Section: Scientific Foundations
Algebraic Geometric Modeling
We are investigating geometric modeling approaches, based on non-discrete models, mainly of semi-algebraic type. Such non-linear models are able to capture efficiently complexes shapes, using few data. However, they required specific methods to handle and solve the underlying non-linear problems.
Effective algebraic geometry is a naturally framework for handling such representations, in which we are developing new methods to solve these non-linear problems. The framework not only provides tools for modeling but also, it makes it possible to exploit the geometric properties of these algebraic varieties, in order to improve this modeling work. To handle and control geometric objects such as parameterised curves and surfaces or their implicit representations, we consider in particular projections techniques. 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 with the polynomials. These tools allows to change geometric representations, computing an implicit model from a parameterised one. We are interested in dedicated methods for solving these type of problems.
The above-mentioned tools of effective algebraic geometry make it possible to analyse in detail and separately the algebraic varieties. We are interested in problems where collections of piecewise algebraic objects are involved. The properties of such geometrical structures are still not well controlled, and the traditional algorithmic geometry methods do not always extend to this context, which requires new investigations. The use of local algebraic representations also raises problems of approximation and reconstruction, on which we are working on.
Many geometric properties are, by nature, independent from the reference one chooses for performing analytic computations. This leads naturally to invariant theory. In addition to the development of symbolic geometric computations that exploit these invariant properties, we are also interested in developing compact representations of shapes, based on algebraic/symbolic descriptions. Our aim is to improve geometric computing performances, by using smaller input data, with better properties of approximation and certified computation.
