Largest Area Ellipse Inscribing an Arbitrary Convex Quadrangle

M. John D. Hayes*, Zachary A. Copeland, Paul J. Zsombor-Murray, Anton Gfrerrer

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


A novel algorithm is presented which employs a projective extension of the Euclidean plane to identify the entire one-parameter family of inscribing ellipses, subject to a set of four linear constraints in the plane of the pencil, and directly identifies the area maximising one given any convex quadrangle. In the algorithm, four specified bounding vertices, no three collinear, determine four line equations describing a convex quadrangle. Considering the quadrangle edges as four polar lines enveloping an ellipse, together with one of the corresponding pole points on the ellipse, we define five bounding constraints on the second order equation revealing a description of the pencil of inscribing line conics. This envelope of line conics is then transformed to its point conic dual for visualisation and area maximisation. The ellipse area is optimised with respect to the single pole point and the maximum area inscribing ellipse emerges.

Original languageEnglish
Title of host publicationMechanisms and Machine Science
PublisherSpringer Netherlands
Number of pages10
Publication statusPublished - 1 Jan 2019

Publication series

NameMechanisms and Machine Science
ISSN (Print)2211-0984
ISSN (Electronic)2211-0992


  • convex quadrangle
  • point and line ellipses
  • pole point and polar line

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering

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