GAMBLE

GAMBLE - 2024

2024Activity reportProject-TeamGAMBLE

RNSR: 201722240E

Research center Inria Centre at Université de Lorraine
In partnership with:Université de Lorraine
Team name: Geometric Algorithms & Models Beyond the Linear & Euclidean realm
In collaboration with:Laboratoire lorrain de recherche en informatique et ses applications (LORIA)
Domain:Algorithmics, Programming, Software and Architecture
Theme:Algorithmics, Computer Algebra and Cryptology

Keywords

Computer Science and Digital Science

A5.5.1. Geometrical modeling
A5.10.1. Design
A7.1. Algorithms
A8.1. Discrete mathematics, combinatorics
A8.3. Geometry, Topology
A8.4. Computer Algebra

1 Team members, visitors, external collaborators

Research Scientists

Olivier Devillers [Team leader, INRIA, Senior Researcher]
Sylvain Lazard [INRIA, Senior Researcher]
Guillaume Moroz [INRIA, Researcher]
Marc Pouget [INRIA, Researcher]
Monique Teillaud [INRIA, Senior Researcher]

Faculty Members

Vincent Despré [UL, Associate Professor]
Laurent Dupont [UL, Associate Professor]
Xavier Goaoc [UL, Professor]
Alba Marina Malaga Sabogal [UL, Associate Professor]

Post-Doctoral Fellow

Niloufar Fuladi [INRIA, Post-Doctoral Fellow]

PhD Students

Marguerite Bin [UL, from Sep 2024]
Loïc Dubois [UNIV GUSTAVE EIFFEL]
Camille Lanuel [UL]
Dorian Perrot [UL, from Sep 2024]
Leo Valque [UL, ATER]
Sarah Wajsbrot [UL]

Interns and Apprentices

Marguerite Bin [UL, Intern, until Mar 2024]
Srijan Chakraborty [UL, Intern, from May 2024 until Jul 2024]
Loïc Nembot Ndeffo [UL, Intern, from May 2024 until Aug 2024]

Administrative Assistants

Nathalie Bethus [CNRS, from Mar 2024]
Cecilia Olivier [INRIA]

External Collaborator

Valentin Feray [CNRS]

2 Overall objectives

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:

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 52, 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:

$•$ 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.

$•$ 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 $ℝ^{d}$ of computational geometry and encompass non-Euclidean spaces.

$•$ 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.

$•$ 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 order type of the input point set. Order types are a subclass of oriented matroids that remains poorly understood: for instance, we do not even know how to sample this space with reasonable bias. One of our goals is to contribute to the development of these foundations by better understanding these discrete geometric structures.

3 Research program

3.1 Non-linear computational geometry

Figure 1.a — Figure 1: Two views of the Whitney umbrella (on the left, the “stick” of the umbrella, i.e., the negative $z$ -axis, is missing). Right picture from [Wikipedia], left picture from [Lachaud et al.].

Figure 1.b — Figure 1: Two views of the Whitney umbrella (on the left, the “stick” of the umbrella, i.e., the negative $z$ -axis, is missing). Right picture from [Wikipedia], left picture from [Lachaud et al.].

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 59, 60, 58, 62, 67). 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 $x^{2} = y^{2} z$ whose origin (the “special” point of the surface) is a vertex of the arrangement induced by the surface while the singular locus is simply the whole $z$ -axis. 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 45, 4614.

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.

3.2 Non-Euclidean computational geometry

Figure 2.a — Figure 2: Left: 3D mesh of a gyroid (triply periodic surface) 70. Right: Simulation of a periodic Delaunay triangulation of the hyperbolic plane 41.

Figure 2.b — Figure 2: Left: 3D mesh of a gyroid (triply periodic surface) 70. Right: Simulation of a periodic Delaunay triangulation of the hyperbolic plane 41.

Triangulations, in particular Delaunay triangulations, in the Euclidean space $ℝ^{d}$ have been extensively studied throughout the 20th century and they are still a very active research topic. Their mathematical properties are now well understood, many algorithms to construct them have been proposed and analyzed (see the book of Aurenhammer et al.38). Some members of Gamble have been contributing to these algorithmic advances (see, e.g. 44, 77, 57, 43); 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 $ℝ^{3}$ under the action of some group of translations 50. This work has already yielded a fruitful collaboration with astrophysicists 63, 78 and new collaborations with physicists are emerging. To the best of our knowledge, our Cgal package 49 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 $d$ -dimensional compact flat manifolds 51. As far as non-compact manifolds are concerned, past approaches, limited to the two-dimensional case, have stayed theoretical 69.

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 48.

Moreover, while our solution for computing triangulations in hyperbolic spaces can be considered as ultimate 41, 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 73, neuromathematics 53, or physics 74. 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 4212.

3.3 Probability in computational geometry

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 71, 72, 47, 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 76 (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 54 and we only obtained in 2015 breakthrough results, but still not definitive 56, 55, 61.

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” 37 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 64. Yet it has practical applications in physics to simulate particules with repulsion such as electrons 68, to simulate the distribution of network antennas 39, or in machine learning 66.

3.4 Discrete geometric structures

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) $n$ -point sets can be studied through the finitely many order types of size $n$ . Yet, this finite space is poorly understood: its estimated size leaves an exponential margin of error, no method is known to sample it without concentrating on a vanishingly small corner, the effect of pattern exclusion or VC dimension-type restrictions are unknown. These are all directions we actively investigate.

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 $A_{1}, A_{2}, ..., A_{n}$ , by the principle of inclusion-exclusion we have

1_{⋃_{i = 1}^{n} A_{i}} = \sum_{I \subseteq {1, 2, ..., n}} {(- 1)}^{| I | + 1} 1_{⋂_{i \in I} A_{i}}

where $1_{X}$ is the indicator function of $X$ . This formula is universal (it applies to any family of sets) but its number of summands grows exponentially with the number $n$ of sets. When the sets are balls, the formula remains true if the summation is restricted to the regular triangulation; we proved that similar simplifications are possible whenever the Venn diagram of the $A_{i}$ is sparse. There is much room for improvements, both for general set systems and for specific geometric settings. Another interesting problem is to combine these simplifications with the inclusion-exclusion algorithms developed, for instance, for graph coloring.

