On local deformations of planar quad-meshes

Planar quad-meshes (meshes with planar quadrilateral faces - PQ-meshes for short) are an important class of meshes (see e.g. Bobenko and Suris [2008]). Although they are often desirable in computer graphics - since planar quads can be rendered with out triangulating them - and architectual geometry (see Pottmann et al. [2007]) - because building with planar tiles is more cost effective - modelling freeform surfaces with planar quadrilaterals is problematic (in fact in practical applications one deforms or subdivides PQ-meshes without obeying the planarity constraint and ensures it afterwards in a global optimization step). In this paper we present a method that allows local deformations of PQ-meshes (with square grid combinatorics) that makes it possible to modify a PQ-mesh while keeping all quadrilaterals planar through the whole process (without a minimization step). In principle the method allows for PQ-mesh subdivision as well.