Re: [Algorithms] decompose onto non-orthogonal vectors

[<ro...@do...>]

Reprise:

In the 3D case:

If  you are certain that p lies in the plane spanned by a and b, then
Peter Dimov's solution is the best.  (assuming you understand dot
products and 2x2 linear systems).

And, in fact,  that is exactly what I use in my code to find the
intersection of a ray with a triangle with given vertices V0, V1, V2:
First find the intersection P of the ray with the plane of the
triangle.  Refer everything, i.e. the intersection point and two of
the vertices, to any one vertex V0, so that, to cast it into Ben's
notation, p = P - V0, a=V1-V0 and b=V2-V0, and we are certain that p
lies in the plane through V0 spanned by a and b.  Form the dot
products as Peter indicates, and solve the resulting 2x2 system for u
and v.  The result V0 + ua + vb will lie in the triangle if and only
if 0<=u, 0<=v and u+v<=1.

But if you do not know a priori that p is a linear combination of a
and b, then the only way to determine that (finding u and v in the
process) is by Gaussian elimination on the underdetermined system as I
described. Of course, you can organize the Gaussian elimination
efficiently by thinking of it as row operations on the matrix of the
system, rather than elementary operations on the equations.  The
matrix approach is simply a tabular organization of the manipulation
of the equations themselves.