Computational Optimization and Applications

, Volume 69, Issue 2, pp 383–402

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

• Siamak Yousefi
• Xiao-Wen Chang
• Henk Wymeersch
• Benoit Champagne
• Godfried Toussaint
Article

## Abstract

In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional 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.

## Keywords

Computational geometry Convex optimization Ellipsoidal outer approximation Intersection of ellipses Intersection of half-planes Minimum volume enclosing ellipsoid

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## Authors and Affiliations

• Siamak Yousefi
• 1
• Xiao-Wen Chang
• 2
• Henk Wymeersch
• 3
• Benoit Champagne
• 1
• Godfried Toussaint
• 4
1. 1.Department of Electrical and Computer EngineeringMcGill UniversityMontrealCanada
2. 2.School of Computer ScienceMcGill UniversityMontrealCanada
3. 3.Department of Signals and SystemsChalmers University of TechnologyGothenburgSweden
4. 4.Faculty of ScienceNew York University Abu DhabiAbu DhabiUnited Arab Emirates