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Double tangent circles and focal properties of sphero-conics. (English) Zbl 1177.51016
Let \(S^2\) be the unit sphere of the real Euclidean \(3\)-space. A sphero-conic is the intersection of \(S^2\) by a quadratic cone whose vertex coincides with the center of \(S^2\).
The author gives two proofs for each of the following two theorems: 5mm
The locus of all points on \(S^2\) such that the absolute sum or difference of tangent distances to two fixed circles \(K_1\), \(K_2\) is constant is a sphero-conic \(C\) with double tangent circles \(K_1\), \(K_2\); the centers of \(K_1\) and \(K_2\) lie on the same axis of \(C\).
If \(C\) is a sphero-conic on \(S^2\) and \(K_1,K_2\subset S^2\) are double tangent circles to \(C\) with centers on the same axis of \(C\), then the absolute sum or difference of tangent distances from points of \(C\) to \(K_1\) or \(K_2\) is constant.
Both proofs cover also the case of purely imaginary tangent distances. The first proof is based on concepts of descriptive and projective geometry. The second proof is algebraic and inspired by G. Salmon [A treatise on conic sections. Chelsea Publishing Company, New York, 6th ed. (1970; Zbl 0211.24002)] who proved the plane version of the two theorems above; this second proof can easily be extended to the hyperbolic case.
Four excellent figures accompany the considerations of the article.

51M09 Elementary problems in hyperbolic and elliptic geometries
51N05 Descriptive geometry
51N25 Analytic geometry with other transformation groups
51M04 Elementary problems in Euclidean geometries
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