Starting in the eighties, the emerging computational geometry community has put
a lot of effort to design and analyze algorithms for geometric problems.
The most commonly used framework was to study
the worst-case theoretical complexity of geometric problems
involving linear objects (points, lines, polyhedra...) in
Euclidean spaces.
This so-called
*classical computational geometry* has some known
limitations:

Objects: dealing with objects only defined by linear equations.

Ambient space: considering only Euclidean spaces.

Complexity: worst-case complexities often do not capture realistic behaviour.

Dimension: complexities are often exponential in the dimension.

Robustness: ignoring degeneracies and rounding errors.

Even if these limitations have already got some attention from the
community ,
a quick look at the flagship conference SoCG

It should be stressed that, in this document, the notion of certified algorithms is to be understood with respect to robustness issues. In other words, certification does not refer to programs that are proven correct with the help of mechnical proof assistants such as Coq, but to algorithms that are proven correct on paper even in the presence of degeneracies and computer-induced numerical rounding errors.

We address several of the above limitations:

** $\u2022$ Non-linear computational geometry. **
Curved objects are ubiquitous in the world we live in. However,
despite this ubiquity and decades of research in several
communities, curved objects
are far
from being robustly and efficiently manipulated by geometric algorithms. Our work on, for instance,
quadric intersections and certified drawing of plane curves has proven that
dramatic improvements can be accomplished when the right mathematics and computer science
are put into motion. In this direction, many problems
are fundamental
and solutions have potential industrial impact in Computer Aided
Design and Robotics for instance.
Intersecting NURBS (Non-uniform rational basis spline) and meshing
singular surfaces in a certified manner
are important examples of such problems.

** $\u2022$ Non-Euclidean computational geometry. **
Triangulations are central
geometric data structures in many areas of science and
engineering. Traditionally, their study has been limited to the
Euclidean setting. Needs for triangulations in non-Euclidean settings have emerged in many areas
dealing with objects whose sizes range from the
nuclear to the astrophysical scale, and both in academia and in industry.
It has become timely to extend the traditional focus on

** $\u2022$ Probability in computational geometry. **
The design of efficient algorithms is driven by the analysis of their
complexity. Traditionally, worst-case input and sometimes uniform distributions
are considered and many results in these settings have had a great influence on
the domain.
Nowadays, it is necessary to be more subtle and to prove new results in between these two extreme settings.
For instance, smoothed analysis, which was introduced for the simplex algorithm and which we applied successfully to
convex hulls, proves that
such promising alternatives exist.

As mentioned above, curved objects are ubiquitous in real world problems modelings and in computer science and, despite this fact, there are very few problems on curved objects that admit robust and efficient algorithmic solutions without first discretizing the curved objects into meshes. Meshing curved objects induces some loss of accuracy which is sometimes not an issue but which can also be most problematic depending on the application. In addition, discretizing induces a combinatorial explosion which could cause a loss in efficiency compared to a direct solution on the curved objects (as our work on quadrics has demonstrated with flying colors , , , , ). But it is also crucial to know that even the process of computing meshes that approximate curved objects is far from being resolved. As a matter of fact there is no algorithm capable of computing in practice meshes with certified topology of even rather simple singular 3D surfaces, due to the high constants in the theoretical complexity and the difficulty of handling degenerate cases. Even in 2D, meshing an algebraic curve with the correct topology, that is in other words producing a correct drawing of the curve (without knowing where the domain of interest is), is a very difficult problem on which we have recently made important contributions , , .

It is thus to be understood that producing practical robust and efficient algorithmic solutions to geometric problems on curved objects is a challenge on all and even the most basic problems. The basicness and fundamentality of two problems we mentioned above on the intersection of 3D quadrics and on the drawing in a topologically certified way of plane algebraic curves show rather well that the domain is still at its infancy. And it should be stressed that these two sets of results were not anecdotical but flagship results produced during the lifetime of Vegas team.