4 Application domains

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 63, 78 (astrophysicist, Groningen, NL) and Michael Schindler 75 (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 53 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.

5 Highlights of the year

Note : Readers are advised that the Institute does not endorse the text in the “Highlights of the year” section, which is the sole responsibility of the team leader.

INRIA management imposed a new Contract of Objectives, Means and Performance (COMP) for the Institute at the end of 2024. The COMP defines the main mission of the Institute as "contributing to the digital sovereignty of the nation through research and innovation", and proposes to change the Institute's missions, organisation and operations in its founding decree. We fear this will have a detrimental impact on our team for the following reasons.

The COMP threatens to reduce the autonomy and independence of scientists and teams in choosing their research topics and collaborations.
The COMP adds new missions for the Institute. Without any new resources, this will reduce resources available for research.
The COMP announces a desire to place all of Inria in a “restricted regime area” (ZRR), which means that the hiring of researchers and interns will be reviewed and possibly vetoed by the Fonctionnaire Sécurité Défense. This creates administrative delays, subjects hiring to opaque criteria, and discourages the hiring of foreign nationals, thus harming research and collaboration.

6 New software, platforms, open data

6.1 New software

6.1.1 pwpoly

Keywords:
Evaluation, Polynomial equations, Complex number
Functional Description:
This software uses a new approach to evaluate and find the roots of polynomials whose coefficients do not all have the same order of magnitude. In particular, after a quasi-linear pre-processing in degree, the evaluation at a point is done in logarithmic time in degree, with the same precision as the evaluation of the original polynomial done with floating-point arithmetic. Moreover, for a well-conditioned polynomial, the calculation of the approximate roots is also performed in quasi-linear time in degree.
Publication:
hal-03980098
Contact:
Guillaume Moroz

6.1.2 voxelize

Keywords:
Visualization, Curve plotting, Implicit surface, Polynomial equations
Functional Description:
Voxelize is a C++ software to visualize the solutions of polynomial equations and inequalities. The software is optimized for high degree curves and surfaces. Internally, polynomials and sets of boxes are stored in the Compressed Sparse Fiber format. The output is either a mesh or a union of boxes written in the standard 3D file format ply.
Release Contributions:
Better parser, and publication of the underlying algorithm.
News of the Year:
The algorithms implemented in voxelize have been detailed and published in the conference CASC 2024 (Computer Algebra in Scientific Computing).
URL:
https://gitlab.inria.fr/gmoro/voxelize
Publication:
hal-04611464
Contact:
Guillaume Moroz

7 New results

7.1 Non-Linear Computational Geometry

Participants: Laurent Dupont, Nuwan Herath Mudiyanselage, Sylvain Lazard, Guillaume Moroz, Marc Pouget.

7.1.1 Fast evaluation and root finding for polynomials with floating-point coefficients

Evaluating or finding the roots of a polynomial $f (z) = f_{0} + \dots + f_{d} z^{d}$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of $f$ , we improved state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of $f$ are given with $m$ significant bits, we provided for the first time an algorithm that finds all the roots of $f$ with a relative condition number lower than $2^{m}$ , using a number of bit operations quasi-linear in the bit-size of the floating-point representation of $f$ . Notably, our new approach handles efficiently polynomials with coefficients ranging from $2^{- d}$ to $2^{d}$ , both in theory and in practice. This work was published at the ISSAC 2023 conference 65 and a journal version was accepted in 2024 for Journal of Symbolic Computation 20 and led to the development of the software pwpoly that was included in Maple 2024.

7.1.2 Towards the Computation of Stabilizing Controllers of Multidimensional Systems

In this paper, we continue the study of the effective computation of stabilizing controllers for internally stabilizable multidimensional systems. Within the algebraic analysis approach to linear systems theory, if $𝒜$ is the ring of multivariate rational functions without poles in the closed complex unit polydisc ${\bar{𝔻}}^{n}$ , then the stabilization problem can be characterized by the fact that a certain finitely presented $𝒜$ -module $ℳ$ , intrinsically associated with the system, is projective, which is equivalent to the fact that the ideal generated by all the minors of a presentation matrix of $ℳ$ is the whole ring $𝒜$ . This last condition can be reduced to the existence of an element $s$ of a polynomial ideal $ℐ$ which has no zero in ${\bar{𝔻}}^{n}$ . According to the Polydisc Nullstellensatz, the latter condition is equivalent to the fact that the common complex zeros of $ℐ$ do not belong to ${\bar{𝔻}}^{n}$ . If the condition for the Polydisc Nullstellensatz is satisfied, then, using cyclic resultants and linear programming, we propose a method to compute such a polynomial $s$ . Using computer algebra methods for handling $ℛ [s^{- 1}]$ -modules, where $ℛ$ is a polynomial ring, we then show how to compute a stabilizing controller. The results are implemented in Maple 26.

In collaboration with Thomas Cluzeau and Alban Quadrat.

7.1.3 Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations

Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods, such as subdivision, homotopy continuation, or marching cube algorithms, polynomial evaluation is treated as a black box, repeating the process for each point. We propose a new approach that partially evaluates the polynomials, allowing us to efficiently reuse computations across multiple points in a grid. Our method leverages the Compressed Sparse Fiber data structure to efficiently store and process subsets of grid points. We integrated our amortized evaluation scheme into a subdivision algorithm. Experimental results show that our approach is efficient in practice. Notably, our software voxelize can successfully enclose curves defined by two trivariate polynomial equations of degree 100, a problem that was previously intractable 28.

