Regular Polytopes; although "n-dimensional sine" might be more appropriate.] Now if the 3 planes in question are the differentials of a parametric surface P(u,v,w) with respect to the components of its homogeneous parameter vector (u,v,w), S turns out to be simply the Gaussian curvature at a general point on the surface; and the Gauss-Bonnet theorem gives the constant sum that Bill is looking for, in the special case that the surface is a polyhedron. I imagine that the analogous result holds in any number of dimensions [although I have not got around to checking this.] Compare the situation for spherical measure, where there is no elementary formula even for the volume of a spherical tetrahedron. Fred Lunnon