There are many
problems in this theme that are expected to have high long-term
impacts. Intersecting NURBS (Non-uniform rational basis spline) in a certified way is an important problem in computer-aided design and
manufacturing. As hinted above, meshing objects in a certified way is important
when topology matters.
The 2D case, that is essentially drawing plane curves with the correct topology,
is a fundamental
problem with far-reaching applications in research or R&D.
Notice that on such elementary problems it is often difficult to predict the
reach of the applications; as an example, we were astonished by the scope of the applications of our
software on 3D quadric intersection

Triangulations, in particular Delaunay triangulations, in the
*Euclidean space* *et al.*
). Some members of Gamble have been contributing to these algorithmic advances
(see, e.g. , , , ); they have also
contributed robust and efficient triangulation packages through the
state-of-the-art Computational Geometry Algorithms Library
Cgal,

It is fair to say that little has been done on non-Euclidean spaces,
in spite of the large number of questions raised by application
domains. Needs for simulations or modeling in a variety of
domains

See
http://

http://

Interestingly, even for the simple case of triangulations on the *sphere*, the software
packages that are
currently
available are far from offering satisfactory solutions in terms of
robustness and efficiency .

Moreover, while our solution for computing triangulations in
hyperbolic spaces can be considered as ultimate , the case
of *hyperbolic manifolds* has hardly been explored. Hyperbolic manifolds are
quotients of a hyperbolic space by some group of hyperbolic
isometries. Their triangulations can be seen as hyperbolic
periodic triangulations. Periodic hyperbolic triangulations and
meshes appear for instance in geometric modeling
, neuromathematics , or physics
. Even the simplest possible case (a surface
homeomorphic to the torus with two
handles)
shows strong mathematical
difficulties , .

In most computational geometry papers, algorithms are analyzed in the worst-case setting. It often yields too pessimistic complexities that arise only in pathological situations that are unlikely to occur in practice. On the other hand, probabilistic geometry gives analyses of great precisions , , , but using hypotheses with much more randomness than in most realistic situations. We are developing new algorithmic designs improving state-of-the-art performances in random settings that are not overly simplified and that can thus reflect many realistic situations.

Twelve years ago, smooth analysis was introduced by Spielman and Teng analyzing the simplex algorithm by averaging on some noise on the data (and they won the Gödel prize). In essence, this analysis smoothes the complexity around worst-case situations, thus avoiding pathological scenarios but without considering unrealistic randomness. In that sense, this method makes a bridge between full randomness and worst case situations by tuning the noise intensity. The analysis of computational geometry algorithms within this framework is still embryonic. To illustrate the difficulty of the problem, we started working in 2009 on the smooth analysis of the size of the convex hull of a point set, arguably the simplest computational geometry data structure; then, only one very rough result from 2004 existed and we only obtained in 2015 breakthrough results, but still not definitive , , .

Another example of problem of different flavor concerns Delaunay triangulations, which are rather ubiquitous in computational geometry. When Delaunay triangulations are computed for reconstructing meshes from point clouds coming from 3D scanners, the worst-case scenario is, again, too pessimistic and the full randomness hypothesis is clearly not adapted. Some results exist for “good samplings of generic surfaces” but the big result that everybody wishes for is an analysis for random samples (without the extra assumptions hidden in the “good” sampling) of possibly non-generic surfaces.

Trade-off between full randomness and worst case may also appear in other forms such as dependent distributions, or random distribution conditioned to be in some special configurations. Simulating these kinds of geometric distributions is currently out of reach for more than few hundred points although it has practical applications in physics or networks.

