The Apollonius contact problem and Lie contact geometry
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.
Mathematics Subject Classification (2000).50D45 51B25 51N10
Key words.affine Minkowski space Apollonius cycle inversive geometry Laguerre geometry Lie geometry light cone Lorentz plane Möbius geometry spear
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