A Completely Generalized and Algorithmic Approach to Identifying the Maximum Area Inscribing Ellipse of an Asymmetric Convex Quadrangle

Abstract

In this thesis, a novel method is presented whereby the maximum area inscribing ellipse, subject to a set of four linear constraints within a two variable system is automatically generated. Two methods currently exist with varying degrees of utility which provide solutions to generate the maximum area inscribing ellipse within a convex quadrangle: projective transformation of the unit square and unit circle to arbitrary parallelograms and trapezoids and corresponding area maximizing inscribing ellipses; as well as a method whereby a non-metric affine coordinate system is constructed for the identification of the area maximizing inscribing ellipse. Problematically, neither of these methods contain a general metric approach for all inscribing ellipses within any given asymmetric convex quadrangle. An algorithm is developed herein using the projective extension of the Euclidean plane which will always generate the entire one-parameter family of inscribing ellipses, and directly identify the area maximizing one of any given convex quadrangle, within a metric space.Given four bounding points, no three of which are collinear, four line equations are generated which describe the convex quadrangle. Alongside the definition of a specific polar point, these five constraints identify a pencil of inscribing line conics, which is then transformed into its point conic dual for visualisation and plotting. The pencil of point conics then has its area optimised with respect to the value of its polar point, at which juncture the maximum area inscribing ellipse may be identified from the pencil of inscribing conics.ii Acknowledgments First, I would like to acknowledge the efforts of Professor Hayes in supplying me with the guidance required to produce a thesis that I am genuinely proud of. Secondly, any of the individuals in my office who endured my geometric ramblings, as well as their perpetual comedic relief and moral support. Kendal Wilson, for enduring the tedium of revising the first draft of this thesis, as well as Fred Noe for invaluable contributions to the creativity of my writing process.iii