Some applications of new spline spaces in computer aided geometric design

Aim of this paper is to describe how the so called Variable Degree Polynomial Spaces can be used for the construction of C spatial curves, approximating or interpolating a given set of data. Their main advantages rely in the easy control on their shape, provided by the variable degrees, and in the low computational cost, comparable with that of standard quintic splines. 1 – Introduction Geometric continuous curves and surfaces based on polynomial or rational splines constitute the main tool of Computer Aided Geometric Design because of their simplicity and because of the easy and intuitive control on their shape provided by the so-called shape parameters. However, in some CAD/CAM applications, as, for instance, in the description of the motion of a milling machine, the physical meaning of the parameter is not negligible and a certain order of analytic continuity is often required; therefore new tools which encompass the new and the old requests would be highly desirable. Aim of this paper is to describe the properties and some applications of new quintic-like spline spaces (called Variable Degree Polynomial Spaces, VDPS for short) which permit the construction of C polynomial (or rational) curves and surfaces with the same simplicity, computational cost and ease of shape control as the classical quintics. Indeed, these spaces are isomorphic to the spaces of C