Fundamental functions for local interpolation of quadrilateral meshes with extraordinary vertices

The aim of this work is to present a general and simple strategy for the construction of compactly supported fundamental spline (piecewise-polynomial) functions for local interpolation, that are defined over quadrangulations of the real plane with extraordinary vertices. The proposed strategy — which extends the univariate framework introduced in Antonelli et al. (Adv Comput Math 40:945–976, 2014) and Beccari et al. (J Comput Appl Math 240:5–19, 2013)—consists in considering a suitable combination of bivariate polynomial interpolants with blending functions that are the natural generalization of odd-degree tensor-product B-splines. These blending functions are constructed as basic limit functions of the bivariate, primal subdivision schemes developed simultaneously in Stam (Comput Aided Geom Des 18:397–427, 2001) and Zorin et al. (Comput Aided Geom Des 18:483–502, 2001). As an application example of our constructive strategy we present the compactly supported C2 fundamental functions for local interpolation that arise by considering as blending functions the basic limit functions of the celebrated Catmull–Clark subdivision scheme proposed in Catmull and Clark (Comput Aided Des 46:103–124, 2016).