Starting in the eighties, the emerging computational geometry community has put
a lot of effort into designing and analyzing 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:

Even if these limitations have already got some attention from the community 42, a quick look at the proceedings of the flagship conference SoCG1 shows that these topics still need a big effort.

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 mechanical 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 concepts
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 splines) 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.

$\u2022$ Discrete geometric structures. Many
geometric algorithms work, explicitly or implicitly, over discrete
structures such as graphs, hypergraphs, lattices that are
induced by the geometric input data. For example, convex hulls or
straight-line graph drawing are essentially based on orientation
predicates, and therefore operate on the so-called

As mentioned above, curved objects are ubiquitous in real world problems
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
a loss of accuracy which is sometimes not an issue but which can also be most problematic
depending on the application. In addition, discretization 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
49, 50, 48, 52, 58).
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 (that is auto-intersecting) 3D surfaces, due to the high constants in the
theoretical complexity and the difficulty of handling degenerate cases.
Part of the difficulty comes from the unintuitive fact that the structure of an algebraic
object can be quite
complicated, as depicted in the Whitney umbrella (see Figure 1), the surface with
equation

Thus 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 the 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 in its infancy. And it should be stressed that these two sets of results were
not anecdotal but flagship results produced during the lifetime of
the Vegas team (the team preceding Gamble).

There are many problems in this theme that are expected to have high long-term impacts. Intersecting NURBS (Non-uniform rational basis splines) 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 intersection2 which was used by researchers in, for instance, photochemistry, computer vision, statistics and mathematics.

Triangulations, in particular Delaunay triangulations, in the
Euclidean spaceet al.29). Some members of Gamble have been contributing to these algorithmic advances
(see, e.g. 34, 68, 47, 33); they have also
contributed robust and efficient triangulation packages through the
state-of-the-art Computational Geometry Algorithms Library
Cgal whose impact extends far
beyond computational geometry.
Application fields include particle physics, fluid dynamics, shape
matching, image processing, geometry processing, computer graphics,
computer vision, shape reconstruction, mesh generation, virtual
worlds, geophysics, and medical
imaging.3

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
domains4
ranging from the infinitely small (nuclear matter, nano-structures,
biological data) to the infinitely large (astrophysics) have led us to
consider 3D periodic Delaunay triangulations, which can be seen
as Delaunay triangulations of the 3D flat torus, i.e., the quotient of
Cgal package 39 is the only publicly available software
that computes Delaunay triangulations of
a 3D flat torus, in the special case where the domain is cubic. This case, although restrictive, is already useful.5
We have also generalized this algorithm
to the case of general

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 38.

Moreover, while our solution for computing triangulations in
hyperbolic spaces can be considered as ultimate 31, 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
64, neuromathematics 43, or physics
65. Even the case of the Bolza surface (a surface
of genus 2, whose fundamental domain is the regular octagon in the
hyperbolic plane) shows mathematical
difficulties 32, 55.

In most computational geometry papers, algorithms are analyzed in the worst-case setting. This often yields too pessimistic complexities that arise only in pathological situations that are unlikely to occur in practice. On the other hand, probabilistic geometry provides analyses with great precision 62, 63, 37, but using hypotheses with much more randomness than in most realistic situations. We are developing new algorithmic designs improving state-of-the-art performance in random settings that are not overly simplified and that can thus reflect many realistic situations.

Sixteen years ago, smooth analysis was introduced by Spielman and Teng analyzing the simplex algorithm by averaging on some noise on the data 67 (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 44 and we only obtained in 2015 breakthrough results, but still not definitive 46, 45, 51.

Another example of a 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” 28 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-offs between full randomness and worst case may also appear in other forms such as dependent distributions, or random distributions conditioned to be in some special configurations. In particular, simulating geometric distributions with repulsive properties, such as the determinantal point process, is currently out of reach for more than a few hundred points 54. Yet it has practical applications in physics to simulate particules with repulsion such as electrons 59, to simulate the distribution of network antennas 30, or in machine learning 56.

Our work on discrete geometric structures develops in several directions, each one probing a different type of structure. Although these objects appear unrelated at first sight, they can be tackled by the same set of probabilistic and topological tools.

A first research topic is the study of Order types. Order types
are combinatorial encodings of finite (planar) point sets, recording
for each triple of points the orientation (clockwise or
counterclockwise) of the triangle they form. This already determines
properties such as convex hulls or half-space depths, and the
behaviour of algorithms based on orientation predicates. These
properties for all (infinitely many)

A second research topic is the study of Embedded graphs and
simplicial complexes. Many topological structures can be
effectively discretized, for instance combinatorial maps record
homotopy classes of embedded graphs and simplicial complexes represent
a large class of topological spaces. This raises many structural and
algorithmic questions on these discrete structures; for example, given
a closed walk in an embedded graph, can we find a cycle of the graph
homotopic to that walk? (The complexity status of that problem is
unknown.) Going in the other direction, some purely discrete
structures can be given an associated topological space that reveals
some of their properties (e.g. the Nerve theorem for
intersection patterns). An open problem is for instance to obtain
fractional Helly theorems for set systems of bounded topological
complexity.

