### 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 language | English (US) |
---|---|

Pages (from-to) | 383-402 |

Number of pages | 20 |

Journal | Computational Optimization and Applications |

Volume | 69 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2018 |

### Fingerprint

### 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

*Computational Optimization and Applications*,

*69*(2), 383-402. https://doi.org/10.1007/s10589-017-9952-3

**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.

Research output: Contribution to journal › Article

*Computational Optimization and Applications*, vol. 69, no. 2, pp. 383-402. https://doi.org/10.1007/s10589-017-9952-3

}

TY - JOUR

T1 - A novel approach for ellipsoidal outer-approximation of the intersection region of ellipses in the plane

AU - Yousefi, Siamak

AU - Chang, Xiao Wen

AU - Wymeersch, Henk

AU - Champagne, Benoit

AU - Toussaint, Godfried

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

KW - Computational geometry

KW - Convex optimization

KW - Ellipsoidal outer approximation

KW - Intersection of ellipses

KW - Intersection of half-planes

KW - Minimum volume enclosing ellipsoid

UR - http://www.scopus.com/inward/record.url?scp=85031902644&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031902644&partnerID=8YFLogxK

U2 - 10.1007/s10589-017-9952-3

DO - 10.1007/s10589-017-9952-3

M3 - Article

VL - 69

SP - 383

EP - 402

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 2

ER -