Modelling of curves in polar and Cartesian coordinates

A new class of spline curves in polar coordinates has been presented in 12] and independently considered in 3]. These are rational trigonometric curves in Cartesian coordinates and can be represented as NURBS. From the relationship existing with the correspondent curves in Cartesian coordinates an alternative way to derive some useful tools for modelling splines in polar coordinates has been provided. In particular an ad hoc algorithm of degree elevation for splines in polar coordinates is presented. x1. Introduction Recently, in 12] a class of spline curves in polar coordinates was proposed. We refer to these curves as p-splines. They have proved to be a generalization of those considered in 10] , which we call p-B ezier curves. The p-splines were independently considered in 3], which he called Focal splines. These classes of curves are interesting because they allow for modelling and interpolation of free forms in polar coordinates with the same facilities as Cartesian splines. In all of the cited papers Sanchez-Reyes emphasizes the fact that the p-spline curves and surfaces are piecewise rational B ezier in Cartesian coordinates and they are not rational splines. Actually, this last assertion is neither proved nor supported by any justiication. In this paper we will provide the algorithm that leads to a representation of these curves as NURBS. In addition to knot insertion, knot removal and subdivision, another known result from Sanchez-Reyes' papers is the possibility of making degree elevation for p-splines from degree n to degree kn, although an explanation of how to achieve this has not been given. In 2] an algorithm for degree elevation for p-B ezier curves has been proposed. In this paper we will suggest how to use it for p-splines, together with tests on eeciency and numerical stability. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.