Theory of constructing closed parametric curves based on manifolds

Abstract

A parametric curve based on a manifold is designed for constructing an accurate closed curve. A circle was defined as the parametric space and a non-uniform B-splines defined on the unit circle were used as base functions. A method to construct knot vectors, control points and corresponding parameters were proposed. A method to determine the coordinates for any point on a curve was also proposed. Some non-uniform rational B-splines (NURBS) control techniques, such as curves with an embedded line, a sharp angle, and so on, were used to verify the proposed method s compatibility with NURBS. Some examples were used to compare the arithmetic with that of NURBS. The results show that the method is not only simple, feasible and reliable but also compatible with a CAD system using NURBS.