Journal for Geometry and Graphics 12 (2008), No. 2, 161--169
Copyright Heldermann Verlag 2008
Double Tangent Circles and Focal Properties of Sphero-Conics
Hans-Peter Schröcker
Unit Geometry and CAD, University Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria
hans-peter.schroecker@uibk.ac.at
We give two proofs for the characterization of a sphero-conic as locus of points such that the absolute value of the sum or difference of tangent distances to two fixed circles is constant. The first proof is based on methods of descriptive and projective geometry, the second is purely algebraic in nature. In contrast to earlier results, our proofs remain valid in case of purely imaginary tangent distances (when the sphero-conic is enclosed by both circles). Minor modifications make the algebraic proof work in the hyperbolic plane as well.
Keywords: Spherical geometry, sphero-conic, double tangent circle, focal property, elliptic geometry, hyperbolic geometry.
MSC: 51M04; 51N05, 51N25
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