Uniform interpolation curves and surfaces based on a family of symmetric splines

Abstract

A method to construct arbitrary order continuous curves, which pass through a given set of data points, is introduced. The method can derive a new family of symmetric interpolating splines with various nice properties, such as partition of unity, interpolation property, local support and second order precision etc. Applying these new splines to construct curves and surfaces, one can adjust the shape of the constructed curve and surface locally by moving some interpolating points or by inserting new interpolating points. Constructing closed smooth curves and surfaces and smooth joining curves and surfaces also become very simple, in particular, for constructing Cr(r⩾1) continuous closed surfaces by using the repeating technique. These operations mentioned do not require one to solve a system of equations. The resulting curves or surfaces are directly expressed by the basis spline functions. Furthermore, the method can also directly produce control points of the interpolating piecewise Bézier curves or tensor product Bézier surfaces by using matrix formulas. Some examples are given to support the conclusions.