From the matrix definition in wikipedia, the freedom of the n-space affine group [its dimension as a manifold in projective matrix space] equals n^2 + n = ( (n+1)^2 - 1 ) - n , suggesting that geometrically, the affine group is simply the projective subgroup fixing the co-point (hyperplane) at infinity. So why would it not be defined this way in the first place?
However the first interpretation seems inconsistent with Cederberg, who defines her "affine" group to fix an "absolute" quadric (whatever that precisely means). The constraint would reduce the freedom to (at most) ( (n+1)^2 - 1 ) - ( (n+2)(n+1)/2 - 1 ) = (n+1)n/2 , consistent with the Euclidean group --- as I originally assumed! Then a whole new can of worms is opened below --- a third proposal apparently inconsistent with either of the above. I rest my case --- affine faffine! WFL On 3/9/11, James Cloos <cloos@jhcloos.com> wrote:
... If I read that correctly, you ask, in effect, for an example of what can be done with rational B-Splines but not by non-rational (to put the q in a form understandable by (computer) graphics artists. Yes?
The classic answer is to transform a model of some object in a manner consistant with what it would look like in real life. A railroad vanishing to the horizon is a typical choice.
The opening text of Star Wars also is a common example.
In short, in the graphics world, affine refers to transforms which can be done with non-rational polynomial parametric models in R²P or R³P.
I don't know whether that matches the orignal meaning of affine, but that *is* what it has come to mean in the graphics world.
-JimC