## Section: New Results

### Non-Linear Computational Geometry

Participants : Sény Diatta, Laurent Dupont, George Krait, Sylvain Lazard, Guillaume Moroz, Marc Pouget.

#### Reliable location with respect to the projection of a smooth space curve

Consider a plane curve $\mathcal{B}$ defined as the projection of the intersection of two analytic surfaces in ${\mathbb{R}}^{3}$ or as the apparent contour of a surface. In general, $\mathcal{B}$ has node or cusp singular points and thus is a singular curve. Our main contribution [6] is the computation of a data structure for answering point location queries with respect to the subdivision of the plane induced by $\mathcal{B}$. This data structure is composed of an approximation of the space curve together with a topological representation of its projection $\mathcal{B}$. Since $\mathcal{B}$ is a singular curve, it is challenging to design a method only based on reliable numerical algorithms.

In a previous work [49], we have shown how to describe the set of singularities of $\mathcal{B}$ as regular solutions of a so-called ball system suitable for a numerical subdivision solver. Here, the space curve is first enclosed in a set of boxes with a certified path-tracker to restrict the domain where the ball system is solved. Boxes around singular points are then computed such that the correct topology of the curve inside these boxes can be deduced from the intersections of the curve with their boundaries. The tracking of the space curve is then used to connect the smooth branches to the singular points. The subdivision of the plane induced by $\mathcal{B}$ is encoded as an extended planar combinatorial map allowing point location. We experimented our method and showed that our reliable numerical approach can handle classes of examples that are not reachable by symbolic methods.

#### Workspace, Joint space and Singularities of a family of Delta-Like Robots

Our paper [7] presents the workspace, the joint space and the singularities of a family of delta-like parallel robots by using algebraic tools. The different functions of the SIROPA library are introduced and used to estimate the complexity representing the singularities in the workspace and the joint space. A Groebner based elimination is used to compute the singularities of the manipulator and a Cylindrical Algebraic Decomposition algorithm is used to study the workspace and the joint space. From these algebraic objects, we propose some certified three-dimensional plotting tools describing the shape of the workspace and of the joint space which will help engineers or researchers to decide the most suited configuration of the manipulator they should use for a given task. Also, the different parameters associated with the complexity of the serial and parallel singularities are tabulated, which further enhance the selection of the different configurations of the manipulator by comparing the complexity of the singularity equations.

*In collaboration with Ranjan Jha, Damien Chablat,
Luc Baron and Fabrice Rouillier.*