[math-fun] Lie (circle) geometry

Is anybody interested in the Laguerre group of equilong transformations --- nowadays apparently subsumed, along with the Moebius group, into something called Lie geometry? I've looked in books by Cecil and by Coolidge, and at numerous web sites, but I'm not much nearer understanding what a Laguerre involution --- let alone a general equilong transformation --- actually _looks_ like! I could say (much) more, but I'll refrain until I find out if there's any takers out there.