Jean-Paul Bécar

On the approximation order of tangent estimators

A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.

Canal surfaces as Bézier curves using mass points

Abstract The paper aims to connect the Bezier curves domain to another known as the Minkowski–Lorentz space for CAGD purposes. The paper details these connections. It provides new algorithms for surface representations and surfaces joining. Some G 1 -blends between canal surfaces illustrate the results with a seahorse sketched. It is well known that rational quadratic Bezier curves define conics. Here, the use of mass points offers the definition of a semi-conic or a branch of hyperbola in the Euclidean plane. Moreover, the choice of an adequate non-degenerate indefinite quadratic form makes a non-degenerate central conic seen as a unit circle. That is not possible in the homogeneous coordinates background. The rational quadratic Bezier curves using mass points provide the modelling of canal surfaces with singular points. These are embedded in the Minkowski–Lorentz space. In that space, these curves are circular arcs which look like ellipse or hyperbola arcs. In the Minkowski–Lorentz space, oriented spheres and oriented planes of R 3 are points on the unit sphere, the points of R 3 and the point at infinity are vectors laying on the light-cone. A particular case of canal surfaces is the cyclides of Dupin. In the Minkowski–Lorentz space, the modelling of any Dupin cyclides patch is completed by a family of algorithms. Each family depends only on the number of singularities known for the Dupin cyclide part. The paper ends with an example of a G 1 connection between two Dupin cyclides. All previous results are finally applied in a seahorse shape design.

DESIGN OF AN EMBEDDED SYSTEM ON A ROBOT TEACHING PLATFORM

Abstract This paper reports a teaching experience. At the end of the academic year, motivated students in electrical engineering and industrial data processing course take part in a national friendly robotic contest. The contestants mainly have to design and program their own autonomous mobile robot for lane tracking. Here, a group of French and Finnish students performed tools for a forthcoming participation. One goal of this project was to make students more sensitive to embedded systems and their real-time applications to classical and fuzzy controllers techniques. Their supervisors applied a known management method both for teamwork and students assessment.