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.

Inria signed a contract for the integration of ISOTOP within Maple.

The project-team Vegas will terminate at the end of 2016. A new project-team Gamble (Geometric Algorithms and Models Beyond the Linear and Euclidean realm) is currently submitted. It intends to extend computational geometry to non-linear objects, non-Euclidean spaces and probabilistic complexities.

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. 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.

Via the Inria ADT FastTrack funding, Eric Biagioli has joined the project in November 2016 for 6 months. He is porting the maple code to C code and enhancing the visualization. This work will prepare for a better diffusion of the software via a webserver and a transfert to Maplesoft with which Inria has signed a contract in April 2016.

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.

The software has been improved and a technical report published .

Contact: Rémi Imbach

Let a smooth real analytic curve embedded in

In previous work, we have shown how to describe the set of singularities

The technical report describes the software SubdivisionSolver (see Section ) used within this project.

Let

This work was done in collaboration with Éric Schost (Waterloo University, Canda).

Rigid motions are fundamental operations in image processing. While
bijective and isometric in

This work was done in collaboration with Kacper Pluta (LIGM - Laboratoire d'Informatique Gaspard-Monge), Yukiko Kenmochi (LIGM - Laboratoire d'Informatique Gaspard-Monge), Pascal Romon (LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées).

Usually, the accuracy of parallel manipulators depends on the architecture of the robot, the design parameters, the trajectory planning and the location of the path in the workspace. This paper reports the influence of static and dynamic parameters in computing the error in the pose associated with the trajectory planning made and analyzed with the Orthoglide 5-axis (Figure ). An error model is proposed based on the joint parameters (velocity and acceleration) and experimental data coming from the Orthoglide 5-axis. Newton and Gröbner based elimination methods are used to project the joint error in the workspace to check the accuracy/error in the Cartesian space. For the analysis, five similar trajectories with different locations inside the workspace are defined using fifth order polynomial equation for the trajectory planning. It is shown that the accuracy of the robot depends on the location of the path as well as the starting and the ending posture of the manipulator due to the acceleration parameters .

This work was done in collaboration with Ranjan Jha (IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes), Damien Chablat (IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes), Fabrice Rouillier (Inria).

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 the transformation of 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.

This work was done in collaboration with Yacine Bouzidi (Inria Lille), Michael Sagraloff (MPII Sarrebruken, Germany) and Fabrice Rouillier (Inria Rocquencourt).

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 work on Delaunay triangulations of flat

We give a definition of the Delaunay triangulation of a point set in a
closed Euclidean

Motivated by applications in various fields, some packages to compute periodic Delaunay triangulations in the 2D and 3D Euclidean spaces have been introduced in the CGAL library and have attracted a number of users. To the best of our knowledge, no software is available to compute periodic triangulations in a hyperbolic space, though they are also used in diverse fields, such as physics, solid modeling, cosmological models, neuromathematics.

This would be a natural extension: 2D Euclidean periodic triangulations can be seen as triangulations of the two-dimensional (flat) torus of genus one; similarly, periodic triangulations in the hyperbolic plane can be seen as triangulations of hyperbolic surfaces. A closed orientable hyperbolic surface is the quotient of the hyperbolic plane under the action of a Fuchsian group only containing hyperbolic translations. Intuition is challenged there, in particular because such groups are non-Abelian in general.

We have obtained some theoretical results on Delaunay triangulations of general closed orientable hyperbolic surfaces, and we have investigated algorithms in the specific case of the Bolza surface, a hyperbolic surface with the simplest possible topology, as it is homeomorphic to a genus-two torus . We are now studying more practical aspects and we propose a first implementation of an incremental construction of Delaunay triangulations of the Bolza surface .

Let

This work was done in collaboration with Nicolas Chenavier (Université Littoral Côte d'Opale ).

Let *locally defined*;
and is shorter than the previously known locally defined path in
Delaunay triangulation such as
the upper path whose expected length is

Let

This work was done in collaboration with Pedro Machado Manhães De Castro (Centro de Informática da Universidade Federal de Pernambuco).