7.1.4 A Subquadratic Algorithm for Computing the $L_{1}$ -distance between Two Terrains

We study the problem of computing the $L_{1}$ -distance between two piecewise-linear bivariate functions $f$ and $g$ , defined over a common bounded domain $𝕄 \subset ℝ^{2}$ , that is, computing the quantity ${∥ f - g ∥}_{1} = \int_{𝕄} | f (x, y) - g (x, y) | d x d y$ . If $f$ and $g$ are defined by linear interpolation over triangulations $𝐓_{f}$ and $𝐓_{g}$ , respectively, of $𝕄$ with a total of $n$ triangles, we show that ${∥ f - g ∥}_{1}$ can be computed in $\tilde{O} (n^{(ω + 1) / 2})$ time, where $Θ (n^{ω})$ is the time required to multiply two $n \times n$ matrices and $\tilde{O}$ notation hides polylogarithmic factors; this bound holds for the currently best known value of $ω$ , which is approximately $2.37$ . The previously best known algorithm for computing ${∥ f - g ∥}_{1}$ takes $Θ (n^{2})$ time in the worst case.

More generally, if the complexity of the overlay of $𝐓_{f}$ and $𝐓_{g}$ is $κ$ , then the runtime of our algorithm is $\tilde{O} (κ^{(ω - 1) / 2} n^{(3 - ω) / 2})$ 31.

In collaboration with Pankaj K. Agarwal and Boris Aronov.

7.1.5 Algebraic data structures and approximations for efficient geometric modeling (HDR, Guillaume Moroz)

This HDR manuscript presents work at the intersection of computer algebra, algorithmic geometry, and their applications in robotics and computational geometry. It focuses on the underlying problems of evaluating polynomials and finding the zero set of systems of equations over the reals. The research is based on three main approaches: dimension reduction techniques for solving systems of equations and describing singular curves; reduction of redundant computations through fast multipoint evaluation algorithms, leading to breakthroughs in problems such as calculating distances between triangulations; and development of certified approximation methods, reducing precision requirements while maintaining result correctness. These contributions have led to the development of dedicated software for robotics researchers, and general-purpose implementations of fast multipoint polynomial evaluation and root-finding algorithms 30.

7.1.6 3D snap rounding (PhD thesis, Léo Valque)

Most algorithms for processing 3D polygonal objects use fixed-precision coordinates for both input and output data. However, geometric operations often produce output coordinates that require higher precision than the input. This discrepancy implies the need for rounding new coordinates to match the precision of the input, while preserving the integrity of the model. The critical problem we address is the removal of self-intersections in 3D models, achieved by subdividing faces along their intersections and rounding the resulting coordinates, while ensuring that the model remains free from self-intersections. This problem is known as the snap rounding problem. In this thesis, we present the first practical and certified local 3D snap rounding algorithm, which successfully eliminates self-intersections in 94% of the self-intersecting models in the Thingi10K dataset, demonstrating its effectiveness in real-world applications. Additionally, we introduce an uncertified yet highly efficient heuristic for this problem, which outperforms previous state-of-the-art methods by successfully resolving all self-intersections in the Thingi10K dataset.

7.1.7 An improved complexity bound for computing the topology of a real algebraic space curve

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 $x$ -critical points have different $x$ -coordinates. The complexity of achieving this sweep generic position is thus no longer a bottleneck in terms of complexity. The bit complexity of our algorithm is $O (d^{18} + d^{17} t)$ where $d$ and $t$ bound the degree and the bitsize of the integer coefficients of the defining polynomials of the curve and polylogarithmic factors are ignored. To the best of our knowledge, this improves upon the best currently known results at least by a factor of $d^{2}$ 17.

7.2 Non-Euclidean Computational Geometry

Participants: Vincent Despré, Loïc Dubois, Camille Lanuel, Alba Marina Málaga Sabogal, Marc Pouget, Monique Teillaud.

7.2.1 $ϵ$ -Net Algorithm Implementation on Hyperbolic Surface

We propose an implementation, using the Cgal library, of an algorithm to compute $ϵ$ -nets on hyperbolic surfaces initially presented in 29. We describe the data structure, detail the implemented algorithm and report experimental results on hyperbolic surfaces of genus 2. The implementation differs from the cited algorithm on several aspects. In particular, we use a different data structure, using a combinatorial map, to represent a triangulation of a surface. Also for the critical step of locating points on the surface, we use the visibility walk and prove its termination in the hyperbolic setting 33.

7.2.2 Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

The Delaunay triangulation of a set of points $P$ on a hyperbolic surface is the projection of the Delaunay triangulation of the set $\tilde{P}$ of lifted points in the hyperbolic plane. Since $\tilde{P}$ is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than $12 g - 6$ with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

In collaboration with Benedikt Kolbe, University of Bonn

7.2.3 Flipping geometric triangulations on hyperbolic surfaces

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We prove that any Delaunay triangulation is geometric, and give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation.

In collaboration with Jean-Marc Schlenker, University of Luxembourg

7.3 Discrete Geometric structures

Participants: Florent Koechlin, Xavier Goaoc, Sarah Wajsbrot.

7.3.1 A Canonical Tree Decomposition for Chirotopes

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. This work was presented at the SoCG 2024 conference 24 .

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

7.3.2 A fractional Helly theorem for set systems with slowly growing homological shatter function

We study parameters of the convexity spaces associated with families of sets in $ℝ^{d}$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$ . Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems 40.

8 Bilateral contracts and grants with industry

8.1 Bilateral contracts with industry

8.1.1 Waterloo Maple Inc.

Participants: Laurent Dupont, Sylvain Lazard, Guillaume Moroz, Marc Pouget, Rémi Imbach.

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 included in Maple 2024.

8.1.2 GeometryFactory

Participants: Vincent Despré, Loïc Dubois, Camille Lanuel, Marc Pouget, Monique Teillaud.