Many domains of science can benefit from the results developed
by Gamble.
Curves and surfaces are ubiquitous in all sciences to
understand and interpret raw data as well as experimental results.
Still, the non-linear problems we address are rather basic and
fundamental, and it is often difficult to predict the impact of
solutions in that area.
The short-term industrial impact is likely to be small because, on basic
problems, industries have used ad hoc solutions for decades and have thus got
used to it.
The example of our work on quadric intersection is typical: even though we were
fully convinced that intersecting 3D quadrics is such an elementary/fundamental problem that it
ought to be useful, we were the first to be astonished by the scope of the applications of our
software

The fact that several of our pieces of software for computing non-Euclidean triangulations have already been requested by users long before they become public is a good sign for their wide future impact once in Cgal. This will not come as a surprise, since most of the questions that we have been studying followed from discussions with researchers outside computer science and pure mathematics. Such researchers are either users of our algorithms and software, or we meet them in workshops. Let us only mention a few names here. We have already referred above to our collaboration with Rien van de Weijgaert , (astrophysicist, Groningen, NL). Michael Schindler (theoretical physicist, ENSPCI, CNRS, France) is using our prototype software for 3D periodic weighted triangulations. Stephen Hyde and Vanessa Robins (applied mathematics and physics at Australian National University) have recently signed a software license agreement with Inria that allows their group to use our prototype for 3D periodic meshing. Olivier Faugeras (neuromathematics, Inria Sophia Antipolis) had come to us and mentioned his needs for good meshes of the Bolza surface before we started to study them. Such contacts are very important both to get feedback about our research and to help us choose problems that are relevant for applications. These problems are at the same time challenging from the mathematical and algorithmic points of view. Note that our research and our software are generic, i.e., we are studying fundamental geometric questions, which do not depend on any specific application. This recipe has made the sucess of the Cgal library.

Probabilistic models for geometric data are widely used to model various situations ranging from cell phone distribution to quantum mechanics. The impact of our work on probabilistic distributions is twofold. On the one hand, our studies of properties of geometric objects built on such distributions will yield a better understanding of the above phenomena and has potential impact in many scientific domains. On the other hand, our work on simulations of probabilistic distributions will be used by other teams, more maths oriented, to study these distributions.

The project-team Vegas terminated at the end of 2016. Our main highlight is actually the creation of the new project-team Gamble (Geometric Algorithms and Models Beyond the Linear and Euclidean realm) on July 1st.

Another highlight of this year is that after two failures, both ANR projects we are coordinating finally won at the ANR lottery with two projects that will start in 2018: ASPAG (ANR-17-CE40-0017) and SoS (ANR-17-CE40-0033).

*Topology and geometry of planar algebraic curves*

Keywords: Topology - Curve plotting - Geometric computing

Functional Description: 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 has been developed since 2007 in collaboration with F. Rouillier from Inria Paris - Rocquencourt. It is based on the method described in [Cheng, J., Lazard, S., Pe

News Of The Year: In 2017, an ADT FastTrack funded a 6 months engineer contract to port the Maple code to C code. In addition, another local engineer from Inria Nancy (Benjamin Dexheimer) implemented a web server to improve the diffusion of our software.

Participants: Elias Tsigaridas, Jinsan Cheng, Luis Penaranda, Marc Pouget and Sylvain Lazard

Contact: Sylvain Lazard

Keywords: Flat torus - CGAL - Geometry - Geometric computing - Voronoi diagram - Delaunay triangulation - Triangulation

Functional Description: This class of CGAL (Computational Geometry Algorithms Library http://www.cgal.org) allows to build and handle periodic regular triangulations whose fundamental domain is a cube in 3D. Triangulations are built incrementally and can be modified by insertion of weighted points or removal of vertices. They offer location facilities for weighted points. The class offers nearest neighbor queries for the additively weighted distance and primitives to build the dual weighted Voronoi diagrams.

Participants: Aymeric Pellé, Mael Rouxel-Labbe and Monique Teillaud

Contact: Monique Teillaud

URL: https://

Keywords: Geometry - Delaunay triangulation - Hyperbolic space

Functional Description: This package implements the construction of Delaunay triangulations in the Poincaré disk model.

Authors: Mikhail Bogdanov, Olivier Devillers and Monique Teillaud

Contact: Monique Teillaud

Publication: Hyperbolic Delaunay Complexes and Voronoi Diagrams Made Practical

URL: https://

Keywords: Geometry - Delaunay triangulation - Hyperbolic space

Functional Description: This module implements the computation of Delaunay triangulations of the Bolza surface.

