The main scientific objective of the VEGAS research team is to contribute to the development of an effective geometric computing dedicated to non-trivial geometric objects. Included among its main tasks are the study and development of new algorithms for the manipulation of geometric objects, the experimentation of algorithms, the production of high-quality software, and the application of such algorithms and implementations to research domains that deal with a large amount of geometric data, notably solid modeling and computer graphics.
Computational geometry has traditionally treated linear objects like line segments and polygons in the plane, and point sets and polytopes in three-dimensional space, occasionally (and more recently) venturing into the world of non-linear curves such as circles and ellipses. The methodological experience and the know-how accumulated over the last thirty years have been enormous.
For many applications, particularly in the fields of computer graphics and solid modeling, it is necessary to manipulate more general objects such as curves and surfaces given in either implicit or parametric form. Typically such objects are handled by approximating them by simple objects such as triangles. This approach is extremely important and it has been used in almost all of the usable software existing in industry today. It does, however, have some disadvantages. Using a tessellated form in place of its exact geometry may introduce spurious numerical errors (the famous gap between the wing and the body of the aircraft), not to mention that thousands if not hundreds of thousands of triangles could be needed to adequately represent the object. Moreover, the curved objects that we consider are not necessarily everyday three-dimensional objects, but also abstract mathematical objects that are not linear, that may live in high-dimensional space, and whose geometry we do not control. For example, the set of lines in 3D (at the core of visibility issues) that are tangent to three polyhedra span a piecewise ruled quadratic surface, and the lines tangent to a sphere correspond, in projective five-dimensional space, to the intersection of two quadratic hypersurfaces.
Effectiveness is a key word of our research project. By requiring our algorithms to be effective, we imply that the algorithms should be robust, efficient, and versatile. By robust we mean algorithms that do not crash on degenerate inputs and always output topologically consistent data. By efficient we mean algorithms that run reasonably quickly on realistic data where performance is ascertained both experimentally and theoretically. Finally, by versatile we mean algorithms that work for classes of objects that are general enough to cover realistic situations and that account for the exact geometry of the objects, in particular when they are curved.
We are interested in the application of our work to virtual prototyping, which refers to the many steps required for the creation of a realistic virtual representation from a CAD/CAM model.
When designing an automobile, detailed physical mockups of the interior are built to study the design and evaluate human factors and ergonomic issues. These hand-made prototypes are costly, time consuming, and difficult to modify. To shorten the design cycle and improve interactivity and reliability, realistic rendering and immersive virtual reality provide an effective alternative. A virtual prototype can replace a physical mockup for the analysis of such design aspects as visibility of instruments and mirrors, reachability and accessibility, and aesthetics and appeal.
Virtual prototyping encompasses most of our work on effective geometric computing. In particular, our work on 3D visibility should have fruitful applications in this domain. As already explained, meshing objects of the scene along the main discontinuities of the visibility function can have a dramatic impact on the realism of the simulations.
Solid modeling, i.e., the computer representation and manipulation of 3D shapes, has historically developed somewhat in parallel to computational geometry. Both communities are concerned with geometric algorithms and deal with many of the same issues. But while the computational geometry community has been mathematically inclined and essentially concerned with linear objects, solid modeling has traditionally had closer ties to industry and has been more concerned with curved surfaces.
Clearly, there is considerable potential for interaction between the two fields. Standing somewhere in the middle, our project has a lot to offer. Among the geometric questions related to solid modeling that are of interest to us, let us mention: the description of geometric shapes, the representation of solids, the conversion between different representations, data structures for graphical rendering of models and robustness of geometric computations.
In the context of drawing plane algebraic curves with the correct topology, we have obtained and submitted this year major results on the resolution of bivariate algebraic systems. In particular, we presented algorithms whose worst-case and expected (Las Vegas) complexities are not likely to be easily improved as such improvments would essentially require to improve bounds on other fundamental problems (such as computing resultants, checking the squarefreeness of univariate polynomials, and isolating their roots) that have hold for decades. See section for details.
Topology and geometry of planar algebraic curves
Keywords: Topology - Curve plotting - Geometric computing
Isotop is a Maple software for computing the topology of an algebraic plane curve, that is, for computing an arrangement of polylines isotopic to the input curve. This problem is a necessary key step for computing arrangements of algebraic curves and has also applications for curve plotting.
This software, registered at the APP in June 2011, has been developed since 2007 in collaboration with F. Rouillier from Inria Paris - Rocquencourt. The distributed version is based on the method described in , which presents several improvements over previous methods. In particular, our approach does not require generic position. This version is competitive with other implementations (such as AlciX and Insulate developed at MPII Saarbrücken, Germany and top developed at Santander Univ., Spain). It performs similarly for small-degree curves and performs significantly better for higher degrees, in particular when the curves are not in generic position.