Company: GeometryFactory

Duration: permanent

Participants: Inria and GeometryFactory

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

9 Partnerships and cooperations

9.1 International research visitors

9.1.1 Visits of international scientists

Other international visits to the team

Victor Chepoi

Status
Researcher
Institution of origin:
Aix-Marseille University
Country:
France
Dates:
Janvier
Context of the visit:
Collaboration on geometric graph theory
Mobility program/type of mobility:
Research stay

Boris Aronov

Status
Researcher
Institution of origin:
New York University
Country:
USA
Dates:
Janvier
Context of the visit:
Workshop on computational geometry
Mobility program/type of mobility:
Lecture series

Nicolas Chenavier

Status
Researcher
Institution of origin:
Université du Littoral Côte d'Opale
Country:
France
Dates:
Mars
Context of the visit:
Research on stochastic geometry
Mobility program/type of mobility:
Research stay

Denys Bulavka

Status
Post-Doc
Institution of origin:
Hebrew University of Jerusalem
Country:
Israel
Dates:
Septembre
Context of the visit:
Research on combinatorial geometry and topology
Mobility program/type of mobility:
Research stay

Sergey Avvakumov

Status
Researcher
Institution of origin:
Tel Aviv University
Country:
Israel
Dates:
Novembre
Context of the visit:
Research on topological combinatorics
Mobility program/type of mobility:
Research stay

Jean Cardinal

Status
Researcher
Institution of origin:
Université libre de Bruxelles
Country:
Belgium
Dates:
Décembre
Context of the visit:
Seminar on geometric algorithms
Mobility program/type of mobility:
Lecture

9.2 National initiatives

9.2.1 ANR JCJC

Participants: Vincent Despré.

ANR Abysm

Title:
Abysm
Duration:
2024 to 2028
Coordinator:
Vincent Despré (Université de Lorraine)
Inria contact:
Vincent Despré
Summary:
The central theme of this project is the study of geometric and combinatorial structures related to hyperbolic surfaces and their moduli from an algorithmic point of view. The needs for hyperbolic geometries are arising, e.g., in crystallography, in geometric modeling, neuromathematics, or physics. The generic need regarding computer science in all those examples is clearly stated in a very recent paper on Nature Communications: "Spaces with negative curvature are difficult to realize and investigate experimentally". In order to solve this issue, our goal is to develop the study of hyperbolic surfaces in computational geometry and make our results readily available for users. We intend to design efficient and algorithms with precise data structures to compute geometrical characteristics of hyperbolic surfaces such as the systole, the diameter and optimal pants decompositions. We also want to study the regularity of the previous parameters while moving through the Teichmüller and moduli spaces. We plan to implement our algorithms and make them publicly available to users.

10 Dissemination

10.1 Promoting scientific activities

10.1.1 Scientific events: selection

Reviewer

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.

10.1.2 Journal

Reviewer - reviewing activities

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.

10.1.3 Software Project

Member of the Editorial Boards

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

10.1.4 Invited talks

Xavier Goaoc gave an invited talk "Intersection patterns of geometric set systems" at the European Conference on Computational Geometry (March, Ioannina, Greece).
Monique Teillaud gave a talk "Triangulations, CGAL, and hyperbolic surfaces" at the seminar of the Applied Algebraic Topology Research Network
Monique Teillaud presented the CGAL project at the Colloque Sciences Ouvertes 2024.

10.1.5 Leadership within the scientific community

Xavier Goaoc was a coorganizer of the Discrete Geometry workshop at the Mathematisches Forschungsinstitut (MFO). (January, Oberwolfach).

Monique Teillaud is a member of the Task Force formed by the Computational Geometry Steering Committee, discussing applied, experimental, and engineering aspects of computational geometry and topology.

10.1.6 Research administration

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

Local

INRIA Commission Information et Édition Scientifique (Laurent Dupont ),
INRIA Comité de centre (Xavier Goaoc ),
LORIA Conseil scientifique (Sylvain Lazard ),
LORIA associate director (Sylvain Lazard ),
Computer science board in the École doctorale IAEM (Sylvain Lazard ,Xavier Goaoc ),
Conseil du Pole scientifique Am2I of University of Lorraine (Xavier Goaoc )
INRIA Comité des utilisateurs des moyens informatiques (chair, Guillaume Moroz )
INRIA Commission de développement technologique (Guillaume Moroz ),
CLHSCT (Guillaume Moroz ),
INRIA and LORIA PhD and postdoc hiring committee (Marc Pouget ),
Member of the mentoring committee at LORIA (Monique Teillaud ),

National

Xavier Goaoc was a member of the steering committee of the conference Geometry and Computing.
INRIA Mission Jeunes Chercheurs (chair, Sylvain Lazard ).

Hiring Committees

Olivier Devillers chaired the hiring committee for a Professor position at Polytech (Université de Lorraine)
Xavier Goaoc chaired the hiring committee for an assistant professor position at LIPN and IUT St Denis (Université Sorbonne Paris Nord)
Xavier Goaoc was vice-chair of the hiring committee for an assistant professor position at LORIA and École des Mines (Université de Lorraine)
Sylvain Lazard chaired the hiring committee for two assistant professor positions at Université de Lorraine (FST-LORIA).

10.1.7 Teaching Committees

Vincent Despre : Head of the Engineer diploma speciality SIR, Systèmes d'Information et Réseaux, Polytech Nancy, Université de Lorraine.
Laurent Dupont : Head of the Bachelor diploma Licence Professionnelle Animateur, Facilitateur de Tiers-lieux Eco-Responsables, Université de Lorraine (not open this year)
Laurent Dupont : Responsible for the course "Création Numérique" of the Bachelor (BUT) "Métiers du Multimédia et de l'Internet"
Laurent Dupont : Responsible for fablab "Charlylab" of I.U.T. Nancy-Charlemagne,
Xavier Goaoc is the chair of the computer science department of l'École des Mines de Nancy.
Xavier Goaoc is a member of the Conseil d'administration de l'École des Mines de Nancy.

