## Section: New Results

### Mesh Generation and Geometry processing

#### Discrete Derivatives of Vector Fields on Surfaces An Operator Approach

Participants : Frédéric Chazal, Maksim Ovsjanikov.

In collaboration with O. Azencot, M. Ben Chen (Technion, Israel Institute of Technology).

Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to representing vector fields, the need arises to compute their derivatives, for example, for solving partial differential equations on surfaces or for designing vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative, that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear operators which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and vector field design, by posing algebraic constraints on the covariant derivative operator.

#### Isotopic Meshing within a Tolerance Volume

Participant : David Cohen-Steiner.

In collaboration with M. Mandad, P. Alliez (Titane Project-team).

We give an algorithm [22] that generates from an input tolerance volume a surface triangle mesh guaranteed to be within the tolerance, intersection free and topologically correct. A pliant meshing algorithm is used to capture the topology and discover the anisotropy in the input tolerance volume in order to generate a concise output. We first refine a 3D Delaunay triangulation over the tolerance volume while maintaining a piecewise-linear function on this triangulation, until an isosurface of this function matches the topology sought after. We then embed the isosurface into the 3D triangulation via mutual tessellation, and simplify it while preserving the topology. Our approach extends to surfaces with boundaries and to non-manifold surfaces. We demonstrate the versatility and efficacy of our approach on a variety of data sets and tolerance volumes.

#### CGALmesh: A Generic Framework for Delaunay Mesh Generation

Participants : Jean-Daniel Boissonnat, Clément Jamin, Mariette Yvinec.

In collaboration with P. Alliez (Titane Project-team).

CGALmesh [21] is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes—surface triangular meshes or volume tetrahedral meshes—from input surfaces, 3D domains, and 3D multidomains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality and on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.