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)


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.


convex quadrangle point and line ellipses pole point and polar line 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boyd, S., Vandenberghe, A.: Convex Optimization. Cambridge University Press, Cambridge, England (2004)Google Scholar
  2. 2.
    Fichtenholz, G.M.: Differential und Integralrechnung II. VEB Deutscher Verlag der Wissenschaften, Altenburg, DE. (1964)Google Scholar
  3. 3.
    Fishback, W.T.: Projective and Euclidean Geometry. John Wiley & Sons, Inc., New York, N.Y., U.S.A. (1969)Google Scholar
  4. 4.
    Gfrerrer, A.: The Area Maximizing Inellipse of a Convex Quadrangle. Private communication (December 19, 2002)Google Scholar
  5. 5.
    Horwitz, A.: Finding Ellipses and Hyperbolas Tangent to two, three, or four Given Lines. Southwest Journal of Pure and Applied Mathematics 1(1) (2002)Google Scholar
  6. 6.
    Horwitz, A.: Ellipses of Maximal Area and of Minimal eccentricity Inscribed in a Convex Quadrilateral. Australian Journal of Mathematical Analysis and Applications 2(1) (January, 2005)Google Scholar
  7. 7.
    Horwitz, A.: Ellipses Inscribed in Parallelograms. Australian Journal of Mathematical Analysis and Applications 9(1) (January, 2012)Google Scholar
  8. 8.
    Klein, F.: Elementary Mathematics from an Advanced Standpoint: Geometry. Dover Publications, Inc., New York, N.Y., U.S.A. (1939)Google Scholar
  9. 9.
    Krut, S., Company, O., Pierrot, F.: “Velocity Performances Indexes for Parallel Mechanisms with Actuation Redundancy”. Robotica 22(2), 129–139 (2004)Google Scholar
  10. 10.
    Marquet, F., Krut, S., Pierrot, F.: ARCHI: A Redundant Mechanism for Machining with Unlimited Rotation Capacities. Proceedings of ICAR (2001)Google Scholar
  11. 11.
    Parker, V.W., Pryor, J.E.: Polygons of Greatest Area Inscribed in an Ellipse. The American Mathematical Monthly 51(4), 205–209 (April, 1944)Google Scholar
  12. 12.
    Plu¨cker, J.: Theorie der algebraischen Curven. Adolph Marcus, Bonn, Germany (1839)Google Scholar
  13. 13.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd Edition. Cambridge University Press, Cambridge, England (1992)Google Scholar
  14. 14.
    Salmon, G.: A Treatise on the Higher Plane Curves. Hodges and Smith, Dublin, Rep. of Ireland (1852)Google Scholar
  15. 15.
    Salmon, G.: A Treatise on Conic Sections, 6th edition. Longmans, Green, and Co., London, England (1879)Google Scholar
  16. 16.
    Strubecker, K.: Einfuhrung in die hohere Mathematik: Grundlagen. Oldenbourg, pp. 356–371 (1956)Google Scholar
  17. 17.
    Yoshikawa, T.: Manipulability of Robotic Mechanisms. International Journal of Robotics Research 4(2) (1985)Google Scholar

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

Personalised recommendations