# The Apollonius contact problem and Lie contact geometry

@article{Knight2005TheAC, title={The Apollonius contact problem and Lie contact geometry}, author={Robert d. Knight}, journal={Journal of Geometry}, year={2005}, volume={83}, pages={137-152} }

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… Expand

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#### References

SHOWING 1-6 OF 6 REFERENCES

Lie sphere geometry

- Physics
- 1992

In this chapter, we give Lie’s construction of the space of spheres and define the important notions of oriented contact and parabolic pencils of spheres. This leads ultimately to a bijective… Expand

A Treatise On The Circle And Sphere

- Mathematics, Philosophy
- 1916

The Circle in Elementary Plane Geometry: Fundamental definitions and notation Inversion Mutually tangent circles Circles related to a triangle The Brocard figures Concurrent circles and concyclic… Expand