10.2 Teaching - Supervision - Juries

10.2.1 Teaching

Licence: Vincent Despre , Algorithmique, 44h, L2 PEIP, Polytech Nancy, France.
Licence: Vincent Despre , Programmation orientée objet, 84h, L3 IA2R, Polytech Nancy, France. (web).
Licence: Laurent Dupont , Web development, 45h, L1, Université de Lorraine, France.
Licence: Laurent Dupont , Web development, 150h, L2, Université de Lorraine, France.
Licence: Laurent Dupont , Web development, 70h, L3, Université de Lorraine, France.
Licence: Laurent Dupont , 3D printing and CAO 40h, L3, Université de Lorraine, France.
Licence : Xavier Goaoc , Algorithms and complexity, 60 HETD, L3, École des Mines de Nancy, France.
Master: Xavier Goaoc , Computer architecture, 32 HETD, M1, École des Mines de Nancy, France.
Master: Xavier Goaoc , Introduction to blockchains, 32 HETD, M1, École des Mines de Nancy, France.
Master: Xavier Goaoc , Réalité augmentée et modèles géométriques pour la vision, 12h, M2 AVR, Université de Lorraine, France
Licence: Alba Marina Malaga Sabogal , Information systems and databases, 86h, L1, Université de Lorraine, France.
Licence: Alba Marina Malaga Sabogal , Content Management Systems, 40h, L1, Université de Lorraine, France.
Master: Guillaume Moroz , Software Engineering, 10h, M1, École des Mines de Nancy, France.
Master: Marc Pouget , Introduction to computational geometry, 10.5h, M2, École Nationale Supérieure de Géologie, France.

10.2.2 Supervision

PhD in progress: Marguerite Bin , Order types: decomposition and complexity, started in Sept. 2024, supervised by Xavier Goaoc and Alfredo Hubard (LIGM, Université Gustave Eiffel).
PhD in progress: Dorian Perrot , Hyperbolic surfaces and computational geometry, started in Sept. 2024, supervised by Vincent Despre and Marc Pouget .
PhD in progress: Loïc Dubois , Untangling graphs on surfaces, started in Oct. 2022, supervised by Vincent Despre and Éric Colin de Verdière (Marne la Vallée).
PhD in progress: Camille Lanuel , A toolbox for hyperbolic surfaces, started in Oct. 2022, supervised by Vincent Despre and Monique Teillaud .
PhD defended in Dec. 2024: Leo Valque , 3D Snap Rounding, supervised by Sylvain Lazard .
PhD in progress: Sarah Wajsbrot , Combinatorial convexity, its generalizations and applications to optimization, started in Oct. 2023, supervised by Xavier Goaoc .
Master internship M2: Marguerite Bin , Homological VC dimension following Kalai and Meshulam, Jan-March 2024.

10.2.3 Juries

Xavier Goaoc chaired the PhD defense committee of Antoine Leudière, Université de Lorraine.
Xavier Goaoc was on the reading and defense committees of the PhD thesis of Florent Tallerie, Université Grenoble Alpes.
Alba Malaga was on the defense committee of the PhD thesis of Florent Tallerie, Université Grenoble Alpes.

10.3 Popularization

10.3.1 Education

Olivier Devillers presented research career in several different classes in highschool within the Chiche program.
Alba Málaga for the math festivals "Printemps des mathématiques" (March, 23-25 2024), "Forum des Mathématiques" (March 17, 2024), "Salon de Culture et Jeux Mathématiques" (May 23-26, 2024) and the maker fair Empower Paris-Saclay (September 27-28, 2024), coordinated and participated in a booth promoting sharing platforms for mathematical communication like Imaginary, or kits math.
Alba Málaga is a member of the scientific board for the association Les maths en scène and the marraine for a high-school math student club in Thionville, le labo Rosa Parks.
Guillaume Moroz is member of the Olympiades committee of the Académie Nancy-Metz.

10.3.2 Interventions

Alba Málaga gave a general audience talk for motivated high school students, «Exposé de géométrie, sur la courbure, illustré par des objets.», at the MATh.en.JEANS, annual South-West meeting, closing conference, Montpellier, 03/05/2024
Alba Marina Malaga Sabogal , Laurent Dupont , Marc Pouget , Dorian Perrot , Leo Valque and Marguerite Bin organized a workshop "Coloriage avec un crayon fin : boostez vos courbes !" for the "Fête de la science" in October 11-12 in Nancy.

11 Scientific production

11.1 Major publications

1 inproceedingsN.Nicolas Bonichon, P.Prosenjit Bose, J.-L.Jean-Lou De Carufel, V.Vincent Despré, D.Darryl Hill and M.Michiel Smid. Improved Routing on the Delaunay Triangulation.ESA 2018 - 26th Annual European Symposium on AlgorithmsHelsinki, Finland2018HAL DOI
2 articleL.Luca Castelli Aleardi and O.Olivier Devillers. Array-based Compact Data Structures for Triangulations: Practical Solutions with Theoretical Guarantees.Journal of Computational Geometry912018, 247-289HAL DOI
3 articleN.Nicolas Chenavier and O.Olivier Devillers. Stretch Factor in a Planar Poisson-Delaunay Triangulation with a Large Intensity.Advances in Applied Probability5012018, 35-56HAL DOI
4 inproceedingsV.Vincent Despré, J.-M.Jean-Marc Schlenker and M.Monique Teillaud. Flipping Geometric Triangulations on Hyperbolic Surfaces.SoCG 2020 - 36th International Symposium on Computational GeometryZurich, Switzerland2020HAL DOI
5 articleO.Olivier Devillers, M.Menelaos Karavelas and M.Monique Teillaud. Qualitative Symbolic Perturbation: Two Applications of a New Geometry-based Perturbation Framework.Journal of Computational Geometry812017, 282--315HAL DOI
6 inproceedingsO.Olivier Devillers, S.Sylvain Lazard and W.William Lenhart. 3D Snap Rounding.Proceedings of the 34th International Symposium on Computational GeometryBudapest, HungaryJune 2018, 30:1--30:14HAL DOI
7 articleO.Olivier Devillers, S.Sylvain Lazard and W.William Lenhart. Rounding meshes in 3D.Discrete and Computational GeometryApril 2020HAL DOI
8 articleX.Xavier Goaoc, P.Pavel Paták, Z.Zuzana Patáková, M.Martin Tancer and U.Uli Wagner. Shellability is NP-complete.Journal of the ACM (JACM)6632019HAL DOI
9 inproceedingsX.Xavier Goaoc and E.Emo Welzl. Convex Hulls of Random Order Types.SoCG 2020 - 36th International Symposium on Computational Geometry16436th International Symposium on Computational Geometry (SoCG 2020)Best paper awardZürich / Virtual, Switzerland2020, 49:1--49:15HAL DOI
10 articleR.Rémi Imbach, G.Guillaume Moroz and M.Marc Pouget. Reliable Location with Respect to the Projection of a Smooth Space Curve.Reliable Computing262018, 13-55HAL
11 articleR.Rémi Imbach, M.Marc Pouget and C.Chee Yap. Clustering Complex Zeros of Triangular Systems of Polynomials.Mathematics in Computer ScienceJune 2020HAL DOI
12 inproceedingsI.Iordan Iordanov and M.Monique Teillaud. Implementing Delaunay Triangulations of the Bolza Surface.33rd International Symposium on Computational Geometry (SoCG 2017)Brisbane, AustraliaJuly 2017, 44:1--44:15HAL DOI back to text
13 articleR.Ranjan Jha, D.Damien Chablat, L.Luc Baron, F.Fabrice Rouillier and G.Guillaume Moroz. Workspace, Joint space and Singularities of a family of Delta-Like Robot.Mechanism and Machine Theory127September 2018, 73-95HAL DOI
14 articleS.Sylvain Lazard, M.Marc Pouget and F.Fabrice Rouillier. Bivariate triangular decompositions in the presence of asymptotes.Journal of Symbolic Computation822017, 123--133HAL DOI back to text
15 articleP.Pedro Machado Manhães De Castro and O.Olivier Devillers. Expected Length of the Voronoi Path in a High Dimensional Poisson-Delaunay Triangulation.Discrete and Computational Geometry6012018, 200--219HAL DOI
16 inproceedingsG.Guillaume Moroz. New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems.2021 IEEE 62nd Annual Symposimum on Foundations of Computer Science (FOCS)FOCS 2021 - 62nd Annual IEEE Symposimum on Foundations of Computer ScienceDenver, United StatesDecember 2021HAL DOI

