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. A sector with wedge angle v will have solid angle v/(2 pi) 4 pi = 2 v. But we can split it perpendicular to its straight edge to make two trihedrals with angles pi/2, pi/2, and v: (c2) 2*solid_angle(%pi/2,%pi/2,v) (d2) 2 v We can dissect a regular n-gonal pyramid of height h and base circumradius r into n congruent tetrahedra with trihedral components v, v, and 2 asin(sin(v) sin(pi/n)), where v = atan(r/h), i.e., half the apex angle. This gives, after much simplification, two expressions for the solid angle of the apex: 2 %pi 2 %pi 2 %pi 2 cos (---) (2 sin (---) cos(v) + cos(-----)) n n n (d3) n acos(--------------------------------------------- 2 %pi 2 2 %pi sin (---) cos (v) + cos (---) n n 2 %pi - cos(-----)) n n = acos((- 1) n - 1 /===\ 2 %pi 2 | | %pi %pi k 2 2 cot (---) n | | (cos(v) + cot(---) cot(-----)) n | | n n k = 1 (1 - ------------------------------------------------------)) 2 2 %pi n (cos (v) + cot (---)) n Shame on me. The product over a half-period of a polynomial in trigs of angles in arithmetic progression always simplifies, in this case, mondomiraculously: n 2 %pi acos((- 1) (1 - 2 sin (n acot(tan(---) cos(v))))) (!) n The (-1)^n looks very strange. But it seems to work, and the succeeding 1-2sin^2 expression oscillates without it. Letting n -> oo, the solid angle of a cone is 2 pi (1 - cos v), where, again, tan v = (base radius)/height. 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. --rwg steradian dentarias