On some CAGD applications

In this talk we will present some recent results and ongoing research of the CAGD team of the University of Valenciennes (www.univ-valenciennes.fr/lamav/cgao/).
Relying on methods from applied mathematics as well as classical geometry we will
focus on the following themes:

· Optimal parameterization of conic sections

In this part we present a simple analytical solution to the problem of determining the optimal parameterization of rational quadratic curves. Optimality is measured with respect to arc length by means of the L2–norm. The presented result is based on a method of Farouki (Optimal parameterizations. Computer Aided Geometric Design,14:153–168, 1997) and Jüttler
(A vegetarian approach to optimal parameterizations. Computer Aided Geometric Design, 14:887–890, 1997), who solve the optimal
parameterization problem analytically in the case of polynomial curves, but suggest a numerical procedure for rational curves.

· Shape preserving and conic reproducing curve subdivision

This part is concerned with the problem of shape preserving interpolatory subdivision.
For arbitrarily spaced, planar input data an efficient non-linear subdivision algorithm
is presented that results in G1 limit curves, reproduces conic sections and respects the convexity properties of the initial data. Significant numerical examples illustrate
the effectiveness of the proposed method.

· Triangular quadric patch construction

An algorithm for constructing arbitrary parametric quadric triangles in rational Bézier
form is presented. The algorithm does not require the knowledge of the underlying quadric, an important property in view of applying this method for the interpolation of triangulated 3D data points. The algorithm consists of four steps starting with the arbitrary choice of the three corner points and corner weights of the patch, by then constructing a certain triangle and a tetrahedron by means of which the remaining
inner control points and weights are obtained guaranteeing the resulting patch to lie on a quadric surface.

· C0 curved triangular interpolants

In this part we provide a unifying comparison of the local parametric C0 curved shape schemes we are aware of, based on a reformulation of their original constructions in terms of polynomial Bézier triangles. With this reformulation we find a geometric interpretation of all the schemes that allows us to analyse their strengths and
shortcomings from a geometrical point of view. Further, we compare the four schemes with respect to their computational costs, their reproduction capabilities of analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their realworld use.