Volume 15, Issue 1
Piecewise Rational Approximations of Real Algebraic Curves
DOI:

J. Comp. Math., 15 (1997), pp. 55-71

Published online: 1997-02

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• Abstract

We use a combination of both algebraic and numerical techniques to construct a $C^1$-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 Pad\'{e} 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 $C^1$ continuity between pieces. The simpler cases of $C^{-1}$ and $C^0$ continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.

• Keywords

@Article{JCM-15-55, author = {}, title = {Piecewise Rational Approximations of Real Algebraic Curves}, journal = {Journal of Computational Mathematics}, year = {1997}, volume = {15}, number = {1}, pages = {55--71}, abstract = { We use a combination of both algebraic and numerical techniques to construct a $C^1$-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 Pad\'{e} 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 $C^1$ continuity between pieces. The simpler cases of $C^{-1}$ and $C^0$ continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9189.html} }
TY - JOUR T1 - Piecewise Rational Approximations of Real Algebraic Curves JO - Journal of Computational Mathematics VL - 1 SP - 55 EP - 71 PY - 1997 DA - 1997/02 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9189.html KW - AB - We use a combination of both algebraic and numerical techniques to construct a $C^1$-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 Pad\'{e} 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 $C^1$ continuity between pieces. The simpler cases of $C^{-1}$ and $C^0$ continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.