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High quality local interpolation by composite parametric surfaces

In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts. With regard to the aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Moreover, recently, the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes. In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness that interpolate quadrilateral meshes of arbitrary topology. In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each edge of the mesh. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline of the mesh at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts. This yields high quality interpolatory surfaces. The proposed surfaces are a generalization of the local univariate spline interpolants introduced in Beccari et al.(2013) and Antonelli et al.(2014), that can have arbitrary continuity and arbitrary order of polynomial reproduction. In particular, these surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh. Moreover, in mesh regions containing vertices of valence other than 4, we suitably define G1- or G2-continuous surface patches that join the neighboring regular ones.