On 3/4/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
Since then, I have encountered numerous rhinoceros' pancreases (or whatever the plural is). ...
However, some more specific clarification now seems to called for --- I must admit I'm royally confused about this entire issue, and it would appear that I may not be alone. My absolute quadric will be c w^2 + x^2 + y^2 + z^2 = 0 . Whether the ground field is real or complex is essentially irrelevant. Reference (A) is Judith N. Cederberg "A course in modern geometries" (2001), section 4.12 on page 299 She discusses 2-space rather than 3-space, and her c corresponds to my 1/c (ouch!); translated, she actually claims that c = oo gives _affine_ (oof!), rather than Euclidean geometry. 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! Reference (B) is H.S.M.Coxeter "Non-Euclidean Geometry" sect 10.94 page 212 He says more cautiously that as c -> 0, the associated polarity degenerates [sect 9.5 p186 ff.], and there is a continuous transition from elliptic to hyperbolic via Euclidean; however as far as I can tell, he nowhere actually claims that fixing this (distinct) quadric determines the symmetry group. While fixing c = 0 obviously excludes translations, it's harder to establish whether this behaviour is a continuous limit ... The more I try to get to grips with what initially appeared to be a simple and elegant concept, the more unsatisfactorily slippery it appears to become! Fred Lunnon