## Section: Scientific Foundations

### Shape approximation and mesh generation

Complex shapes are ubiquitous in robotics (configuration spaces),
computer graphics (animation models) or physical simulations (fluid
models, molecular systems). In all these cases, no natural *shape
space* is available or when such spaces exist they are not easily
dealt with. When it comes to performing calculations, the objects
under study must be discretized. On the other hand, several
application areas such as Computer Aided Geometric Design or medical
imaging require reconstructing 3D or 4D shapes from samples.

The questions aforementioned fall in the realm of *geometric
approximation theory* , a topic Geometrica is actively involved in. More
precisely, the generation of samples, the definition of differential
quantities (e.g. curvatures) in a discrete setting, the geometric and
topological control of approximations, as well as multi-scale
representations are investigated. Connected topics of interest are
also the progressive transmission of models over networks and their
compression.

Surface mesh generation and surface reconstruction have received a
great deal of attention by researchers in various areas ranging from
computer graphics through numerical analysis to computational
geometry. However, work in these areas has been mostly heuristic and
the first theoretical foundations have been established only recently.
Quality mesh generation amounts to finding a partition of a domain
into linear elements (mostly triangles or quadrilaterals) with
topological and geometric properties. Typically, one aims at
constructing a piecewise linear (PL) approximation with the ``same''
topology as the original surface (same topology may have several
meanings). In some contexts, one wants to simplify the topology in a
controlled way. Regarding the geometric distance between the surface
and its PL approximation, different measures must be considered:
Hausdorff distance, errors on normals, curvatures, areas etc. In
addition, the shape, angles or size of the elements must match certain
criteria. We call *remeshing* the techniques involved when the
input domain to be discretized is itself discrete. The input mesh is
often highly irregular and non-uniform, since it typically comes as
the output of a surface reconstruction algorithm applied to a point
cloud obtained from a scanning device. Many geometry processing
algorithms (e.g. smoothing, compression) benefit from remeshing,
combined with uniform or curvature-adapted sampling. Geometrica intends to
contribute to all aspects of this matter, both in theory and in
practice.

Volumetric mesh generation consists in triangulating a given
three-dimensional domain with tetrahedra having a prescribed size and
shape. For instance, the tetrahedra in a general purpose mesh should
be as regular as possible (isotropic case), whereas they should be
elongated in certain directions for problem specific
meshes. Volumetric mesh generation often makes use of surface mesh
generation techniques (*e.g.* to approximate the boundary of the
domain or interfaces between two media). Thanks to its strong
experience with Delaunay triangulations, Geometrica recently made several
contributions to the generation of volumetric meshes, and intends to
pursue in this direction.