Degree Elevation for Single-valued Curves in Polar Coordinates

A new class of single-valued curves in polar coordinates obtained by a transformation of a subset of rational B ezier curves into Cartesian coordinates has recently been presented in (SS anchez-Reyes, 1990), and independently considered by P.de Casteljau, who called these curves focal B ezier. These curves are trigonometric polynomials that can be represented by a basis similar to the Bernstein polynomial basis. From their deenition and expression in terms of the Fourier basis it is obvious that every curve of degree n can be expressed as a curve of degree kn, for any natural value k. In this paper, two alternative formulae for degree elevation from degree n to kn are presented and relative proofs are given. A simple and eecient implementation is provided and its stability is numerically proved.