Jorge Caravantes

Computing the topology of an arrangement of quartics

We analyze how to compute in an efficient way the topology of an arrangement of quartic curves. We suggest a sweeping method that generalizes the one presented by Eigenwillig et al. for cubics. The proposed method avoids working with the roots of the involved resultants (most likely algebraic numbers) in order to give an exact and complete answer. We only treat in detail the cases of one and two curves because we do not introduce any significant variation in the several curves case with respect to Eigenwilligs paper.

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

Improving the topology computation of an arrangement of cubics

This paper is devoted to improve the efficiency of the algorithm introduced in [A. Eigenwillig, L. Kettner, E. Schomer, N. Wolpert, Exact, efficient and complete arrangement computation for cubic curves, Computational Geometry 35 (2006) 36-73] for analyzing the topology of an arrangement of real algebraic plane curves by using deeper the well-known geometry of reducible cubics instead of relying on general algebraic tools.

Analyzing group based matrix multiplication algorithms

We review different group based algorithms for matrix multiplication and discuss the relations between the combinatorial properties of the used group and the complexity of these algorithms. We introduce a variant of an algorithm based on the ideas exposed in [4] well-adapted for experimentation. Finally we show how this approach can also be used for matrix multiplication over a field with characteristic different from 2.

On the Interference Problem for Ellipsoids: Experiments and Applications

The problem of detecting when two moving ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc. By analysing symbolically the sign of the real roots of the characteristic polynomial of the pencil defined by two ellipsoids \(\mathcal {A}\) and \(\mathcal {B}\) we use and analyse the new closed formulae introduced in [9] characterising when \(\mathcal {A}\) and \(\mathcal {B}\) overlap, are separate and touch each other externally for determining the interference of two moving ellipsoids. These formulae involves a minimal set of polynomial inequalities depending only on the entries of the matrices A and B (defining the ellipsoids \(\mathcal {A}\) and \(\mathcal {B}\)), need only to compute the characteristic polynomial of the pencil defined by A and B and do not require the computation of the intersection points between them. This characterisation provides a new approach for exact collision detection of two moving ellipsoids since the analysis of the univariate polynomials (depending on the time) in the previously mentioned formulae provides the collision events between them.

Solving positioning problems with minimal data

Global and local positioning systems (LPS) make use of nonlinear equations systems to calculate coordinates of unknown points. There exist several methods, such as Sturmfels’ resultant, Groebner bases and least squares, for dealing with this kind of equations. We introduce two methods for solving this problem with the aid of symbolic techniques relying on closed-form solutions for the solution set of a system of linear equations. We suppose the receiver just detects or chooses minimal data, i.e., four satellites in global positioning systems (GPS) or three stations in LPS. Both methods proceed by parameterizing the line joining two solution points to later solve a nonlinear univariate equation, either quadratic or with degree smaller than 6. The first one uses the Generalized Cramer Identities, which is a different presentation of the generalized Moore–Penrose inverse, and ends with a degree 6 univariate equation for GPS and a degree 4 univariate equation for LPS. The second one solves the system by dealing with a more geometric way, ending with a quadratic equation. Our approach covers all possible cases with a finite number of solutions, while Bancroft’s method cannot be applied when the four satellites, taking the clock bias as fourth coordinate, and the origin lay in the same hyperplane of

Computing the topology of an arrangement of parametric or implicit algebraic curves in the Lagrange basis

{\mathbb{R}}^{4}

Bertini-type theorems for formal functions in Grassmannians

R4, and the method by Grafarend and Shan fails when the four satellites are in the same plane in