Observation
Computer Aided Geometric Design (CAGD) is the discipline dealing with mathematical models for the computer representation of curves, surfaces and solids. CAGD has numerous applications, such as automobile, ship or aircraft design, modelling of human organs, or experimental data fitting. In particular, the research team has focus its work on a central issue in CAGD, the so-called Geometry Processing, whose goal is the calculation of geometric properties of already constructed geometric objects. In its broader meaning, this term includes those algorithms that are applied to already existing geometric entities, for instance the computation of offset curves and surfaces, arc-length parameterizations, and conversions between different curve and surface formats. A key point in Geometry Processing is the representation and manipulation of geometry in the NURBS standard (Non-Uniform Rational B-splines), which allows the representation of both free-form and classical analytic (planes, quadrics, tori) curves or surfaces. NURBS are also employed in data exchange formats, such as IGES. We conduct research on solving the limitations of this standard, which are basically two-fold. A first shortcoming stems from its inability to encompass transcendental curves and surfaces, some of them (helix, catenary, spiral) widely used in Engineering. As a second limitation, the highest degree admitted by current CAD programs is bounded by efficiency reasons. Therefore, certain higher degree entities, such as surfaces resulting from a global spatial deformation, can be neither incorporated into the NURBS standard.