We study the following problem: Given

This work was done in collaboration with David Bremner (U. New Brunswick), Marc Glisse (Inria Datashape), Giuseppe Liotta (U. Perugia), Tamara Mchedlidze (Karlsruhe Institute for Technology), Sue Whitesides (U. Victoria), and Stephen Wismath (U. Lethbridge).

A standard way to approximate the distance between
two vertices *Farthest Point Sampling* (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources.

We analyzed the stretch factor

This work was done in collaboration with Pegah Kamousi (Université Libre de Bruxelles), Anil Maheshwari (Carleton University), and Stefanie Wuhrer (Inria Grenoble Rhône-Alpes).

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable .

This work was done in collaboration with Dominique Attali (CNRS, Grenoble) and Marc Glisse (Inria Datashape).

A two years licence and cooperation agreement was signed on April 1st, 2016 between WATERLOO MAPLE INC., Ontario, Canada (represented by Laurent Bernardin, its Executive Vice President Products and Solutions) and Inria. On the Inria side, this contract involves the teams VEGAS and OURAGAN (Paris), and it is coordinated by Fabrice Rouillier (OURAGAN).

F. Rouillier and VEGAS are the developers of the ISOTOP software for the computation of topology of curves. One objective of the contract is to transfer a version of ISOTOP to WATERLOO MAPLE INC.

We organized, with IECL, a «journée Charles Hermite» about geometry and probability. A regular working group on the topic was started in november.

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 ended in March 2016. It is coordinated by Xavier Goaoc who moved from the Vegas team to Marne-la-Vallée university in 2013.

Project website: https://

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.

The project funded the postdoc position of Rémi Imbach from November 2014 until October 2016. We organized two workshops in 2016 with the OPTI team in Nantes, on certified surface continuation.

Project website:
https://

The objectives of the *ASsociate Team On Non-ISH
euclIdeaN Geometry* is to study various structures and algorithms in
non-Euclidean spaces, from a computational geometry
viewpoint. Proposing algorithms operating in such spaces requires a
prior deep study of the mathematical properties of the objects
considered, which raises new fundamental and difficult questions that
we want to tackle.

A key characteristic of the project is its interdisciplinarity: it gathers approaches, knowledge, and tools in mathematics and computer science. A mathematical study of the considered objects will be performed, together with the design of algorithms when applicable. Algorithms will be analyzed both in theory and in practice after prototype implementations. In the long term, implementations should be improved whenever it makes sense to target longer-term integrations into CGAL, in order to disseminate our results to end-users.

The partners are the Johann Bernouilli Institute of Mathematics and Computer Science of University of Groningen, the Mathematics Research Unit of University of Luxembourg, and the Talgo team of École Normale Supérieure. The project is coordinated by Monique Teillaud and supported by Inria Nancy - Grand Est.

Project website:
https://

Gert Vegter, Professor at Univerity of Groningen, was awarded an invited professor position by University of Lorraine and spent one month in the group in May. He is coordinating the NEAT Astonishing on the Dutch side.

Sény Diatta, Senegalese PhD student co-advised by Guillaume Moroz, Daouda Niang Diatta (Ziguinchor) and Marie-Françoise Roy (Rennes), obtained a bourse Eiffel from Campus France, which includes a salary for 10 months to visit LORIA.

Iordan Iordanov spent one month at University of Luxembourg in June. The visit was partially supported by by University of Luxembourg and by the NEAT Astonishing.

Sylvain Lazard organized with S. Whitesides (Victoria University) the 15th Workshop on Computational Geometry at the Bellairs Research Institute of McGill University in Feb. (1 week workshop on invitation).

Monique Teillaud co-organized the workshop *20 years of
CGAL*, with Efi Fogel, Michael Hoffmann, and Emo Welzl, Zurich,
Switzerland, September 10-11, and she gave a talk.

Monique Teillaud was a member of the program committee of
EuroCG, *European Workshop on Computational Geometry*.

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.

