


Paper's Title:
Ellipses Inscribed in Parallelograms
Author(s):
A. Horwitz
Penn State University,
25 Yearsley Mill Rd.
Media, PA 19063
U. S. A.
alh4@psu.edu
Abstract:
We prove that there exists a unique ellipse of minimal eccentricity, E_{I}, inscribed in a parallelogram, Ð. We also prove that the smallest nonnegative angle between equal conjugate diameters of $E_{I} equals the smallest nonnegative angle between the diagonals of Ð. We also prove that if E_{M} is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then E_{M} is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let Ð be any convex quadrilateral. In previous papers, the author proved that there is a unique ellipse of minimal eccentricity, E_{I}, inscribed in Ð, and a unique ellipse, E_{O}, of minimal eccentricity circumscribed about Ð. We defined Ð to be bielliptic if E_{I }and E_{O} have the same eccentricity. In this paper we show that a parallelogram, Ð, is bielliptic if and only if the square of the length of one of the diagonals of Ð equals twice the square of the length of one of the sides of Ð .
Paper's Title:
Ellipses of Maximal Area and of Minimal Eccentricity Inscribed in a Convex Quadrilateral
Author(s):
Alan Horwitz
Penn State University,
25 Yearsley Mill Rd., Media, Pa 19063
alh4@psu.edu
Url: www.math.psu.edu/horwitz
Abstract:
Let Ð be a convex quadrilateral in the plane and let M1 and M2 be the midpoints of the diagonals of Ð. It is well–known that if E is an ellipse inscribed in Ð, then the center of E must lie on Z, the open line segment connecting M1 and M2 . We use a theorem of Marden relating the foci of an ellipse tangent to the lines thru the sides of a triangle and the zeros of a partial fraction expansion to prove the converse: If P lies on Z, then there is a unique ellipse with center P inscribed in Ð. This completely characterizes the locus of centers of ellipses inscribed in Ð. We also show that there is a unique ellipse of maximal area inscribed in Ð. Finally, we prove our most signifigant results: There is a unique ellipse of minimal eccentricity inscribed in Ð.
Paper's 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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