Section: Other Grants and Activities

Bilateral actions

Associated team CALAMATA

Participants : Ioannis Emiris, Athanasios Kakargias, Bernard Mourrain [ contact person ] , Nikos Pavlidis, Jean-Pierre Técourt, Elias Tsigaridas, Michael Vrahatis.

The team of Geometric and Algebraic Algorithms at the National University of Athens, Greece, has been associated with GALAAD since 2003. See its web site .

This bilateral collaboration is entitled CALAMATA (CALculs Algebriques, MATriciels et Applications). The Greek team (http://www.di.uoa.gr/~erga/ ) is headed by Ioannis Emiris. The focus of this project is the solution of polynomial systems by matrix methods. Our approach leads naturally to problems in structured and sparse matrices. Real root isolation, either of one univariate polynomial or of a polynomial system, is of special interest, especially in applications in geometric modeling, CAGD or nonlinear computational geometry. We are interested in computational geometry, actually, in what concerns curves and surfaces.

In 2006, we had the visit of B. Mourrain, J.P. Pavone and J. Wintz to Athens, to work on univariate polynomial subdivision solvers and arrangement problems. M. Vrahatis and K. E. Parsopoulos visited GALAAD, and worked on Particule Sworm Optimisation methods with applications to geometric and algebraic problems.

NSF-INRIA collaboration

Participants : Laurent Busé, André Galligo, Mohamed Elkadi, Bernard Mourrain [ contact person ] , Jean-Pierre Técourt.

The objective of this collaboration between GALAAD and the Geometric Modeling group at Rice University in Houston, Texas (USA) is to investigate techniques from Effective Algebraic Geometry in order to solve some of the key problems in Geometric Modeling and Computational Biology. The two groups have similar interests and complementary strengths. Effective Algebraic Geometry is the branch of Algebraic Geometry that pursues concrete algorithms rather than abstract proofs. It deals mainly with practical methods for representing polynomial curves and surfaces along with robust techniques for solving systems of polynomial equations. Many applications in Geometric Modeling and Computational Biology require fast robust methods for solving systems of polynomial equations. Here we concentrate our collective efforts on solving standard problems such as implicitization, inversion, intersection, and detection of singularities for rational curves and surfaces. To aid in modeling, we shall also investigate some novel approaches to represent shape. In contemporary Computational Biology, many problems can be reduced to solving large systems of low degree polynomial equations. We plan to apply our polynomials solvers together with new tools for analysing complex shapes to help study these currently computationally intractable problems.

PAI Procore collaboration

Participants : Laurent Busé, Stéphane Chau, Yi-King Choi [ Hong Kong Univ. ] , André Galligo, Yang Liu [ Hong Kong Univ. ] , Wenping Wang [ Hong Kong Univ. ] , Julien Wintz.

The objective of this collaboration is to conduct research in algebraic techniques for solving problems in geometric modeling. We will investigate the use of implicit models, for compact and efficient shape representation and processing. The application domains are Computer Aided Geometric Design, Robotics, Shape compression, Computer Biology. We will focus on algebraic objects of small degree such as quadrics, with the aim to extend the approach to higher degree. In particular, we are interested in the following problems:

• Shape segmentation and representation using quadrics.

• Shape processing using quadrics.

• Collision detection for objects defined by quadric surfaces.

Experimentation and validation will lead to joint open source software implementation, dedicated to quadric manipulations. A package collecting these tools will be produced.