11.2 Publications of the year

International journals

17 articleJ.-S.Jin-San Cheng, K.Kai Jin, M.Marc Pouget, J.Junyi Wen and B.Bingwei Zhang. An improved complexity bound for computing the topology of a real algebraic space curve.Journal of Symbolic Computation125November 2024HAL DOI back to text
18 articleV.Vincent Despré, B.Benedikt Kolbe and M.Monique Teillaud. Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains.Discrete and Computational GeometryMay 2024HAL DOI
19 articleV.Vincent Despré, J.-M.Jean-Marc Schlenker and M.Monique Teillaud. Flipping geometric triangulations on hyperbolic surfaces.Journal of Computational Geometry2024HAL DOI
20 articleR.Rémi Imbach and G.Guillaume Moroz. Fast evaluation and root finding for polynomials with floating-point coefficients.Journal of Symbolic Computation127March 2025, 102372HAL DOI back to text
21 articleF.Florent Koechlin and P.Pablo Rotondo. The effects of semantic simplifications on random BST-like expression-trees.Discrete Mathematics3475May 2024, 113906HAL DOI
22 articleC.Camille Lanuel, F.Francis Lazarus and R.Rudi Pendavingh. A linear bound for the Colin de Verdière parameter for graphs embedded on surfaces.SIAM Journal on Discrete Mathematics3832024, 2289-2296HAL DOI

International peer-reviewed conferences

23 inproceedingsL. C.Luca Castelli Aleardi and O.Olivier Devillers. SCARST: Schnyder Compact and Regularity Sensitive Triangulation Data Structure.Proceedings of the 40th International Symposium on Computational Geometry40th International Symposium on Computational Geometry (SoCG 2024)Athens, Greece2024HAL DOI
24 inproceedingsM.Mathilde Bouvel, V.Valentin Féray, X.Xavier Goaoc and F.Florent Koechlin. A Canonical Tree Decomposition for Chirotopes.40th International Symposium on Computational Geometry (SoCG 2024)40th International Symposium on Computational Geometry (SoCG 2024)Leibniz International Proceedings in Informatics (LIPIcs)Athens, GreeceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024HAL DOI back to text
25 inproceedingsD.Denys Bulavka, É.Éric Colin de Verdière and N.Niloufar Fuladi. Computing shortest closed curves on non-orientable surfaces.40th International Symposium on Computational Geometry (SoCG 2024)International Symposium on Computational GeometryAthens, GreeceSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024HAL DOI
26 inproceedingsT.Thomas Cluzeau, G.Guillaume Moroz and A.Alban Quadrat. Towards the Computation of Stabilizing Controllers of Multidimensional Systems.IFAC-Papers onlineMTNS 2024 - 6th International Symposium on Mathematical Theory of Networks and Systems5817Cambdridge, United KingdomElsevierOctober 2024HAL DOI back to text
27 inproceedingsÉ.Éric Colin de Verdière, V.Vincent Despré and L.Loïc Dubois. Untangling Graphs on Surfaces.Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete AlgorithmsSODA 2024Alexandria, United StatesSociety for Industrial and Applied Mathematics2024, 4909-4941HAL DOI
28 inproceedingsG.Guillaume Moroz. Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations.Proceedings of the 26th International Workshop on Computer Algebra in Scientific ComputingComputer Algebra in Scientific ComputingRennes, FranceSeptember 2024HAL back to text

Conferences without proceedings

29 inproceedingsV.Vincent Despré, C.Camille Lanuel and M.Monique Teillaud. Computing an -net of a closed hyperbolic surface.EuroCG'24 - 40th European Workshop on Computational GeometryIoannina, Greece2024HAL back to text

Doctoral dissertations and habilitation theses

30 thesisG.Guillaume Moroz. Algebraic data structures and approximations for efficient geometric modeling.Université de LorraineDecember 2024HAL back to text

