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Largest Area Ellipse Inscribing an Arbitrary Convex Quadrangle

  • M. John D. Hayes
  • Zachary A. Copeland
  • Paul J. Zsombor-Murray
  • Anton Gfrerrer
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)

Abstract

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.

Keywords

convex quadrangle point and line ellipses pole point and polar line 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • M. John D. Hayes
    • 1
  • Zachary A. Copeland
    • 1
  • Paul J. Zsombor-Murray
    • 2
  • Anton Gfrerrer
    • 3
  1. 1.Department of Mechanical and Aerospace EngineeringCarleton UniversityOttawaCanada
  2. 2.Department of Mechanical EngineeringMcGill UniversityMontrealCanada
  3. 3.Institute for GeometryGraz University of TechnologyGrazAustria

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