The Apollonius contact problem and Lie contact geometry

Abstract

A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries-Laguerre geometry and oriented Möbius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model. (2000): 50D45, 51B25, 51N10. Mathematics Subject Classification

A Euclidean Area Theorem via Isotropic Projection

“…We thus obtain a 1-1 correspondence between singular planes in M 3 and spears in E, called the isotropic projection of the singular planes of M 3 onto the spears of E (or of the spears of E onto the singular planes of M 3 ). For more on isotropic projection, the reader is referred to [4] and [5].…”

Simple and branched skins of systems of circles and convex shapes

“…Others prefer simplicity of skinning algorithms; our approach belongs to this category. For instance, an elegant and purely geometric method based on constructing Apollonius circles [22] has been used recently for finding skin touching points on input circles/spheres [4]. Furthermore, different algorithms use different admissible input data sets [23].…”

Geometric constructions on cycles in $\mathbb{R}^n$

“…In [6], Lie geometry was used to analyze the existence and properties of solutions of geometric constructions associated to the Apollonius construction in R n . In [9] the special case of the Apollonius problem in the plane was considered. In [7], simple algorithms for symbolic solutions of a number of such geometric constructions were given.…”