On 7/13/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
Which gives me yet another excuse to inflict some determinant geometry on everybody ...
As well as an opportunity for my personal misprint demon to reduce my peroration to wombat's do's. This should have read:
Given 3 planes in 3-space (or indeed n primes in n-space) with equations \sum_j a_ij x_j = b_i, for i = 1,...,n, normalised so that \sum_j (a_ij)^2 = 0 for i = 1,...,n, consider the determinant S = |a_ij|.
Now if the 3 planes in question are the differentials of the tangent plane 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.
Fred Lunnon