A Novel Approach for Ellipsoidal Outer-Approximation of the Intersection Region of Ellipses in the Plane

TL;DR

This paper introduces a new method for tightly approximating the intersection of multiple ellipses in 2D by constructing a convex polygon and then fitting the smallest enclosing ellipse, improving accuracy over traditional methods.

Contribution

The paper presents a novel algorithm that efficiently constructs a tight outer-approximation of ellipse intersections using tangent lines and convex optimization, outperforming conventional techniques.

Findings

The proposed method yields a tighter approximation than traditional techniques.

It effectively detects and handles non-convex or unbounded intersection regions.

Numerical experiments confirm improved approximation accuracy with manageable computational cost.

Abstract

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 half-planes is obtained, whose intersection forms a polygon. To find the polygon more efficiently, the points are given an order and the intersection of the half-planes corresponding to every two neighbouring points is calculated. If the 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.