Reports & preprints

31 miscP. K.Pankaj K. Agarwal, B.Boris Aronov and G.Guillaume Moroz. A Subquadratic Algorithm for Computing the L1-distance between Two Terrains.January 2025HAL back to text
32 miscM.Mathilde Bouvel, V.Valentin Féray, X.Xavier Goaoc and F.Florent Koechlin. A canonical tree decomposition for order types, and some applications.March 2024HAL
33 miscV.Vincent Despré, C.Camille Lanuel, M.Marc Pouget and M.Monique Teillaud. ε-Net Algorithm Implementation on Hyperbolic Surfaces.December 2024HAL back to text
34 miscV.Vincent Despré, C.Camille Lanuel and M.Monique Teillaud. Computing an -net of a closed hyperbolic surface.February 2024HAL
35 miscS.Sylvain Lazard and L.Leo Valque. Removing self-intersections in 3D meshes while preserving floating-point coordinates.January 2025HAL

Scientific popularization

36 inbookG.Guillaume Coiffier, S.Sewade Ogun, L.Leo Valque and P.Priyansh Trivedi. STATE OF THE ART.THINK BEFORE LOADING2024HAL

11.3 Cited publications

37 inproceedingsD.Dominique Attali, J.-D.Jean-Daniel Boissonnat and A.André Lieutier. Complexity of the Delaunay triangulation of points on surfaces: the smooth case.Proceedings of the 19th Annual Symposium on Computational Geometry2003, 201--210URL: http://dl.acm.org/citation.cfm?id=777823DOI back to text
38 bookF.Franz Aurenhammer, R.Rolf Klein and D.-T.Der-Tsai Lee. Voronoi diagrams and Delaunay triangulations.World Scientific2013, URL: http://www.worldscientific.com/worldscibooks/10.1142/8685back to text
39 articleR.Rémi Bardenet, F.Frédéric Lavancier, X.Xavier Mary and A.Aurélien Vasseur. On a few statistical applications of determinantal point processes.ESAIM: Procs602017, 180-202URL: https://doi.org/10.1051/proc/201760180DOI back to text
40 miscM.Marguerite Bin. A fractional Helly theorem for set systems with slowly growing homological shatter function.2024, URL: https://arxiv.org/abs/2411.18605back to text
41 articleM.Mikhail Bogdanov, O.Olivier Devillers and M.Monique Teillaud. Hyperbolic Delaunay complexes and Voronoi diagrams made practical.Journal of Computational Geometry52014, 56--85HAL back to text back to text
42 inproceedingsM.Mikhail Bogdanov, M.Monique Teillaud and G.Gert Vegter. Delaunay triangulations on orientable surfaces of low genus.Proceedings of the 32nd International Symposium on Computational Geometry2016, 20:1--20:15HAL DOI back to text
43 inproceedingsJ.-D.Jean-Daniel Boissonnat, O.Olivier Devillers and S.Samuel Hornus. Incremental construction of the Delaunay graph in medium dimension.Proceedings of the 25th Annual Symposium on Computational Geometry2009, 208--216HAL back to text
44 articleJ.-D.Jean-Daniel Boissonnat, O.Olivier Devillers, R.René Schott, M.Monique Teillaud and M.Mariette Yvinec. Applications of random sampling to on-line algorithms in computational geometry.Discrete and Computational Geometry81992, 51--71HAL back to text
45 techreportY.Yacine Bouzidi, S.Sylvain Lazard, G.Guillaume Moroz, M.Marc Pouget, F.Fabrice Rouillier and M.Michael Sagraloff. Improved algorithms for solving bivariate systems via Rational Univariate Representations.INRIAFebruary 2015HAL back to text
46 articleY.Yacine Bouzidi, S.Sylvain Lazard, M.Marc Pouget and F.Fabrice Rouillier. Separating linear forms and Rational Univariate Representations of bivariate systems.Journal of Symbolic Computation68May 2015, 84-119HAL DOI back to text
47 phdthesisP.Pierre Calka. Tessellations, convex hulls and Boolean model: some properties and connections.Université René Descartes - Paris V2009HAL back to text
48 inproceedingsM.Manuel Caroli, P. M.Pedro M. M. de Castro, S.Sébastien Loriot, O.Olivier Rouiller, M.Monique Teillaud and C.Camille Wormser. Robust and Efficient Delaunay Triangulations of Points on or Close to a Sphere.Proceedings of the 9th International Symposium on Experimental Algorithms6049Lecture Notes in Computer Science2010, 462--473HAL back to text
49 incollectionM.Manuel Caroli and M.Monique Teillaud. 3D Periodic Triangulations.CGAL User and Reference Manual3.5CGAL Editorial Board2009, URL: http://doc.cgal.org/latest/Manual/packages.html#PkgPeriodic3Triangulation3SummaryDOI back to text
50 inproceedingsM.Manuel Caroli and M.Monique Teillaud. Computing 3D Periodic Triangulations.Proceedings of the 17th European Symposium on Algorithms5757Lecture Notes in Computer Science2009, 59--70back to text
51 inproceedingsM.Manuel Caroli and M.Monique Teillaud. Delaunay Triangulations of Point Sets in Closed Euclidean $d$ -Manifolds.Proceedings of the 27th Annual Symposium on Computational Geometry2011, 274--282HAL DOI back to text
52 incollectionB.Bernard Chazelle and others. Application challenges to computational geometry: CG impact task force report.Advances in Discrete and Computational Geometry223Contemporary MathematicsProvidenceAmerican Mathematical Society1999, 407--463back to text
53 articleP.Pascal Chossat, G.Grégory Faye and O.Olivier Faugeras. Bifurcation of hyperbolic planforms.Journal of Nonlinear Science212011, 465--498URL: http://link.springer.com/article/10.1007/s00332-010-9089-3DOI back to text back to text
54 inproceedingsV.Valentina Damerow and C.Christian Sohler. Extreme points under random noise.Proceedings of the 12th European Symposium on Algorithms2004, 264--274URL: http://dx.doi.org/10.1007/978-3-540-30140-0_25DOI back to text
55 inproceedingsO.Olivier Devillers, M.Marc Glisse and X.Xavier Goaoc. Complexity analysis of random geometric structures made simpler.Proceedings of the 29th Annual Symposium on Computational GeometryJune 2013, 167-175HAL DOI back to text
56 inproceedingsO.Olivier Devillers, M.Marc Glisse, X.Xavier Goaoc and R.Rémy Thomasse. On the smoothed complexity of convex hulls.Proceedings of the 31st International Symposium on Computational GeometryLipics2015HAL DOI back to text
57 articleO.Olivier Devillers. The Delaunay hierarchy.International Journal of Foundations of Computer Science132002, 163-180HAL back to text
58 articleL.Laurent Dupont, D.Daniel Lazard, S.Sylvain Lazard and S.Sylvain Petitjean. Near-Optimal Parameterization of the Intersection of Quadrics: III. Parameterizing Singular Intersections.Journal of Symbolic Computation4332008, 216--232HAL DOI back to text
59 articleL.Laurent Dupont, D.Daniel Lazard, S.Sylvain Lazard and S.Sylvain Petitjean. Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm.Journal of Symbolic Computation4332008, 168--191HAL DOI back to text
60 articleL.Laurent Dupont, D.Daniel Lazard, S.Sylvain Lazard and S.Sylvain Petitjean. Near-optimal parameterization of the intersection of quadrics: II. A classification of pencils.Journal of Symbolic Computation4332008, 192--215HAL DOI back to text
61 articleM.Marc Glisse, S.Sylvain Lazard, J.Julien Michel and M.Marc Pouget. Silhouette of a random polytope.Journal of Computational Geometry712016, 14HAL back to text
62 articleM.Michael Hemmer, L.Laurent Dupont, S.Sylvain Petitjean and E.Elmar Schömer. A complete, exact and efficient implementation for computing the edge-adjacency graph of an arrangement of quadrics.Journal of Symbolic Computation4642011, 467-494HAL DOI back to text
63 inproceedingsJ.Johan Hidding, R.Rien van de Weygaert, G.Gert Vegter, B. J.Bernard J.T. Jones and M.Monique Teillaud. Video: the sticky geometry of the cosmic web.Proceedings of the 28th Annual Symposium on Computational Geometry2012, 421--422back to text back to text
64 articleJ. B.J. B. Hough, M.M. Krishnapur, Y.Y. Peres and B.B. Virág. Determinantal processes and independence.Probab. Surv.32006, 206-229back to text
65 inproceedingsR.Rémi Imbach and G.Guillaume Moroz. Fast evaluation and root finding for polynomials with floating-point coefficients.ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic ComputationTromsøJuly 2023HAL back to text
66 articleA.Alex Kulesza and B.Ben Taskar. Determinantal Point Processes for Machine Learning.Foundations and Trends® in Machine Learning52–32012, 123-286URL: http://dx.doi.org/10.1561/2200000044DOI back to text
67 articleS.Sylvain Lazard, L.Lui sM. Peñaranda and S.Sylvain Petitjean. Intersecting quadrics: an efficient and exact implementation.Computational Geometry: Theory and Applications351-22006, 74--99HAL back to text
68 articleO.Odile Macchi. The coincidence approach to stochastic point processes.Advances in Applied Probability711975, 83–122DOI back to text
69 articleM.Marisa Mazón and T.Tomás Recio. Voronoi diagrams on orbifolds.Computational Geometry: Therory and Applications81997, 219--230back to text
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GAMBLE - 2024

