71 research outputs found
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
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
Sur les solutions périodiques du mouvement plan de libration des satellites et des planètes
Ce papier présente une étude analytique du mouvement plan de rotation des satellites (et des planètes) dans leurs mouvements orbitaux. Les trois familles des solution périodiques sont obtenues par la méthode du prolongement analytique de Poincaré. Ensuite, la stabilité de ces solutions périodiques est discutée, et les équations approchées des courbes limites de stabilité sont données jusqu'au quatrième ordre. This paper presents an analytical study of the rotational motion of the satellites (and the planets) in their orbital planes. The three families of periodic solutions are obtained by the method of analytical continuation as formulated by Poincaré. The stability of these solutions are analyzed, and the approximate equations of the transition curves are obtained to the fourth order.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42563/1/10569_2005_Article_BF01227807.pd
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