[Paper Review] Ellipses of minimal area and of minimal eccentricity circumscribed about a convex quadrilateral

Alan Horwitz|arXiv (Cornell University)|Jul 13, 2007

Point processes and geometric inequalities1 references4 citations

TL;DR

This paper resolves long-standing gaps in Steiner's characterization of the most nearly circular ellipse (minimal eccentricity) circumscribed about a convex quadrilateral. Using algebraic geometry and optimization, it proves the existence and uniqueness of both the minimal-eccentricity and minimal-area circumscribed ellipses, and introduces the novel concept of bielliptic quadrilaterals—those where the inscribed and circumscribed ellipses of minimal eccentricity share the same eccentricity, even when not bicentric.

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, D. 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 D. We also show that there exists an ellipse which passes through the vertices of D 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 D. Finally, we call a convex quadrilateral, D, 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.

Motivation & Objective

• To close gaps in Steiner's classical proof regarding the ellipse of minimal eccentricity circumscribed about a convex quadrilateral.
• To establish the existence and uniqueness of the ellipse of minimal area passing through the vertices of a convex quadrilateral.
• To introduce and characterize a new class of convex quadrilaterals—bielliptic quadrilaterals—where the minimal-eccentricity inscribed and circumscribed ellipses have equal eccentricity.
• To investigate the geometric relationship between inscribed and circumscribed ellipses in convex quadrilaterals, generalizing the bicentric circle case.

Proposed method

• Derives the general conic equation for ellipses passing through the four vertices of a convex quadrilateral in an oblique coordinate system.
• Uses algebraic conditions (positive definite quadratic form and non-degeneracy) to define the parameter space of valid ellipses.
• Applies Lemma 1 to express the semi-axes and eccentricity of a conic in terms of its coefficients, enabling optimization over the parameter space.
• Proves uniqueness of the minimal-eccentricity ellipse via critical point analysis of the eccentricity function over the valid parameter interval.
• Introduces the concept of directional constants $M_1$ and $M_2$ associated with conjugate diameters and proves their existence and invariance across all circumscribed ellipses.
• Constructs a system of equations to find when equal conjugate diameters align with $M_1$ and $M_2$, proving existence of such a minimal-eccentricity ellipse.

Experimental results

Research questions

• RQ1 Does a unique ellipse of minimal eccentricity always exist for any convex quadrilateral?
• RQ2 Can the existence and uniqueness of the minimal-area circumscribed ellipse be rigorously proven?
• RQ3 Are there convex quadrilaterals for which the minimal-eccentricity inscribed and circumscribed ellipses have the same eccentricity, even if not bicentric?
• RQ4 Is there a geometric relationship between the minimal-eccentricity inscribed and circumscribed ellipses in a convex quadrilateral, generalizing the bicentric circle case?

Key findings

• There exists a unique ellipse of minimal eccentricity circumscribed about any convex quadrilateral, resolving a gap in Steiner’s original proof.
• There exists a unique ellipse of minimal area passing through the vertices of a convex quadrilateral, which is also the ellipse of minimal eccentricity.
• A new class of convex quadrilaterals—bielliptic quadrilaterals—is introduced, where the minimal-eccentricity inscribed and circumscribed ellipses have equal eccentricity.
• The paper proves the existence of a bielliptic convex quadrilateral that is not bicentric, demonstrating that biellipticity is a strictly weaker condition than bicentricity.
• A bielliptic trapezoid that is not bicentric is explicitly constructed, showing the concept extends beyond symmetric quadrilaterals.
• It is proven that no ellipse inscribed in a convex quadrilateral and no ellipse circumscribed about it can share the same center, a nontrivial geometric constraint.