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 43, 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
51, 52, 50, 54, 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.30). Some members of Gamble have been contributing to these algorithmic advances
(see, e.g. 35, 68, 48, 34); 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 40 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 39.

Moreover, while our solution for computing triangulations in
hyperbolic spaces can be considered as ultimate 32, 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 44, 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 337.

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, 38, 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 45 and we only obtained in 2015 breakthrough results, but still not definitive 47, 46, 53.

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” 29 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 56. Yet it has practical applications in physics to simulate particules with repulsion such as electrons 59, to simulate the distribution of network antennas 31, or in machine learning 57.

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 55, 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 44 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 CGAL project, in which we are deeply involved, received a SoCG Test of Time award. Monique Teillaud and Andreas Fabri (GeometryFactory) gave the (video) presentation at the award ceremony.

Evaluating or finding the roots of a polynomial pwpoly that is in the process of being included in Maple 2024.

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

This paper exhibits and analyzes 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 runs in polynomial time, in terms of the initial perimeter and the genus of the surface 15.

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

The Colin de Verdière graph parameter

In collaboration with
Francis Lazarus (CNRS, G-SCOP) and
Rudi Pendavingh ( TU/e - Eindhoven University of Technology).

Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices and edges of the graph on the surface (avoiding the obstacles) to remove all crossings; in other words, whether the drawing is homotopic to an embedding. While the problem boils down to planarity testing when the surface is the sphere or the disk (or equivalently the plane without any obstacle), the other cases have never been studied before, except when the input graph is a cycle, in an abundant literature in topology and more recently by Despré and Lazarus [SoCG 2017, J. ACM 2019], who gave a near-linear algorithm for this problem. Our algorithm runs in

In collaboration with
Éric Colin de Verdière (CNRS, Laboratoire d'Informatique Gaspard-Monge).

We consider infinite staircase translation surfaces with varying step sizes. We show that for typical step sizes, up to scaling, the translation flow has a unique invariant, non-atomic, ergodic Radon measure in almost every direction 25.

In collaboration with Serge Troubetzkoy (Institut de Mathématiques de Marseille).

We present two lower bounds that hold with high probability for random point sets. We first give a new, and elementary, proof that the classical models of random point sets (uniform in a smooth convex body, uniform in a polygon, Gaussian) have a superconstant number of extreme points with high probability. We next prove that any algorithm that determines the orientation of all triples in a planar set of

In collaboration with
Denys Bulavka (Charles University, Prague) ,
Philippe Duchon (LaBRI, Bordeaux), and
Marc Glisse (Inria-DATASHAPE).

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):

(a) The number of extreme points in an

(b) The (labeled) order types read off a set of

Result (a) generalizes to arbitrary dimension

For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of Journal of the ACM 13.

In collaboration with Emo Welzl (ETH Zürich).

We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Bárány and Kalai regarding weak Discrete & Computational Geometry 12.

In collaboration with Otfried Cheong (SCALGO) and Andreas Holmsen (KAIST).

We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets, and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.

In collaboration with Mathilde Bouvel (INRIA project Team MOCQUA) and Valentin Feray (IECL, Nancy).

We consider the problem of designing fast and compact solutions for representing the connectivity information of triangle meshes. While traditional data structures (Half-Edge, Corner Table) are fast and user-friendly, they tend to be memory-expensive. On the other hand, compression schemes, while meeting information-theory lower bounds, lack the ability to facilitate rapid navigation within the mesh structure. Compact representations provide an advantageous balance for representing large meshes, enabling a judicious compromise between memory consumption and fast implementation of navigational operations. We propose new representations that are sensitive to the regularity of the graph while still having worst case guarantees. For all our data structures we have both an interesting storage cost, typically 2 or 3 r.p.v. (references/vertex) in the case of very regular triangulations and provable upper bounds in the worst case scenario. One of our solution has a worst case cost of 3.33 r.p.v. which is currently the best-known bound improving the previous 4 r.p.v. 28. In terms of running time, our representations are slightly slower (factor 1.5 to 4) than classical data structures. In our experiments we compare on various meshes runtime and memory performance of our representations with those of the most efficient existing solutions 21.

In collaboration with Luca Castelli Aleardi (LIX)

We present a drawing of

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. This led to the inclusion of hefroots in Maple 2023, and to the development of a improved software pwpoly that is planned to be included in Maple 2024.

Company: GeometryFactory

Duration: permanent

Participants: Inria and GeometryFactory

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

SoS is co-funded by ANR (ANR-17-CE40-0033) and FNR (INTER/ANR/16/11554412/SoS) as a PRCI (Projet de Recherche Collaborative Internationale).

The central theme of this project is the study of geometric and combinatorial structures related to surfaces and their moduli. Even though they work on common themes, there is a real gap between communities working in geometric topology and computational geometry and SoS aims to create a long-lasting bridge between them. Beyond a common interest, techniques from both ends are relevant and the potential gain in perspective from long-term collaborations is truly thrilling.

In particular, SoS aims to extend the scope of computational geometry, a field at the interface between mathematics and computer science that develops algorithms for geometric problems, to a variety of unexplored contexts. During the last two decades, research in computational geometry has gained wide impact through CGAL, the Computational Geometry Algorithms Library. In parallel, the needs for non-Euclidean geometries are arising, e.g., in geometric modeling, neuromathematics, or physics. Our goal is to develop computational geometry for some of these non-Euclidean spaces and make these developments readily available for users in academia and industry.

To reach this aim, SoS follows an interdisciplinary approach, gathering researchers whose expertise cover a large range of mathematics, algorithms and software. A mathematical study of the objects considered is performed, together with the design of algorithms when applicable. Algorithms are analyzed both in theory and in practice after prototype implementations, which are improved whenever it makes sense to target longer-term integration into CGAL.

Our main objects of study are Delaunay triangulations and circle patterns on surfaces, polyhedral geometry, and systems of disjoint curves and graphs on surfaces.

The MinMax projet is funded by ANR under number ANR-19-CE40-0014

This collaborative research project aims to bring together researchers from various areas – namely, geometry and topology, minimal surface theory and geometric analysis, and computational geometry and algorithms – to work on a precise theme around min-max constructions and waist estimates.

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.

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.

Rencontres Arithmétiques du GdR Informatique Mathématique (October, Nancy).

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