Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines

Catmull-Rom splines have local control, can be either approximating or interpolating, and are efficiently computable. 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 or surface independently of the defining control vertices. Thus it is desirable to construct a subclass of the Catmull-Rom splines that has shape parameters. We present such a class, some members of which are interpolating and others approximating. As was done for the Beta-spline, shape parameters are introduced by requiring geometric rather than parametric continuity. Splines in this class are defined by a set of control vertices and a set of shape parameter values. The shape parameters may be applied globally, affecting the entire curve, or they may be modified locally, affecting only a portion of the curve near the corresponding joint. We show that this class results from combining geometrically continuous (Beta-spline) blending functions with a new set of geometrically continuous interpolating functions related to the classical Lagrange curves. We demonstrate the practicality of several members of the class by developing efficient computational algorithms. These algorithms are based on geometric constructions that take as input a control polygon and a set of shape parameter values and produce as output a sequence of Bézier control polygons that exactly describes the original curve. A specific example of shape design using a low-degree member of the class is given.