Another research topic is that of Sparse inclusion-exclusion formulas.
For any family of sets

where

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 6 (which was the first and still is the only one —to our knowledge— to compute robustly and efficiently the intersection of 3D quadrics) which has been used by researchers in, for instance, photochemistry, computer vision, statistics, and mathematics. Our work on certified drawing of plane (algebraic) curves falls in the same category. It seems obvious that it is widely useful to be able to draw curves correctly (recall also that part of the problem is to determine where to look in the plane) but it is quite hard to come up with specific examples of fields where this is relevant. A contrario, we know that certified meshing is critical in mechanical-design applications in robotics, which is a non-obvious application field. There, the singularities of a manipulator often have degrees higher than 10 and meshing the singular locus in a certified way is currently out of reach. As a result, researchers in robotics can only build physical prototypes for validating, or not, the approximate solutions given by non-certified numerical algorithms.

The fact that several of our
pieces of software for computing non-Euclidean triangulations
had already been requested by users
long before they become public in Cgal is a good sign for their wide future
impact. 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. Rien van de
Weijgaert 53, 69 (astrophysicist,
Groningen, NL) and Michael Schindler 66
(theoretical physicist, ENSPCI, CNRS, France) used our
software for 3D periodic weighted triangulations.
Stephen Hyde and Vanessa Robins (applied mathematics and
physics at Australian National University) used our package 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 43 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 poisonous atmosphere maintained by the management of Inria towards the researchers creates an unhealthy climate not favourable to the "serene and efficient research" that is advocated by the COP.

We address the problem of computing a drawing of high
resolution of a plane curve defined by a bivariate polynomial equation

One of the challenges for computing drawings on a high-resolution grid is to minimize the complexity due to the evaluation of the input polynomial. Most state-of-the-art approaches focus on bounding the number of independent evaluations. Using state-of-the-art Computer Algebra techniques, we design new algorithms that amortize the evaluations and improve the complexity for computing such drawings.

Our main contribution is to use a non-uniform grid based on the Chebyshev nodes to take advantage of multipoint evaluation techniques via the Discrete Cosine Transform. We propose two new algorithms that compute drawings and compare them experimentally on several classes of high degree polynomials. Notably, one of those approaches is faster than state-of-the-art drawing software. 18.

We present a new data structure to approximate accurately and
efficiently a polynomial

Given a polynomial Python/NumPy that is an order of magnitude faster than the
state-of-the-art solver MPSolve for high degree polynomials
with random coefficients. This result was presented in 2022 at the
FOCS 2021 conference 20.

The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm finishes in polynomial time, in terms of the initial perimeter length and the genus of the surface 27.

In collaboration with Benedikt Kolbe (now at University of Bonn) and Hugo Parlier (University of Luxembourg).

We proved that the size of the Delaunay triangulation of set of points

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):

Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points

15.
In collaboration with Emo Welzl (ETH Zürich)

We prove that for any set

In collaboration with Otfried Cheong and Andreas Holmsen (KAIST)

We prove that there exist no weak

In collaboration with Otfried Cheong and Andreas Holmsen (KAIST)

This article extends the work of Flajolet on the relation between generating series and inherent ambiguity. We first propose an analytic criterion to prove the infinite inherent ambiguity of some context-free languages, and apply it to give a purely combinatorial proof of the infinite ambiguity of Shamir’s language. Then we show how Ginsburg and Ullian’s criterion on unambiguous bounded languages translates into a useful criterion on generating series, which generalises and simplifies the proof of the recent criterion of Makarov. We then propose a new criterion based on generating series to prove the inherent ambiguity of languages with interlacing patterns, like

Company: Waterloo Maple Inc.

Duration: 2 years, renewable

Participants: Gamble and Ouragan Inria teams

Abstract:
A renewable two-years licence and cooperation agreement was signed on April 1st, 2018
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 Gamble and Ouragan (Paris), and it is coordinated by Fabrice
Rouillier (Ouragan).

F. Rouillier and Gamble are the developers of the Isotop software for the computation
of topology of curves. The transfer of a version of Isotop to Waterloo Maple Inc. should be done on the long run.

This contract was amended last year to include the new software hefroots for the isolation of the complex roots of a univariate polynomial. The transfer of hefroots to Waterloo Maple Inc. started at the end of 2021 with the help of the independent contractor Rémi Imbach. Rémi Imbach was then hired for one year by Inria through the ADT program and hefroots will be included in Maple 2023.

Company: GeometryFactory

Duration: permanent

Participants: Inria and GeometryFactory

Abstract:
Cgal packages developed in Gamble are commercialized by GeometryFactory.

Monique Teillaud coordinated the organization of the workshop Structures on Surfaces (CIRM, Marseille, May 2-6), which gathered more than 60 participants.

David Eppstein (University of California, Irvine, USA) was invited to give a talk in the framework of the (virtual) SoS seminar.

Xavier Goaoc was co-chair of SoCG'2022, the flagship conference of compuational geometry.

Monique Teillaud was a member of the program committee of EuroCG, the 38th European Workshop of Computational Geometry (Perugia, Italy, March 14-16). She was also a member of the committee for the Young Researchers Forum of CGWeek, the Computational Geometry Week (Berlin, Germany, June 7–10).

All members of the team are regular reviewers for the conferences of our field, namely Symposium on Computational Geometry (SoCG), European Symposium on Algorithms (ESA), Symposium on Discrete Algorithms (SODA), International Symposium on Symbolic and Algebraic Computation (ISSAC), etc.

Monique Teillaud is a managing editor of JoCG, Journal of Computational Geometry.

All members of the team are regular reviewers for the journals of our field, namely Discrete and Computational Geometry (DCG), 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.

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

Monique Teillaud was invited to give a (virtual) talk at the NYC Geometry Seminar “Flipping Geometric Triangulations on Hyperbolic Surfaces”.

Team members are involved in various committees managing the scientific life of the lab or at a national level.