The Apollonius contact problem and Lie contact geometry (Q811824)

The paper essentially deals with the Apollonius problem in the classical real Laguerre plane, that is, determinining the number of circles that touch three given points or circles in the geometry of plane sections of an elliptic cylinder in 3-dimensional real affine space. Of course the solution to this problem is well known and the problem has been comprehensively dealt with more generally, for example, for all 2- and 4-dimensional circle planes in a unified way by \textit{A. E. Schroth} [Topological circle planes and topological quadrangles, Longman (1995; Zbl 0839.51013), Chapter 7], by using associated Lie geometries. In his approach the author uses two models of the real Laguerre plane, the spear-cycle model in the extended Euclidean plane and, what he calls, the affine Minkowski model on the 3-dimensional real affine space endowed with a metric of signature 1, which is the affine part of the classical antiregular generalized quadrangle associated with the Laguerre plane. Since the latter model does not represent the full Lie geometry of the Laguerre plane, the author has to distinguish between the different forms of Lie cycles in his models (spears, oriented circles, finite points or the additional point at infinity in the spear-cycle model), when they touch each other and how they are transformed. The result is a classification of Lie triads, that is, triples of Lie cycles. The author shows that the class of a Lie cycle is invariant under Lie transformations and that there are precisely five classes a Lie triad can have. Furthermore, the number of solutions to the Apollonius problem for a Lie triad (0, 1, 2 or a continuum of solutions) is uniqely determined by the class of the Lie triad. The classification is achieved by a detailed analysis of which Lie triads can be taken by Möbius, Laguerre or Lie transformations to a Laguerre triad (triples comprised of oriented circles or finite points in the spear-cycle model) with given signature.