Geometric Continuity and Shape Parameters for Catmull-Rom Splines

摘要

Polynomial Catmull-Rom splines have local control, can be either approximating or interpolating, and are efficiently computable. Practical experience with Beta-splines has shown that it is useful to endow a spline with shape parameters, used to modify the shape of the curve of surface independent of the defining control vertices. Thus, it is desirable to construct a subclass of the Catmull-Rom splines which has shape parameters. This document presents such a class, some members of which are interpolating and other approximating. As was done for the Beta-spline, shape parameters are introduced by requiring geometric rather than algebraic continuity. Splines in this class are defined by a set of control vertices and a set of shape parameter values associated with the joints of the curve. The shape parameters may be applied globally, affecting the entire curve, or they may be modified locally, affecting only a portion of the curve or surface near the corresponding joint. The interpolating members of the class are new in that no previous local interpolating technique possesses locally variable shape parameters. It is shown that this class results from combining geometric continuous (Beta-spline) blending functions with a new set of geometric continuous interpolating functions. The interpolating functions are shown to be a geometric continuous generalization of the classical Lagrange polynomoids. Keywords: Curves and Surfaces, Computer-aided geometric design. (Author)