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New descriptions of conics via twisted cylinders, focal disks, and directors

Apostol, Tom M. and Mnatsakanian, Mamikon A. (2008) New descriptions of conics via twisted cylinders, focal disks, and directors. American Mathematical Monthly, 115 (9). pp. 795-812. ISSN 0002-9890.

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Conics have been investigated since ancient times as sections of a circular cone. Surprising descriptions of these curves are revealed by investigating them as sections of a hyperboloid of revolution, referred to here as a twisted cylinder. We generalize the classical focus-directrix property of conics by what we call the focal disk-director property (Section 2). We also generalize the classical bifocal properties of central conics by the bifocal disk property (Section 5), which applies to all conics, including the parabola. Our main result (Theorem 5) is that each of these two generalized properties is satisfied by sections of a twisted cylinder, and by no other cures. Although some of these results are mentioned in Salmon's treatise [6] and a not by Ferguson [4], they are not widely known, and we go far beyond these earlier treatments.

Item Type:Article
Subject Keywords:conics; twisted cylinders; focal disks; directors; mathematics
Issue or Number:9
Record Number:CaltechAUTHORS:APOamm08
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13613
Deposited By: Arun Sannuti
Deposited On:13 May 2009 19:35
Last Modified:03 Oct 2019 00:41

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