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New descriptions of conics via twisted cylinders, focal disks, and directors. (English) Zbl 1172.51009

The article contains a thorough study of two characterizations of conics in the Euclidean plane that generalize their well-known focal properties. In the usual focal definitions of conics, focal points are replaced by circles and distances by tangent distances (square roots of powers) with respect to these circles. The following two main results can be stated:
1)
The locus of all points in the Euclidean plane such that ratio of the distance to a fixed line and tangent distance to a given circle is constant is a conic section.
2)
The locus of all points in the Euclidean plane such that either the sum or absolute difference of tangent distances to two given circles is constant is a conic section.
The converse of these statements is also true. In fact, any real circle in double contact with the given conic can serve as a focal disc.
These results are not new and can be traced back to the 19th century, see for example E.-E. Bobillier [Correspondance Mathématique et Physique III, 270–274 (1827)], J. Steiner [J. Reine Angew. Math. 45, 189–211 (1853; ERAM 045.1227cj)] or G. Salmon [A treatise on conic sections. 6th ed. New York, N. Y.: Chelsea Publishing Company (1970; Zbl 0211.24002)]. The author’s treatment of this topic is, however, noteworthy. They present a generalization of the classic Dandelin proof for the usual focal properties of conics. The cone of revolution is replaced by a hyperboloid of revolution and the inscribed spheres are allowed to intersect the plane of the conic. Only in this setting the proofs are valid for all combinations of conics and focal disks, in particular for the “abnormal” configuration of a hyperbola with double tangent circles in its exterior.
The article contains a careful discussion of many special cases and numerous telling illustrations.
Versions of the main results for the elliptic and hyperbolic plane can be found in [J. Kaczmarek and C. Pretki, Zesz. Nauk., Geom. 21, 41–50 (1995; Zbl 0868.51026) and H.-P. Schröcker, J. Geom. Graph. 12, No. 2, 161–169 (2008; Zbl 1177.51016)].

MSC:

51M04 Elementary problems in Euclidean geometries
51N20 Euclidean analytic geometry
97G70 Analytic geometry, vector algebra (educational aspects)
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