A Class of Rational Quartic Splines and their Local Tensor Product Extensions

Abstract

This work describes a new class of rational quartic splines with local shape parameters as well as their local tensor product extensions. The important local shape preserving properties of the new rational quartic spline curves, including monotonicity-preserving and convexity-preserving, are discussed in detail. With a simple knot vector, the resulting rational quartic spline curves are G2 continuous at the knots in general, and by setting the values of local shape parameters, the spline curves can be C2 or C2∩FC3 continuous at the specified knots. Local Bézier curve segments representation form with respect to the new spline curves is developed, based on which exact formulations for representing conic segments and straight line segments are given. And the local subdivision algorithm is also developed. Based on the presented rational quartic splines, tensor product rational splines and local tensor product rational splines are constructed. The non-negativity conditions for the local tensor product rational splines are deduced. Without solving a linear system, the resulting local tensor product spline surfaces can interpolate arbitrary selected control points partly or entirely by specifying some values of local shape parameters. Some experiments of fitting 3D triangular mesh models demonstrate that the tensor product and local tensor product rational spline surfaces have better performances concerning energy-loss functional involving accuracy and smoothness than the classical bi-cubic B-spline surfaces.