We are currently working on an improved version integrating a new bivariate polynomial solver based on several of our recent results published in , , . This version is not yet distributed.
Contact: Sylvain Lazard & Marc Pouget
Keywords: Numerical solver - Polynomial or analytical systems
The software SubdivisionSolver solves square systems of analytic equations on a compact
subset of a real space of any finite dimension. SubdivisionSolver is a numerical
solver and as such it requires that the solutions in the subset are isolated and
regular for the input system (i.e. the Jacobian must not vanish). SubdivisionSolver is
a subdivision solver using interval arithmetic and multiprecision arithmetic to
achieve certified results. If the arithmetic precision required to isolate
solutions is known, it can be given as an input parameter of the process,
otherwise the precision is increased on-the-fly. In particular, SubdivisionSolver can
be interfaced with the Fast_Polynomial library
(https://
The software is based on a classic branch and bound algorithm using interval arithmetic: an initial box is subdivided until its sub-boxes are certified to contain either no solution or a unique solution of the input system. Evaluation is performed with a centered evaluation at order two, and existence and uniqueness of solutions is verified thanks to the Krawczyk operator.
SubdivisionSolver uses two implementations of interval arithmetic: the C++ boost library that provides a fast arithmetic when double precision is enough, and otherwise the C mpfi library that allows to work in arbitrary precision. Considering the subdivision process as a breadth first search in a tree, the boost interval arithmetic is used as deeply as possible before a new subdivision process using higher precision arithmetic is performed on the remaining forest.
We used SubdivisionSolver for the experiments in , , see Section .
Contact: Rémi Imbach
The CGAL library offers a package to compute the 3D periodic Delaunay triangulation of a
point set in
We have been extending this package in three directions:
First, a few new small functions have been added to the Delaunay triangulation class and integrated in CGAL 4.7.
We have developed and documented some new classes allowing to compute weighted periodic Delaunay triangulations. They have been submitted to the CGAL editorial board and accepted for inclusion in CGAL. The code still needs some polishing, and the testsuite must be completed, before a public distribution in CGAL.
We have continued our work to use this package together with the 3D mesh generation package of CGAL , in order to propose a construction of meshes of periodic volumes. Although last year's preliminary results were already convincing , , the work is not ready yet for being submitted to CGAL: the code requires to be completed, documented, and extensively tested.
Contact: Monique Teillaud
In collaboration with Aymeric Pellé (Geometrica project-team)
This work was done in the framework of the Inria ADT
(Action de Développement Technologique) OrbiCGAL http://
In a classical Symbolic Perturbation scheme, degeneracies are handled
by substituting some polynomials in
This work was done in collaboration with Menelaos Karavelas (Univ. of Crete).
We show that the memoryless routing algorithms Greedy Walk, Compass Walk,
and all variants of visibility walk based on orientation predicates are asymptotically optimal
in the average case on the Delaunay triangulation.
More
specifically, we consider the Delaunay triangulation of an unbounded Poisson point
process of unit rate and demonstrate that the worst-case path between any two
vertices inside a domain of area
This work was done in collaboration with Ross Hemsley (formerly in Inria Geometrica).
We establish an upper bound on the smoothed complexity of convex
hulls in
We also analyze the expected complexity of the convex hull of
This work was done in collaboration with Xavier Goaoc (Univ. Marne la Vallée), Marc Glisse and Remy Thomasse (Inria Geometrica).
In the context of our algorithm Isotop for computing the topology of plane algebraic curves (see Section ), we work on the problem of solving a system of two bivariate polynomials. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. But we are also interested in computing, as intermediate symbolic objects, a Rational Univariate Representation (RUR) that is, roughly speaking, a univariate polynomial and two rational functions that map the roots of the univariate polynomial to the two coordinates of the solutions of the system. RURs are relevant symbolic objects because they allow to turn many queries on the system into queries on univariate polynomials. However, such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.
We published this year results showing that, given two polynomials of degree at most
However, during the publishing process, we have substentially improved these results. We have
presented for these three sub-problems new algorithms that have worst-case bit complexity
This work was done in collaboration with Yacine Bouzidi (Inria Saclay), Michael Sagraloff (MPII Sarrebruken, Germany) and Fabrice Rouillier (Inria Rocquencourt). It is published in the research report and submitted to a journal.
