[boost] Re: Probing interest in fixed dimension matrix class

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.