Variational Design with Parametric Meshes of Arbitrary Topology

Many mathematical problems in geometric modeling are merely due to the difficulties of handling piecewise polynomial parameterizations of surfaces (e.g., smooth connection of patches, evaluation of geometric fairness measures). Dealing with polygonal meshes is mathematically much easier although infinitesimal smoothness can no longer be achieved. However, transferring the notion of fairness to the discrete setting of triangle meshes allows to develop very efficient algorithms for many specific tasks within the design process of high quality surfaces. The use of discrete meshes instead of continuous spline surfaces is tolerable in all applications where (on an intermediate stage) explicit parameterizations are not necessary. We explain the basic technique of discrete fairing and give a survey of possible applications of this approach.