A key ingredient of the above work is the classical triangular
decomposition algorithm via subresultants on which we obtain two results
of independent interest. First, we improved by a factor
This work was done in collaboration with Fabrice Rouillier (Inria Rocquencourt). It is published in the research report and submitted to a journal.
Let a smooth real analytic curve embedded in
On the same topic but with a different approach, we improved our research report by including experimental data using SubdivisionSolver (see Section ) and submitted this work to a journal.
In collaboration with F. Rouillier, D. Chablat and our PhD student Ranjan Jha, we analyzed the singularities and the workspace of different families of robots.
The first result is a certified description of the workspace and the singularities of a Delta like family robot . Workspace and joint space analysis are essential steps in describing the task and designing the control loop of the robot, respectively. This paper presents the descriptive analysis of a family of delta-like parallel robots by using algebraic tools to induce an estimation about the complexity in representing the singularities in the workspace and the joint space. A Gröbner based elimination is used to compute the singularities of the manipulator and a Cylindrical Algebraic Decomposition algorithm is used to study the workspace and the joint space. From these algebraic objects, we propose some certified three dimensional plotting describing the shape of workspace and of the joint space which will help the engineers or researchers to decide the most suited configuration of the manipulator they should use for a given task. Also, the different parameters associated with the complexity of the serial and parallel singularities are tabulated, which further enhance the selection of the different configurations of the manipulator by comparing the complexity of the singularity equations.
The second result is an algebraic method to check the singularity-free
paths for parallel robots . Trajectory planning is a
critical step while programming the parallel manipulators in a robotic
cell. The main problem arises when there exists a singular configuration
between the two poses of the end-effectors while discretizing the path with
a classical approach. This paper presents an algebraic method to check the
feasibility of any given trajectories in the workspace. The solutions of
the polynomial equations associated with the trajectories are projected in
the joint space using Gröbner based elimination methods and the remaining
equations are expressed in a parametric form where the articular variables
are functions of time
We addressed the problem of finding the reflection point on quadric
mirror surfaces, especially ellipsoid, paraboloid or hyperboloid of two
sheets, of a light ray emanating from a 3D point
source
The white ANR grant PRESAGE brings together computational geometers (from the VEGAS and GEOMETRICA projects of Inria) and probabilistic geometers (from Universities of Rouen, Orléans and Poitiers) to tackle new probabilistic geometry problems arising from the design and analysis of geometric algorithms and data structures. We focus on properties of discrete structures induced by random continuous geometric objects.
The project, with a total budget of 400kE, started on Dec. 31st, 2011 and will end in march 2016. It is coordinated by Xavier Goaoc who moved from the Vegas team to Marne-la-Vallée university in 2013.
Project website: http://
The objective of the young-researcher ANR grant SingCAST is to intertwine further symbolic/numeric approaches to compute efficiently solution sets of polynomial systems with topological and geometrical guarantees in singular cases. We focus on two applications: the visualization of algebraic curves and surfaces and the mechanical design of robots.
After identifying classes of problems with restricted types of singularities, we plan to develop dedicated symbolic-numerical methods that take advantage of the structure of the associated polynomial systems that cannot be handled by purely symbolic or numerical methods. Thus we plan to extend the class of manipulators that can be analyzed, and the class of algebraic curves and surfaces that can be visualized with certification.
This is a 3.5 years project, with a total budget of 100kE, that started on March 1st 2014, coordinated by Guillaume Moroz.
In 2015, the project funded the postdoc position of Rémi Imbach.
Project website:
https://
Monique Teillaud was invited at the Workshop on Computational
Geometric and Algebraic Topology, Mathematisches
Forschungsinstitut Oberwolfach, where she presented CGAL,
the Computational Geometry Algorithms
Library. https://
Sylvain Lazard organized with S. Whitesides (Victoria
University) the 14th Inria - McGill - Victoria Workshop on Computational
Geometry
Guillaume Moroz co-organized the Journées Nationales de Calcul Formel 2015 (JNCF).
Monique Teillaud co-organized the Dagstuhl Seminar on
Computational Geometry, with Otfried Cheong (KAIST, Korea) and Jeff
Erickson (University of Illinois at Urbana-Champaign), Leibniz-Zentrum für Informatik, Germany, March 9-13.
http://
Monique Teillaud co-organized the Workshop on Computational
Geometry in non-Euclidean Spaces, with Éric Colin de Verdière (CNRS,
ENS Paris) and Jean-Marc Schlenker (Mathematics Research Unit,
U. Luxembourg), Inria, Loria, Nancy, August 26-28. The slides of the
talks are available on the workshop website
http://
Monique Teillaud and Marc Pouget organized the 39th CGAL Developer Meeting, Sept 28 - Oct 2 at Inria Nancy - Grand Est.
