TY - CHAP

T1 - Largest Area Ellipse Inscribing an Arbitrary Convex Quadrangle

AU - Hayes, M. John D.

AU - Copeland, Zachary A.

AU - Zsombor-Murray, Paul J.

AU - Gfrerrer, Anton

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - convex quadrangle

KW - point and line ellipses

KW - pole point and polar line

UR - http://www.scopus.com/inward/record.url?scp=85067545468&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-20131-9_24

DO - 10.1007/978-3-030-20131-9_24

M3 - Chapter

AN - SCOPUS:85067545468

T3 - Mechanisms and Machine Science

SP - 239

EP - 248

BT - Mechanisms and Machine Science

PB - Springer Netherlands

ER -