On 7/15/07, Bill Gosper <gosper@alum.mit.edu> wrote:
WFL> normalised so that \sum_j (a_ij)^2 = 0 for i = 1,...,n, Now that was a normalization liberals can only dream of.
Oh dear. Don't listen to what I say --- listen to what I mean --- \sum_j (a_ij)^2 = 1, of course!
My turn: ... Is someone out there who can say if Todhunter has the solid angle from vertex angles formula
cos(c) + cos(b) + cos(a) + 1 (d1) 2 acos ---------------------------- ? a b c 4 cos(-) cos(-) cos(-) 2 2 2
In spherical trig. terms, this gives the area of a triangle in terms of its sides (rather than its angles) --- or effectively, the area of its polar / dual triangle in terms of its angles. I don't recall ever seeing such a formula, but I'm not a serious student of this topic. Warning: information about polar triangles is sparse on the web; and what is available is frequently plain wrong! Some of the confusion probably arises from the fact that a "spherical triangle" in this context is more properly regarded as a set of 8 triangles in congruent pairs. Todhunter's book is available for free download also from project Gutenberg, at http://www.gutenberg.org/files/19770/19770-pdf.pdf Fred Lunnon