Refinable polycube G-splines

Abstract

Polycube G-splines form a 2-manifold guided by a mesh of quadrilateral faces such that at most six quads meet at each vertex. In particular, this replicates the layout of the quad faces of a polycube. Polycube G-splines are piecewise bicubic and polycube G-spline surfaces are almost everywhere tangent-continuous (G1) based on rational linear reparameterization. They can be constructed in two different ways. One construction interprets the quad mesh vertices in the fashion of C2 bicubic splines – this provides for good shape; the other interprets the 2×2 inner Bézier coefficients of each bicubic as C1 bicubic B-spline coefficients – this offers four degrees of freedom per patch and enables adaptive refinement so that the resulting G-spline spaces are nested, i.e. any G-spline surface can be exactly re-represented at different levels of refinement.