Just an observation - why quote Wikipedia when Affine Space, Functions etc. are defined at mathworld.wolfram.com ?

Light dawns at last [I now suspect that I learnt this stuff as a student, and --- like everybody else, it seems --- promptly forgot it]. Klein's insight [in his Erlanger Programm presumably --- my German is rusty, and my eyesight no longer up for struggling with gothic Fraktur type] really is simple and elegant after all. For simplicity I'll consider real 3-space: the generalisation to n-space should be obvious. Question: what do the terms "affine" and "absolute" actually mean? Step(0): start with "Projective" geometry, defined in the familiar fashion by invariance under linear transformations PGL(4) acting via matrix product on points, represented homogeneously by 4-vectors [w, x, y, z] modulo the reals. Step(1): choose an arbitrary plane, the "plane at infinity", and a coordinate frame in which it is represented by 1.w + 0.x + 0.y + 0.z = 0 . "Affine" geometry is invariant under the subgroup fixing this plane: so the corresponding row of the matrix must equal [1, 0, 0, 0]. Step(2): choose an arbitrary quadric, the "absolute quadric", which can then be reduced to the equivalent form (w^2)/d + x^2 + y^2 + z^2 = 0 ; the geometry whose subgroup further fixes this quadric is: "Hyperbolic" when d < 0, so the quadric is real and nonsingular; "Elliptic" when d > 0, so the quadric is complex and nonsingular; "Euclidean" when d = 0, so the quadric degenerates to a conic at infinity [the associated polarity approaches a non-invertible limit as d -> 0]. Question: what practical use is affine geometry anyway? Well, apart from Klein's classification, not an awful lot --- which perhaps explains why the term has fallen into such disrepute. One nice theorem mentioned in Wikipedia is that under affine transformation, volumes remain in the same proportions --- this having been established in affine geometry, it follows immediately for elliptic, hyperbolic and Euclidean. Big deal! Proof: the volume of a simplex is given by (1/6) times the determinant of the 4 points at its corners, normalised so that w = 1. Under product with an affine matrix, the points remain normalised; and their determinant is multiplied by that of the matrix, a constant. Question: why was it so hard to get to the bottom of this? A veritable quagmire of confusing typos, ambiguities and omissions lurks in waiting for the unwary seeker after truth --- particularly one whose inbuilt compass guides him unerringly to tumble into one such pitfall after another. [Would that I could home in on my own errors so reliably!] Coxeter sect 10.94 p212 (uncharacteristically) muddles curvature with radius; Cederberg sect 4.12 p299 (incomprehensibly) muddles "affine" with "Euclidean"; Pedoe "Geometry" sect 80.2 p351 discusses the plane at infinity, but makes no mention anywhere of the absolute quadric; Wikipedia fails to mention fixing the plane at infinity at all; Mathworld mentions fixing infinity briefly, but gives no context. (Why didn't I quote Mathworld earlier? In a word, Google!) Question: if mathematicians cannot remember what "affine" means, why is it plastered gaily over the computing literature? I had to ponder his post for a while, but concluded that what James Cloos was attempting to articulate is that in computer graphics etc, "affine" has come to be used in a rather vague way as an antonym for projective ("rational"): to describe the representation of a transformation in a Cartesian fashion using GL(n), as opposed to homogeneously using GL(n+1). In this context the geometry is pretty well always Euclidean rather than affine, whichever representation is in use. Exceptions are mostly unintentional --- see eg. Ken Shoemake and Tom Duff "Matrix Animation and Polar Decomposition" Proc. Graphics Interface '92 (April 1992) 258--264 http://www.cs.wisc.edu/graphics/Courses/838-s2002/Papers/polar-decomp.pdf At this juncture I feel some empathy with Moliere's M. Jourdain, on discovering that he had been speaking prose all his life without knowing it. I don't know who invented "affine" --- Expect the b****r must be laffin'! And this, I'm sure we all sincerely hope, will be af-final word. Fred Lunnon