Re: [Algorithms] decompose onto non-orthogonal vectors

[Peter D. <pd...@mm...>]

> >> how to you get scalars (u,v) such that u*a + v*b = p?
> >> Ie. decompose p onto a and b.
> >
> >Easy. dot the equation with a and b, correspondingly:
> >
> >u * (a dot a) + v * (b dot a) = p dot a;
> >v * (a dot b) + v * (b dot b) = p dot b.
> >
> >Solve the system.
>
> In the 2D case this is more complicated than it need be, but will give
> the correct solution (provided, of course that one knows dot product
> and how to solve a 2x2 system).
>
> In the 3D case, this will give the correct solution only if p is
> indeed in the plane of a and b.  It will not detect that there is no
> solution if p is not coplanar with a and b.

This works for any number of dimensions. Whenever a solution exists, the (u,
v) pair determined by this approach will have the property u * a + v * b =
p. (Easy to prove.)

--
Peter Dimov
Multi Media Ltd.