Piecewise Rational Approximations of Real Algebraic Curves

Abstract

We use a combination of both algebraic and numerical techniques to construct a C¹-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Padé approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve C¹-continuity between pieces. The simpler cases of C - and C°-continuity are also handled in a similar manner.