High-quality meshing is a prerequisite for most algorithms working on triangulations or tetrahedral partitions.  This is true in particular for the construction of splines where badly shaped triangles can lead to severe numerical problems.  Delaunay refinement is a popular greedy method for 2D isotropic meshing with many advantageous properties like bounds on angles, smooth grading and optional size of elements.  Extensions to surface meshing exist, again with guarantees and providing great flexibility in the representation of the input shape.  The central question will be whether it is possible to improve these methods such that the quality of the mesh (number of elements, shape of elements, approximation) is optimized.  The goal is to develop such optimizing meshing methods, which would supersede the need for auxiliary tools like global or local parameterizations used in state-of-the-art algorithms and which hence provide more flexibility.  As meshing is a central tool in shape modeling, there is a variety of applications which directly benefit from results.

At the same time I was continuing work on parametrization, which lead to

Rhaleb Zayer, Christian Rossl and Hans-Peter Seidel
Curvilinear Spherical Parameterization
In: Shape Modeling International (SMI), Matsushima, Japan,
IEEE, Los Alamitos, 2006