Re: [Algorithms] decompose onto non-orthogonal vectors

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

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.
>> ...

>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.
>

Actually you don't need them expressed in an orthonormal basis, any
basis will do. (Although the solution is shorter when you have an
orthonormal basis and have learned the concept of "dot product", but
you might not get that until one of the later chapters.

>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.

Again, you don't need matrices (maybe chapter 3) but just high school
algebra.