From: Deane Yang (deane_yang_at_[hidden])
Date: 2005-06-02 13:18:52
Andy Little wrote:
>
> Ideally the two expressions should be equivalent:
>
> 2/3 *A +1/3 * B == A + (1/3)*(B - A) = (2 * A + B ) / 3 etc
>
> If no multiplication or addition is allowed on points, this is not the
> case. (Welll.. its possible to work it at compile time, but I believe
> this is too restrictive) The points implementation is getting in the way
> of the expression. Expressions will fail for no good reason. OTOH These
> calculations work correctly if A and B are 'position-vectors'.
>
What is the difference between a "point" and a "position-vector"?
Unfortunately, it appears that everyone uses similar words but with
different specific meanings.
For me, a "point" is an element in n-dimensional flat affine space. A
"vector" is an element in the associated n-dimensional vector space
(sometimes called the "tangent space"). So the allowable operations are
the standard vector space operations for vectors (vector addition and
scalar multiplication) plus
point + vector -> point
This should be enough to do any computation that anybody would ever
need. And, unless computational speed is a real issue, I think it's much
clearer to always code one's computations in this form. In my mind, the
meaning of
A + (1/3)*(B-A) ("start at A and go 1/3 the distance towards B")
is much more transparent than
(2/3)A + (1/3)B ("rescale A by 2/3; rescale B by 1/3; add")
However, I do acknowledge that for intensive computations you might not
want to constrain yourself in this way.