Modelling by Shepard-type curves and surfaces

Near interpolating parametric curves of Shepard type are introduced and studied. The maximal distance between such curves and their control polygon is bounded in terms of the maximal absolute first order difference of the control points. Progressive iterative approximation and weighted progressive iterative approximation processes for Shepard-type curves are studied, giving two intuitive and straightforward schemes to generate sequences of curves with finer and finer precision for data point fitting. We show that these curves overcome some of the original Shepard’s drawbacks, have some advantages with respect to the Bézier case and are optimal in some sense. An application to an interproximation type problem is also discussed. Finally the results are extended to tensor product surfaces.