We use a combination of both algebraic and numerical techniques to construct a C~1-continuous, piecewise (m,n) rational approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass preparation Theorem and Newton power series factorization, based on the technique of Hensel lifting. These, together with modified rational Pade approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve.
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