Discrete Geometry for Reliable Surface Quad-Remeshing

In this overview paper we will glimpse how new concepts from discrete differential geometry help to provide a unifying vertical path through parts of the geometry processing pipeline towards a more reliable interaction. As an example, we will introduce some concepts from discrete differential geometry and the QuadCover algorithm for quadrilateral surface parametrization. QuadCover uses exact discrete differential geometric concepts to convert a pair (simplicial surface, guiding frame field) to a global quad-parametrization of the unstructured surface mesh. Reliability and robustness is an omnipresent issue in geometry processing and computer aided geometric design since its beginning. For example, the variety of incompatible data structures for geometric shapes severely limits a reliable exchange of geometric shapes among different CAD systems as well as a unifying mathematical theory. Here the integrable nature of the discrete differential geometric theory and its presented application to an effective remeshing algorithm may serve an example to envision an increased reliability along the geometry processing pipeline through a consistent processing theory.