


Paper Title:
Ellipses of Minimal Area and of Minimal Eccentricity Circumscribed About a Convex Quadrilateral
Author(s):
Alan Horwitz
Penn State University,
25 Yearsley Mill Rd.,
Media, PA 19063,
U.S.A
alh4@psu.edu
Abstract:
First, we fill in key gaps in Steiner's nice characterization of the most nearly circular ellipse which passes through the vertices of a convex quadrilateral, . Steiner proved that there is only one pair of conjugate directions, M_{1} and M_{2}, that belong to all ellipses of circumscription. Then he proves that if there is an ellipse, E, whose equal conjugate diameters possess the directional constants M_{1} and M_{2}, then E must be an ellipse of circumscription which has minimal eccentricity. However, Steiner does not show the existence or uniqueness of such an ellipse. We prove that there is a unique ellipse of minimal eccentricity which passes through the vertices of . We also show that there exists an ellipse which passes through the vertices of and whose equal conjugate diameters possess the directional constants M_{1} and M_{2}. We also show that there exists a unique ellipse of minimal area which passes through the vertices of . Finally, we call a convex quadrilateral, , bielliptic if the unique inscribed and circumscribed ellipses of minimal eccentricity have the same eccentricity. This generalizes the notion of bicentric quadrilaterals. In particular, we show the existence of a bielliptic convex quadrilateral which is not bicentric.
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