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.
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Knight, R.D. The Apollonius contact problem and Lie contact geometry. J. geom. 83, 137–152 (2005). https://doi.org/10.1007/s00022-005-0009-x
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DOI: https://doi.org/10.1007/s00022-005-0009-x