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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, M1 and M2, 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 M1 and M2,
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 M1 and M2. 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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