Re: [Algorithms] decompose onto non-orthogonal vectors

[Will P. <wi...@cs...>]

I prefer this answer to mine from a theory standpoint, because it
expresses all of the gotchas that I had to list explicitly... code wise, I
wouldn't want to actually make a matrix to solve it (i'm sure you wouldn't
either).

One thing I didn't mention, and I haven't proven it to myself
mathematically yet:  if vectors a and b are linearly independent (which
they don't seem to be from the original thread author's drawing), then
there's only one solution for u and v. For example, in the "normal"
orthonormal basis that we use ((1,0,0)(0,1,0)(0,0,1), there's only one
linear combination of the basis that will give a point p.

i'm assuming the basis vectors span the subspace for those mathematically
inclined people who want to jump on me. :)

Will

----
Will Portnoy
http://www.cs.washington.edu/homes/will

On Fri, 14 Jul 2000, Jonathan Blow wrote:

> "Discoe, Ben" wrote:
> > 
> > I thought it would be a simple problem, but all usual sources failed to
> > answer, so perhaps it will be obvious to someone on this list.
> > 
> > Given a point p and two unit vectors a and b like this:
> > 
> >       b
> >      /
> >    v/----p
> >    /    /
> >   /    /
> >  /    /
> > ---------a
> >     u
> > 
> > how to you get scalars (u,v) such that u*a + v*b = p?
> > Ie. decompose p onto a and b.
> 
> This is one of those "fundamental linear algebra things"
> that Ron and I were just talking about.
> 
> Your original coordinates of 'p' are in the coordinate
> system that we are used to, that is, two basis vectors
> that are orthonormal.  We will call this space R.
> 
> You want the coordinates of 'p' in a system in which
> the vectors are unit, but not orthogonal.  We will call
> this space S.
> 
> Just build a transformation matrix from R to S, and
> call that matrix T.  Then p' = Tp and you are done.
> 
> For instructions on how to make a matrix that goes from
> one space to another, given the basis vectors of each 
> space, consult any one of a trillion graphics books.
> 
>    -J.
> 
> 
> > I promise i consulted an academic
> 
> What kind of academic was that?
> 
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