# Project : galaad

## Section: Scientific Foundations

### Geometry

We are interested in geometric modeling problems, based on non-discrete models, mainly of semi-algebraic type. Our activities focus in particular on the following points:

#### Geometry of algebraic varieties

In order to solve effectively an algebraic problem, a preprocessing analyzing
step is often mandatory. From such study, we will be able to deduce the
method of resolution that is best suited to and thus produce an efficient
solver, dedicated to a certain class of systems. The effective algebraic
geometry provides us tools for analysis and makes it possible to
exploit the geometric properties of these algebraic varieties. For this
purpose, we focus on new formulations of resultants allowing us to produce
solvers from linear algebra routines, and adapted to the solutions
we want to compute.
Among these formulations, we study in particular *residual*
and *toric * resultant theory. The latter approach relates
the generic properties of the solutions of polynomial equations,
to the geometry of the Newton polytope associated to the polynomials.

#### Geometric algorithms for curved arcs and surface patches

The above-mentioned tools of effective algebraic geometry make it possible to analyze in detail and separately the algebraic varieties. On the other hand, traditional algorithmic geometry deals with problems whose data are linear objects (points, segments, lines) but in very great numbers. Combining these two points of view, we concentrate on problems where collections of piecewise algebraic objects are involved. The properties of such geometrical structures are still not well known, and the traditional algorithmic geometry methods do not always extend to this context, which requires new investigations.

#### Geometry of singularities and topology

The analysis of singularities for a (semi)-algebraic set provides a better understanding of their structure. As a result, it may help us better apprehend and approach modeling problems. We are particularly interested in applying singularity theory to cases of implicit curves and surfaces, silhouettes, shadows curves, moved curves, medial axis, self-intersections, appearing in algorithmic problems in CAGD and shape analysis.

#### Geometry, groups, and invariants

The objects in geometrical problems are points, lines, planes, spheres, quadrics, .... Their properties are, by nature, independent from the reference one chooses for performing analytic computations. Which leads us to methods from invariant theory. In addition to the development of symbolic geometric computations that exploit these invariants, we are also interested in developing more synthetic representations for handling those expressions.