Authors: Iordan Iordanov and Monique Teillaud

Contact: Monique Teillaud

Publication: Implementing Delaunay Triangulations of the Bolza Surface

URL: https://

Consider a plane curve

It is known that the internal stabilizability of a
multidimensional system can be investigated by studying a certain
polynomial ideal

The Cgal library offers software packages to compute Delaunay
triangulations of the (flat) torus of genus one in two and three
dimensions. To the best of our knowledge, there is no available
software for the simplest possible extension, i.e., the Bolza surface,
a hyperbolic manifold homeomorphic to a torus of genus two. We present
an implementation based on the theoretical results and the incremental
algorithm proposed recently. We describe the representation of the
triangulation, we detail the different steps of the algorithm, we
study predicates, and report experimental results . The implementation
is publicly available in the development branch of Cgal on
`github`

A good sample is a point set such that any ball of radius

This work was done in collaboration with Marc Glisse (Project-team Datashape).

The complexity of the Delaunay triangulation of

We present a new oblivious walking strategy for convex subdivisions. Our walk is faster than the straight walk and more generally applicable than the visiblity walk. To prove termination of our walk we use a novel monotonically decreasing distance measure .

This work was done in collaboration with Wouter Kuijper and Victor Ermolaev (Nedap Security Management).

Let

This work was done in collaboration with William J. Lenhart (Williams College, USA).

We consider the problem of designing space efficient solutions for representing
triangle meshes.
Our main result is a new explicit data structure for compactly representing
planar triangulations: if one is allowed
to permute input vertices, then a triangulation with

This work was done in collaboration with Luca Castelli Aleardi (LIX).

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 colleagues of the mathematics department (Institut Elie Cartan Nancy) a regular working group about geometry and probability.

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.

The project has a total budget of 100kE. It started on March 1st 2014 and will finished in August 2018. It is coordinated by Guillaume Moroz, with a participation of 60%, and Marc Pouget with a participation of 40%.

Project website:
https://

Title: ASsociate Team On Non-ISH euclIdeaN Geometry

International Partners (Institution - Laboratory - Researcher):

University of Groningen (Netherlands) - Johann Bernouilli Institute of Mathematics and Computer Science - Gert Vegter

University of Luxembourg - Mathematics Research Unit - Jean-Marc Schlenker

Université Paris Est Marne-la-Vallée - Laboratoire d'Informatique Gaspard Monge - Éric Colin de Verdière

Start year: 2017

See also: https://

Some research directions in computational geometry have hardly been explored. The spaces in which most algorithms have been designed are the Euclidean spaces

Gert Vegter spent three weeks in Gamble in the framework of the Astonishing associate team.

Olivier Devillers spent one month at Computational Geometry Lab
of Carleton University http://

Sylvain Lazard organized with S. Whitesides (Victoria University) the 16th Workshop on Computational Geometry at the Bellairs Research Institute of McGill University in Feb. (1 week workshop on invitation).

Monique Teillaud co-organized with Claire Mathieu Celebrating Claude Puech's birthday, Paris, June 12.

Monique Teillaud co-organized the workshop Geometric Aspects of Materials Science with Vanessa Robins and Ileana Streinu, Brisbane, Australia, July 4–5.

Monique Teillaud co-organized with the Astonishing partners the Astonishing workshop at Loria/Inria nancy, September 25–26.

Sylvain Lazard was a member of the program committee of SoCG,
*Symposium on Computational Geometry*.

Monique Teillaud was a member of the program committee of WADS,
*Algorithms and Data Structures Symposium*.

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* and a member of the editorial board 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.

Monique Teillaud was an invited speaker of CATS, *Computational & Algorithmic Topology*, Sydney, Australia, June 27 – July 1st.