Monique Teillaud was a member of the program committee of the 2015 Symposium on Computational Geometry SoCG'15.
All members of the team are regular reviewers for the conferences of our field, namely the Symposium on Computational Geometry (SoCG) and the International Symposium on Symbolic and Algebraic Computation (ISSAC) and also SODA, CCCG, EuroCG.
Olivier Devillers is a member of the Editorial Board of Graphical Models.
Monique Teillaud is a managing editor of JoCG, Journal of Computational Geometry. She is also a member of the Editorial Boards of CGTA, Computational Geometry: Theory and Applications, and of IJCGA, International Journal of Computational Geometry and Applications.
Marc Pouget and Monique Teillaud are members of the CGAL editorial board.
All members of the team are regular reviewers for the journals of our field, namely Discrete and Computational Geometry (DCG), Computational Geometry. Theory and Applications (CGTA), Journal of Computational Geometry (JoCG), International Journal on Computational Geometry and Applications (IJCGA), Journal on Symbolic Computations (JSC), SIAM Journal on Computing (SICOMP), Mathematics in Computer Science (MCS), etc.
Guillaume Moroz was invited to give a talk at the Fields Institute in Toronto for the workshop on Algebra, Geometry and Proofs in Symbolic Computation. He was also invited to give a talk at the IHP for the seminar on algorithmic and computational geometry and at Supelec.
S. Lazard was president of the hiring committee for a Professor position (UL/ENSEM/LORIA).
M. Teillaud is a member of the Steering Committee of the European Symposium on Algorithms (ESA).
L. Dupont is a member of “Commission Pédagogique Nationale” (CPN) Information-Communication / Métiers du Multimédia et de l'Internet.
M. Teillaud is a member of the Scientific Board of the Société Informatique de France (SIF).
M. Teillaud is a member of the working group for the BIL, Base d'Information des Logiciels of Inria.
S. Lazard: Head of the PhD and Post-doc hiring committee for Inria Nancy-Grand Est (since 2009). Member of the Bureau de la mention informatique of the École Doctorale IAE+M (since 2009). Head of the Mission Jeunes Chercheurs for Inria Nancy-Grand Est (since 2011). Head of the Department Algo at LORIA (since 2014). Member of the Conseil Scientifique of LORIA (since 2014).
M. Teillaud is a member of the CDT, Commission de développement technologique, of Inria Nancy - Grand Est.
M. Teillaud is maintaining the Computational Geometry Web Pages
http://
Master : Guillaume Moroz, Géométrie algébrique et applications en
robotique, 12h, M1, École Mathématque Africaine, Sénégal.
Master: Marc Pouget, Introduction to computational geometry, 10.5h, M2, École Nationale Supérieure de Géologie, France.
Licence: Laurent Dupont, Algorithmique, 72h, L1, Université de Lorraine, France.
Licence: Laurent Dupont, Web development, 75h, L2, Université de Lorraine, France.
Licence: Laurent Dupont, Traitement Numérique du Signal, 10h, L2, Université de Lorraine, France.
Licence: Laurent Dupont, Data structures, 40h, L1, Université de Lorraine, France.
Licence: Sylvain Lazard, Algorithms and Complexity, 25h, L3, Université de Lorraine, France.
PhD: Rémy Thomasse, Complexity analysis of random convex hulls, defended December 18th 2015. The student is hosted in EPI geometrica but supervised by O. Devillers .
PhD in progress : Ranjan Jha, Étude de l’espace de travail des mécanismes à boucles fermées, started in Oct. 2013, supervised by Damien Chablat, Fabrice Rouillier and Guillaume Moroz.
PhD in progress : Sény Diatta, Complexité du calcul de la topologie d'une courbe dans l'espace et d'une surface, started in Nov. 2014, supervised by Daouda Niang Diatta, Marie-Françoise Roy and Guillaume Moroz.
Postdoc: Rémy Imbach, Topology and geometry of singular surfaces with numerical algorithms, supervised by Guillaume Moroz and Marc Pouget.
Postdoc: Laurent Veysseire, Probabilistic analysis of geometric structures, supervised by Olivier Devillers.
O. Devillers was a member (advisor) of the PhD defense committee of Rémy Thomasse (Univ. Nice - Sophia Antipolis).
S. Lazard was an external examiner (rapporteur) of the Ph.D. of Cuong Tran (Univ. Pierre et Marie Curie).
Guillaume Moroz is a member of the organizing committee of the Olympiades académiques de mathématiques.