Why cross fields are not equivalent to quadrilateral meshes

Isogeometric Analysis (IGA) necessitates quadrilateral meshing owing to the utilization of NURBS or other locally tensor-product spline functions as the foundation for its analysis. It is a widely held misconception that the generation of cross fields over the surface domain – a common precursor to quad-mesh construction – establishes equivalence between cross fields and quad-meshes. In this study, we dispel this notion by presenting theorems that distinctly delineate the singularity configurations of both cross fields and quad-meshes. It is posited that these can be viewed as characteristic classes of different fiber bundles, where cross field singularity aligns with the cross fiber bundle and structured mesh singularity corresponds to holomorphic line bundles. As a result, the conditions necessitating quad-mesh singularities are decidedly more stringent than those for cross field singularities. This explains a notable query in the field: the absence of a 3-5 quad-mesh (a quad-mesh with two singularities with topological valence 3 and 5 respectively) on a torus. Additionally, the theoretical framework is broadened to incorporate other types of structured meshes. The formulated theorems give rise to practical algorithms for cross field construction and structured mesh generation. The efficacy and applicability of these algorithms are corroborated through numerical examples that not only validate the theoretical findings but also underscore their practical usability.