Guillaume Moroz was invited to give a talk at the Effective Geometry and Algebra seminar at IRMAR.

Monique Teillaud is chairing the Steering Committee of the Symposium on Computational Geometry (SoCG). She was a member of the Steering Committee of the European Symposium on Algorithms (ESA) until September.

Monique Teillaud is a member of the Scientific Board of the *Société Informatique
de France* (SIF).

Monique Teillaud acted as a reviewer for the DFG, *Deutsche Forschungsgemeinschaft* (German Research Foundation).

Olivier Devillers was the representative of LORIA in the hiring committee for an Associate Professor (MCF) position (IUT St Dié/LORIA) and composed the committee with the president.

L. Dupont is the secretary of *Commission Pédagogique
Nationale Carrières Sociales / Information-Communication / Métiers du Multimédia
et de l'Internet* (since May).

M. Teillaud is a member of the working group for the BIL,
*Base d'Information des Logiciels* of Inria.

O. Devillers: Elected member to *Pole AM2I* the council that
gathers labs in mathematics, computer science, and control theory at
*Université de Lorraine*.

L. Dupont Instigator (June 2016) and head of the Bachelor diploma
*Licence Professionnele Animation des Communautés et Réseaux Socionumériques*, Université de Lorraine.

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: Olivier Devillers, *Synthèse, image et géométrie*, 12h (academic year 2017-18), IPAC-R, Université de Lorraine.
https://

Master: Olivier Devillers and Monique Teillaud, *Computational Geometry*, 24h (academic year 2017-18), Master2 Informatique, ENS Lyon
https://

Licence: Sény Diatta, *Algorithme et Programmation*, 54h, L1, Université de Lorraine, France.

Licence: Sény Diatta, *Outils Informatiques et Internet*, 42h, L1, Université de Lorraine, France.

Licence: Charles Duménil, *Mathématiques*, 42h, L2, Université de Lorraine, France.

Licence: Charles Duménil, *Logiciel*, 20h, L2, Université de Lorraine, France.

Licence: Charles Duménil, *Algorithmique et programmation avancée*, 34h, M2, 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 *Databases* 30h L3, Université de Lorraine, France,

Licence: Laurent Dupont *Web devloppment and Social networks* 80h L3, Université de Lorraine, France.

Licence: Iordan Iordanov, *Algorithmique et Programmation*, 64h, L1, Université de Lorraine, France.

Licence: Iordan Iordanov, *Systèmes de gestion de bases de données*, 20h, L2, Université de Lorraine, France.

Licence: Iordan Iordanov, *Algorithmique et développement web*, 28h, L2, Université de Lorraine, France.

Licence: Iordan Iordanov, *Programmation objet et événementielle*, 16h, L3, Université de Lorraine, France.

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

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

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.

PhD in progress: George Krait, Topology of singular curves and surfaces, applications to visualization and robotics, started in Nov. 2017, supervised by Sylvain Lazard, Guillaume Moroz and Marc Pouget.

Postdoc: Vincent Despré, Triangulating surfaces with complex projective structures, started in Nov. 2017, supervised by Monique Teillaud.

Jian Qian, from Ècole Normale Supérieure Paris, did a L3 internship from Jul 2017 until Aug 2017 co-advised by Guillaume Moroz and Marc Pouget on a topic of ANR SingCAST.

Guillermo Alfonso Reyes Guzman, from Université de Lorraine, did a Master internship from March 2017 until July 2017 advised by O. Devillers on deletion in 3D Delaunay triangulation.

Camille Truong-Allie (Master 1, “research path”, École des Mines de Nancy), Lloyd algorithm in the flat torus, started in October, supervised by Monique Teillaud.

L. Dupont participated to several days of popularization of computerscience: Open Bidouille Camp March, 26th 2017, popularization of programming, general audience ; ISN day March, 30th 2017, popularization of computerscience for high-school teachers ; Fête de la Science 14th October 2017 Inria event, general audience, and Google Day in Nancy 21st October 2017, general audience.