GAMBLE - 2024

2024Activity reportProject-TeamGAMBLE

Keywords

Computer Science and Digital Science

Other Research Topics and Application Domains

1 Team members, visitors, external collaborators

Research Scientists

Faculty Members

Post-Doctoral Fellow

PhD Students

Interns and Apprentices

Administrative Assistants

External Collaborator

2 Overall objectives

3 Research program

3.1 Non-linear computational geometry

3.2 Non-Euclidean computational geometry

3.3 Probability in computational geometry

3.4 Discrete geometric structures

4 Application domains

5 Highlights of the year

6 New software, platforms, open data

6.1 New software

6.1.1 pwpoly

6.1.2 voxelize

7 New results

7.1 Non-Linear Computational Geometry

7.1.1 Fast evaluation and root finding for polynomials with floating-point coefficients

7.1.2 Towards the Computation of Stabilizing Controllers of Multidimensional Systems

7.1.3 Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations

7.1.4 A Subquadratic Algorithm for Computing the L1-distance between Two Terrains

7.1.5 Algebraic data structures and approximations for efficient geometric modeling (HDR, Guillaume Moroz)

7.1.6 3D snap rounding (PhD thesis, Léo Valque)

7.1.7 An improved complexity bound for computing the topology of a real algebraic space curve

7.2 Non-Euclidean Computational Geometry

7.2.1 ϵ-Net Algorithm Implementation on Hyperbolic Surface

7.2.2 Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

7.2.3 Flipping geometric triangulations on hyperbolic surfaces

7.3 Discrete Geometric structures

7.3.1 A Canonical Tree Decomposition for Chirotopes

7.3.2 A fractional Helly theorem for set systems with slowly growing homological shatter function

8 Bilateral contracts and grants with industry

8.1 Bilateral contracts with industry

8.1.1 Waterloo Maple Inc.

8.1.2 GeometryFactory

9 Partnerships and cooperations

9.1 International research visitors

9.1.1 Visits of international scientists

Other international visits to the team

Victor Chepoi

Boris Aronov

Nicolas Chenavier

Denys Bulavka

Sergey Avvakumov

Jean Cardinal

9.2 National initiatives

9.2.1 ANR JCJC

ANR Abysm

10 Dissemination

10.1 Promoting scientific activities

10.1.1 Scientific events: selection

Reviewer

10.1.2 Journal

Reviewer - reviewing activities

10.1.3 Software Project

Member of the Editorial Boards

10.1.4 Invited talks

10.1.5 Leadership within the scientific community

10.1.6 Research administration

Local

National

Hiring Committees

10.1.7 Teaching Committees

10.2 Teaching - Supervision - Juries

10.2.1 Teaching

10.2.2 Supervision

10.2.3 Juries

10.3 Popularization

10.3.1 Education

7.1.4 A Subquadratic Algorithm for Computing the $L_{1}$ -distance between Two Terrains

7.2.1 $ϵ$ -Net Algorithm Implementation on Hyperbolic Surface