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GALAAD2 Research team

Géométrie , Algèbre, Algorithmes

  • Leader : Bernard Mourrain
  • Research center(s) : CRI Sophia Antipolis - Méditerranée
  • Field : Algorithmics, Programming, Software and Architecture
  • Theme : Algorithmics, Computer Algebra and Cryptology

Team presentation

Our day life environment is increasingly interacting with a digital world, populated by captors, sensors, or devices used to simplify or improve some of our activities. Computing is becoming ubiquitous and this evolution raises new challenges to represent, analyze and transform this digital information. From this perspective, geometry is playing an important role. There is a strong interaction between physical and digital worlds through geometric modeling and analysis. Understanding a physical phenomena can be done by analyzing numerical simulations on a digital representation of the geometry. Conversely developing digital geometry is nowadays used to produce devices that are part our day life or routinely used. Within this context, our research program aims at developing new and efficient methods for modeling geometry with algebraic representations. We are interested in developing algebraic models which provide compact and precise representations of the geometry and to develop efficient algorithms on these representations for accurate and powerful modeling, computation and analysis.

Research themes

Our main scientific objectives are the following: * Algebraic algorithms for geometric computing: our goal is to develop algebraic algorithms for geometry: intersection or self-intersection locus of algebraic surface patches, detect and analyze singularities, compute offsets, envelopes. * Symbolic-numeric methods for analysis: The objective is to devise adapted tools for analyzing the geometric properties of the algebraic models taking into account noisy data. This includes topology certification of intersection curves or singular loci or arrangement computations for semi-algebraic sets. * Algebraic representations for geometric modeling: Compact, efficient and structured descriptions of shapes are required in many scientific computations in engineering, such as "Isogeometric'' Finite Elements methods or point cloud fitting problems. Our objective is to investigate new algebraic representations (or improve the existing ones) together with their analysis and implementations.

Keywords: Algbra; geometry; algorithm; effective algebraic geometry; geometric modeling;