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 , a quick look at the proceedings of the flagship conference SoCG 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
, , , , ).
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.
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 ), 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 intersection which was used by researchers in, for instance, photochemistry, computer vision, statistics and mathematics.

Triangulations, in particular Delaunay triangulations, in the
Euclidean spaceet 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
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.

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
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 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.
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 .

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 case of the Bolza surface (a surface
of genus 2, whose fundamental domain is the regular octagon in the
hyperbolic plane) shows mathematical
difficulties , .

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 , , , 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 (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 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” 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. Simulating these kinds of geometric distributions is currently out of reach for more than a few hundred points although it has practical applications in physics or networks.

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 (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 , (astrophysicist,
Groningen, NL) and Michael Schindler
(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 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.

We developed a breakthrough method for the two fundamental problems of evaluating a polynomial on multiple points and finding its complex roots, with a bit complexity quasi-linear in the degree. In particular, our approach solves a problem that has been open for 50 years. It was selected to the prestigious conference FOCS , and the practical efficiency of this method led to a transfer contract with the private company Waterloo Maple Inc.

We propose a new algorithm to compute the topology of a real algebraic space curve. The novelties of this algorithm are a new technique to achieve the lifting step which recovers points of the space curve in each plane fiber from several projections and a weakened notion of generic position. As opposed to previous work, our sweep generic position does not require that

Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. This type of curve appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory. We also provide a semi-algorithm to check their validity. Finally, we present experiments, some of which are not reachable by other methods, and discuss the efficiency of our method , .

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

Given a polynomial

We call the condition number associated with
a perturbation of the

Our algorithms are well suited for random polynomials since

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 .

Guided by insights on the mapping class group of a surface, we give experimental evidence that the upper bound recently proven on the diameter of the flip graph of a surface by Despré, Schlenker, and Teillaud is largely overestimated. To obtain this result, we propose a set of techniques allowing us to actually perform experiments. We solve arithmetic issues by proving a density result on rationally described genus two hyperbolic surfaces, and we rely on a description of surfaces allowing us to propose a data structure on which flips can be efficiently implemented .

The Delaunay triangulation of a set of points

The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincaré disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider generalized Bolza surfaces In collaboration with Matthijs Ebbens and Gert Vegter (Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence, University of Groningen).

Given a set of points far neighbors in such a graph
when

Intersection patterns of convex sets in

In collaboration with Andreas Holmsen (KAIST) and Zuzana Patáková (Charles University of Praha)

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in

In collaboration with
Andreas Holmsen (KAIST) and
Cyril Nicaud (LIGM)

Company: Waterloo Maple Inc.

Duration: 2 years

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 is amended this 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 this year with the help of the independent contractor Rémi Imbach.

Company: GeometryFactory

Duration: permanent

Participants: Inria and GeometryFactory

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

A. Málaga co-organized a Fabrikathon in Marseille, October 11-13, 2021; a workshop on design and production of mathematical objects.

Five virtual seminars were organized in the framework of the SoS project

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), SIAM Journal on Discrete Mathematics (SIDMA), etc.

Members of the team were invited to give a talk at the following venues: Stochastic Geometry Days (O. Devillers, ), Séminaire francilien de géométrie algorithmique et combinatoire (M. Teillaud, ), Journées Nationales de Calcul Formel 2021 (G. Moroz, ), Copenhagen-Jerusalem combinatorics seminar (X. Goaoc, ), DGeCo seminar (X. Goaoc, ), Séminaire de systèmes dynamiques de l'IMT (Alba Málaga, ), Séminaire commun de géometrie de l'IECL (Alba Málaga), Séminaire virtuel francophone Groupes et Géométrie (Alba Málaga, ), as well as the seminars of the INRIA project teams LFANT (X. Goaoc) and OURAGAN (M. Teillaud).

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

Members of Gamble are occasionally reviewing proposals or applications for foreign research agencies (e.g., FWF, NWO, NSERC, ISF, etc.) or foreign universities. We are not giving more details here to preserve anonymity.

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

A. Málaga co-authored the winning image in the competition marking the 40th anniversary of the mathematical meeting center CIRM, related to the non-euclidean geometry research questions treated in the Gamble team.