Monique Teillaud is a managing editor of JoCG, *Journal of
Computational Geometry*. She is also a member of the Editorial
Board of IJCGA, *International Journal of Computational Geometry
and Applications*. She resigned from the Editorial Board of CGTA,
*Computational Geometry: Theory and Applications*, after
unsuccessfully trying to convince the Editorial Board to leave
Elsevier and move to a free (libre and gratis) open-access model.

Marc Pouget and Monique Teillaud are members of the CGAL editorial board.

Olivier Devillers resigned from the Editorial Board of Graphical Models (Elsevier) after discussion to move to a free open-access model.

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.

Olivier Devillers was invited to give a talk at the geometry week organized by GipsaLab in Grenoble.

Guillaume Moroz was invited to give talks at the LIGM seminar in Marne-la-Vallée university, at the SpecFun team seminary in Inria Saclay and at the MSDOS workshop in CIRM.

Monique Teillaud was invited to give a talk talk at the seminar
*Computer Science meets Mathematics* of the University of
Luxembourg, February 8: “CGAL, geometry made practical”. She was
invited to give a talk at the *Mittagsseminar* of Institute of
Theoretical Computer Science of ETH Zürich on September 8:
“Delaunay triangulations on orientable surfaces of low genus”.

We invited:

Kacper Pluta (LIGM - Laboratoire d'Informatique Gaspard-Monge),

Mickaël Buchet (Tohoku University).

Andrew Yarmola (University of Luxembourg).

M. Teillaud has been elected Chair of the Steering Committee of the Symposium on Computational Geometry (SoCG). She is a member of the Steering Committee of the European Symposium on Algorithms (ESA).

Sylvain Lazard was president of the hiring committee for a Professor position (UL/École des Mines/LORIA).

Monique Teillaud was the representative of LORIA in the hiring committee for an Associate Professor (MCF) position (École des Mines/LORIA) and composed the committee with the president. She was also a member of the Inria CR2 Nancy - Grand Est interview committee and of the hiring committee for a Professor position (FST/LORIA).

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).

G. Moroz is member of the Mathematics Olympiades committee of the
Nancy-Metz academy. G. Moroz is member of the *Comité des
utilisateurs des moyens informatiques*

M. Pouget is elected at the *Comité de centre*, and member
of the board of the Charles Hermite federation of labs. M. Pouget is
secretary of the board of *AGOS-Nancy*.

M. Teillaud is a member of the BCP, *Bureau du Comité des
Projets* and of the CDT, *Commission de
développement technologique* of Inria Nancy - Grand Est.

M. Teillaud is maintaining the Computational Geometry Web Pages
http://

Master : O. Devillers,
*Synthèse, image et géométrie
*, 12h (academic year 2015-16) and 12h (academic year 2016-2017), IPAC-R, Université de Lorraine.
https://

Master: Marc Pouget, *Introduction to computational geometry*, 10.5h, M2,
École Nationale Supérieure de Géologie, France.

Licence: Sylvain Lazard, *Algorithms and Complexity*, 25h, L3, Université de Lorraine, France.

Licence: Laurent Dupont, *Algorithmique*, 78h, 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.

PhD : Ranjan Jha, Étude de l’espace de travail des mécanismes à boucles fermées, defended in Jul. 2016, 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.

PhD in progress : Charles Duménil, Probabilistic analysis of geometric structures, started in Oct. 2016, supervised by Olivier Devillers.

PhD in progress : Iordan Iordanov, Triangulations of Hyperbolic Manifolds, started in Jan. 2016, supervised by Monique Teillaud.

Postdoc: Rémy Imbach, Topology and geometry of singular surfaces with numerical algorithms, supervised by Guillaume Moroz and Marc Pouget.

O. Devillers was president of the PhD defense committee of Vincent Despré (Univ. Grenoble-Alpes).

G. Moroz was in the PhD defense committee of Ranjan Jha (IRCCyN).

Licence: Laurent Dupont, creation and opening of L3 (Licence Professionnelle) « Animation des Communautés et Réseaux Socionumériques », Université de Lorraine, France.