A novel approach for ellipsoidal outer-approximation of the intersection region of ellipses in the plane

In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional (2D) space is proposed. First, the vertices of a tight polygon that contains the convex intersection of the ellipses are found in an efficient manner. To do so, the intersection points of the ellipses that fall on the boundary of the intersection region are determined, and a set of points is generated on the elliptic arcs connecting every two neighbouring intersection points. By finding the tangent lines to the ellipses at the extended set of points, a set of halfplanes is obtained, whose intersection forms a polygon. To find the polygon more efficiently, the points are given an order and the intersection of the halfplanes corresponding to every two neighbouring points is calculated. If the Funding for this work was provided in parts by research grants from the Natural Sciences and Engineering Research Council of Canada S. Yousefi Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada E-mail: [email protected] X.-W. Chang School of Computer Science, McGill University, Montreal, QC, Canada E-mail: [email protected] H. Wymeersch Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden E-mail: [email protected] B. Champagne Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada E-mail: [email protected] G. Toussaint Computer Science Program, New York University Abu Dhabi, United Arab Emirates E-mail: [email protected] 2 Siamak Yousefi et al. polygon is convex and bounded, these calculated points together with the initially obtained intersection points will form its vertices. If the polygon is non-convex or unbounded, we can detect this situation and then generate additional discrete points only on the elliptical arc segment causing the issue, and restart the algorithm to obtain a bounded and convex polygon. Finally, the smallest area ellipse that contains the vertices of the polygon is obtained by solving a convex optimization problem. Through numerical experiments, it is illustrated that the proposed technique returns a tighter outer-approximation of the intersection of multiple ellipses, compared to conventional techniques, with only slightly higher computational cost.