Publication

New algebraic conditions for the identification of the relative position of two coplanar ellipses

Journal Article (2017)

Journal

Computer Aided Geometric Design

Doc link

http://dx.doi.org/10.1016/j.cagd.2017.03.013

File

Download the digital copy of the doc pdf document

Abstract

The identification of the relative position of two real coplanar ellipses can be reduced to the identification of the nature of the singular conics in the pencil they define and, in general, their location with respect to these singular conics in the pencil. This latter problem reduces to find the relative location of the roots of univariate polynomials. Since it is usually desired that all generated expression are algebraic to simplify further analysis, including the case in which the ellipses undergone temporal variations, all recent methods available in the literature rely mathematical tools such as Sturm-Habicht sequences or subresultant sequences. This paper presents an alternative based on more elementary tools which results in a binary decision tree to classify the relative location of two ellipses in 12 different classes. The decision at each node is taken based on the sign of a set of algebraic/rational expressions on the ellipses coefficients, the most complex of them being third and second order polynomial discriminants.

Categories

automation, pattern recognition.

Author keywords

Ellipses, pencils of conics, interference detection, positional relationships.

Scientific reference

M. Alberich-Carramiñana, B. Elizalde and F. Thomas. New algebraic conditions for the identification of the relative position of two coplanar ellipses. Computer Aided Geometric Design, 2017, to appear.