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.