Recursive polynomial curve schemes and computer-aided geometric design

Abstract

A class of polynomial curve schemes is introduced that may have widespread application to CAGD (computer-aided geometric design), and which contains many well-known curve schemes, including Bezier curves, Lagrange polynomials, B-spline curve (segments), and Catmull-Rom spline (segments). The curves in this class can be characterized by a simple recursion formula. They are also shown to have many properties desirable for CAGD; in particular they are affine invariant, have the convex hull property, and possess a recursive evaluation algorithm. Further, these curves have shape parameters which may be used as a design tool for introducing such geometric effects as tautness, bias, or interpolation. The link between probability theory and this class of curves is also discussed.