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

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)383-402
Number of pages20
JournalComputational Optimization and Applications
Volume69
Issue number2
DOIs
StatePublished - Mar 1 2018

Fingerprint

Outer Approximation
Intersection
Polygon
Half-plane
Set of points
Arc of a curve
Convex polygon
Restart
Convex optimization
Ellipse
Convex Optimization
Tangent line
Computational Cost
Numerical Experiment
Optimization Problem

Keywords

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

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

Cite this

A novel approach for ellipsoidal outer-approximation of the intersection region of ellipses in the plane. / Yousefi, Siamak; Chang, Xiao Wen; Wymeersch, Henk; Champagne, Benoit; Toussaint, Godfried.

In: Computational Optimization and Applications, Vol. 69, No. 2, 01.03.2018, p. 383-402.

Research output: Contribution to journalArticle

Yousefi, Siamak ; Chang, Xiao Wen ; Wymeersch, Henk ; Champagne, Benoit ; Toussaint, Godfried. / A novel approach for ellipsoidal outer-approximation of the intersection region of ellipses in the plane. In: Computational Optimization and Applications. 2018 ; Vol. 69, No. 2. pp. 383-402.
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