Duality and Inscribed Ellipses
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Duality and Inscribed Ellipses Mahesh Agarwal1 · John Clifford1 · Michael Lachance1
Received: 13 January 2015 / Revised: 27 May 2015 / Accepted: 8 June 2015 / Published online: 7 July 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract We give a constructive proof for the existence of inscribed families of ellipses in triangles and convex quadrilaterals; a unique ellipse exists in a convex pentagon. The techniques employed are based upon duality principles. One by-product of this approach is that the ellipse inscribed in a pentagon and that inscribed in its diagonal pentagon are algebraically related; they are as intrinsically linked as are the pentagon and its diagonal pentagon. Keywords
Linfield’s theorem · Ellipses
Mathematics Subject Classification
Primary 14H52 · 51E10
1 Introduction The goal of this paper is to show how duality of curves lends itself to investigating families of inscribed ellipses in convex n-gons. This question was inspired by a sequence
Dedicated to Professor Ed Saff on his 70th birthday, with admiration and gratitude Communicated by Doron Lubinsky.
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Michael Lachance [email protected] Mahesh Agarwal [email protected] John Clifford [email protected]
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Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI 48128, USA
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of articles by Kalman, Gorkin, and Minda and Phelps (see [2,3,6]). The result hinges upon a geometric insight in the case of quadrilaterals; in the case of pentagons a surprising connection between the unique ellipses inscribed, respectively, in the convex pentagon and its diagonal pentagon is exposed. Our main result is: Theorem 1.1 (Main theorem) The following statements concerning inscribed ellipses in non-degenerate convex n-gons are valid. • • • •
In triangles, there exists a unique two-parameter family of inscribed ellipses. In quadrilaterals, there exists a unique one-parameter family of inscribed ellipses. In pentagons, there exists precisely one inscribed ellipse. For n ≥ 6, there exist n-gons for which there are no inscribed ellipses; whenever there is an inscribed ellipse, it is unique.
The parameters referenced above may be taken to be a prescribed point of contact on any two sides of a triangle, or on any single side of a quadrilateral. The specificity concerning ellipses in the main theorem is in contrast to the general observation that three distinct tangent lines determine a two-parameter family of conics, four tangent lines a one-parameter family, and five a single conic. In our proof, we borrow heavily from duality arguments made by Linfield and Marden (see [4,5]). In those works, the authors were very much concerned about the location of critical points of complex functions. Here, we exploit some of their techniques and notation, primarily those of Linfield. The mere existence of inscribed ellipses follows from an application of Brianchon’s theorem (see [1]) to degenerate hexagons. We illustrate one such application in the case of triangles. Theorem 1.2 (Brianchon’s theorem) If a hexa
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