In various applications, meshes with quadrilateral elements are preferred over popular triangle meshes due to their distinct characteristics. To facilitate the (semi-)automatic generation of such meshes for either planar domains or curved surfaces, a variety of algorithmic approaches have been proposed.

Over the past decade particular progress has been made in one particular, relatively novel category of quadrilateral meshing algorithms. These make use of vector fields, or more generally cross and frame fields, as well as quantized global parametrizations of the domain to be meshed. When constructed properly, these objects from differential geometry impose semi-regular grid structures on the surface via their streamlines and isolines, respectively, well-suited for quad mesh generation purposes.

We take a glimpse at the underlying basic ideas, key algorithms and recent developments, as well as future challenges.