On 3/6/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Now I had belatedly noticed that fixing the (degenerate) quadric c = oo permits dilations, as well as rotations; but she seems not to have realised that it also permits translations, which are not affine! ...
A well-wisher who prefers to remain anonymous has politely pointed out that I don't know my affine from my elbowine. Now I think about it, I have actually complained on several past occasions of not knowing what other people meant by this term (privately suspecting that they didn't know either) --- this time I carelessly assumed it must refer to the Cartesian linear group. But now he refers me to the fount of all wisdom, at which I may bathe and finally become cleansed (perhaps) --- http://en.wikipedia.org/wiki/Affine_transformation [I have been previously been taken sniffily to task for citing wikipedia --- but have to aver that they seem pretty reliable, if often unpolished and on the terse side, regarding mathematical topics I feel competent to judge.] An "affine transformation", it sez 'ere, is x -> A x + b where A, b are fixed matrix, vector and x is a (finite dimensional) vector variable. Ah: I knew there was something dodgy about this notion --- these transformations are not closed under composition! So that has to be patched up by immediately abandoning the Cartesian coordinates motivating it, and moving homogeneous coordinates instead. Which perhaps we should have been using all along. But then, of course, we'd never have even considered coming up with such a clunky definition in the first place. What conceivable application does it have? Has anybody ever come across a problem which could be solved using affine geometry, rather than Cartesian, Euclidean, projective, elliptic, hyperbolic ... you know, all the intuitively reasonable options? A brave attempt is made to refute such intemperate criticism at http://en.wikipedia.org/wiki/Affine_geometry which quotes in particular the theorem about the concurrency of the medians of a triangle. I find this example unconvincing: surely it is essentially (a special case of) a theorem of projective geometry? I advise all you youngsters out there to cultivate a deep suspicion of anyone using this term in the future! Including me. Especially me. Fred Lunnon