Pattern mapping with quad-pattern-coverable quad-meshes

We show that for every surface of positive genus, there exist many quadrilateral manifold meshes that can be texture-mapped with locally translated copies of a single square-texture pattern. This implies, for instance, that every positive-genus surface can be covered seamlessly with any of the 17 plane symmetric wallpaper patterns. We identify sufficient conditions for meshes to be classified as “quad-pattern-coverable”, and we present several methods to construct such meshes. Moreover, we identify some mesh operations that preserve the quad-pattern-coverability property. For instance, since vertex insertion remeshing, which is the remeshing operation behind Catmull–Clark subdivision, preserves quad-pattern-coverability, it is possible to cover any surface of positive genus with iteratively finer versions of the same texture.

Graphical AbstractFigure optionsDownload full-size imageDownload high-quality image (891 K)Download as PowerPoint slideHighlights► QPC meshes exist on any surface with positive genus. ► A QPC mesh can be covered seamlessly with all 17 wallpaper groups. ► We identify sufficient and necessary conditions for QPC meshes. ► We present two methods to construct such meshes. ► We present some mesh operations that preserve the QPC property.