A local method for the construction of high-quality interpolatory surfaces of arbitrary topology

In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh or a curve network is definitely a challenging task. A high-quality surface must satisfy both criteria of analytical nature (regularity of the surface) and aesthetic concepts (shape quality). In particular, with regard to the aesthetic quality, it is well known that an interpolatory surface may show an excessively oscillating behavior in the intermediate areas in between the locations of the constraints. To fix the problem, a recent trend is exploiting approaches based on a non-uniform parameterization (centripetal or chordal), which has proven to be effective both in the parametric 2D setting and, for regular meshes, in the context of subdivision schemes. In this talk, we present a construction for analytical surfaces of high quality that interpolate quadrilateral meshes of arbitrary topology, that is meshes possibly containing extraordinary vertices. Two are the key ingredients of the approach: i) an innovative parameterization technique that allows the parameters to smoothly vary between one mesh face and another, and ii) the spline interpolants proposed by the authors in two previous papers, which have local support and non-uniform parameterization. We also show that the proposed approach can be easily applied in order to interpolate a given arbitrary network of curves.