On 12/12/05, dasimov@earthlink.net <dasimov@earthlink.net> wrote:

The Moebius group PSL(2,C) of conformal automorphism of S^2 is pretty darn fascinating, so whatever this is about sounds promising.

Can you tell us at least a bit more about what these equilong thingies are (even if the def. is somewhat opaque)?

--Dan

Oh lor', I was rather hoping somebody was going to tell me ... As in Moebius geometry, the basic geometric objects in Laguerre geometry are "cycles", including in 2-D, circles, lines, points; or in 3-D, spheres, planes, points, etc. However, unlike Moebius, cycles are (in general) "oriented" --- think of interiors or exteriors of discs and half-planes, rather than circles and lines. [This sounds a rather trivial distinction at first, but patience ...] "Equilong" (ghastly word --- why not say "commensal"?) transformations preserve cycles, and distance: specifically, the length of that common tangent cone which joins a pair of spheres "compatibly", its interior touching an inside-oriented sphere, its exterior touching an outside- oriented sphere. The catch is that points_ are not preserved as such: only cycles. Neither are loci preserved, so that two cycles with the same locus but distinct orientations may transform to distinct loci. The fundamental generators of the group [corresponding to reflections in lines in 2-D or planes in 3-D for isometries] are "Laguerre" involutions, the definition of which is a tad unfriendly: given a base-line K in 2-D and scalar "power" p, LI(K,p) transforms an arbitrary (directed) line L into the line M concurrent with L and K, such that the half-angles u,v between L,M resp. and K satisfy tan(u/2)tan(v/2) = p. The special cases when L and K are parallel or anti-parallel require a limiting argument. [I can supply a Maple procedure which does the business in ordinary 2-D homogeneous line coordinates L1.x + L2.y = -L0: I mention this because it took a good deal of work to make reasonably clean and correct!]. A spectacular example of an equilong transformation takes a polygon and forms its offset by a given margin in both directions: it's an instructive exercise to express this transformation as a product of (the minimum number of) involutions! Some references: Besides numerous XIX-th century tomes in German, the only books in English I know of are T.~E.~Cecil \sl Lie Sphere Geometry \rm Springer (1992). J.~L.~Coolidge \sl A Treatise on the Circle and the Sphere \rm Clarendon Press (1916); Chelsea (1971). Papers that seem relevant, but only tweak the curtain slightly: Edward Kasner, John De Cicco \sl Equilong and Conformal Transformations of Period Two \rm Proc Natl Acad Sci U S A. (1940) 26(7): 471–476; online at http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1078212 B.~J.~Zlobec \& N.~M.~Kosta \sl Geometric Constructions on Cycles \rm Rocky Mt. J. Math. \bf 34 (2004) 1565--1585; along with at least two more very similar; also online. Two papers which I haven't managed to see yet, by J.~F.~Rigby and I.~M.~Yaglom in C.~Davis, B.~Gr\"unbaum, F.~A.~Scherk \sl The Geometric Vein, the Coxeter Festschrift \rm Springer (1981).