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

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.

Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

“…Therefore, the characteristic polynomial decomposes as f (λ) = (λ + a 2 )g(λ). We use Lemma 5, Lemma 7 and Lemma 12 as before, together with Theorem 11 to conclude that all the roots are negative in α (1). Note that g(λ) does not have any double root, but −a 2 could be a root of g(λ) (−a 2 and −b 2 are the only possible double roots without an associated tangency, see Theorem 11).…”

“…The previous algorithm just intends to illustrate the possibility of reasonable applications from Table 2. A deeper analysis in the development of algorithms should take care of the efficiency in the implementation and can follow the line of techniques already used to detect the positional relationship between conics or ellipsoids, we refer to [1,7] and references therein.…”

“…To prove assertion (2) we construct a path so that the sphere moves without intersecting the paraboloid. For example, for a general S and P, the sum of the paths 1], ensures that there is not contact between the surfaces and the center at the end of the path is located in the negative part of the OZ-axis (see Figure 3 (a)). Now, a similar argument to that given in assertion (1) applies to prove assertion (2).…”

“…There exists an extensive literature about contact detection between quadric surfaces (see [14] for a classical reference and references therein such as [21]), but one of the main tools relies on the use of a characteristic polynomial associated to the pencil of the quadrics to treat the collision detection problem. Indeed that was done in previous works considering conic curves (see [1,6,7,15]) and other quadric surfaces, especially ellipsoids, in Euclidean or Projective spaces (see [10,22,23]).…”

Complete Classification and Efficient Determination of Arrangements Formed by Two Ellipsoids

“…This condition is translated in [Jia et al 2011] in terms of more elementary computations which apply to continuous collision detection of two moving ellipsoids. Similar problem of characterizing the relative position of two ellipses has been considered by [Alberich-Carraminana et al 2017;Etayo et al 2006], where algebraic conditions are used.…”