Abstract
This paper deals with some mathematical objects that the authors have named ∈-points (see [8]), and that appear in the problem of parametrizing approximately algebraic curves. This type of points are used as based points of the linear systems of curves that appear in the parametrization algorithms, and they play an important role in the error analysis. In this paper, we focus on the general study of distance properties of ∈-points on algebraic plane curves, and we show that if P⋆ is an ∈-point on a plane curve C of proper degree d, then there exists an exact point P on C such that its distance to P⋆ is at most \(\sqrt \varepsilon\) if P⋆ is simple, and O(\(\sqrt \varepsilon\)1/2d) if P⋆ is of multiplicity r > 1. Furthermore, we see how these results particularize to the univariate case giving bounds that fit properly with the classical results in numerical analysis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arrondo E., Sendra J., Sendra J.R., (1997). Parametric Generalized Offsets to Hypersurfaces. J. of Symbolic Computation vol. 23, pp 267–285.
Bulirsch, R., Stoer, J., (1993). Introduction to Numerical Analysis. Springer Verlag, New York.
Emiris, I.Z., Galligo, A., Lombardi, H., (1997). Certified Approximate Univariate GCDs J. Pure and Applied Algebra, Vol.117 and 118, pp. 229–251.
Farouki, R.T., Rajan V.T., (1988). On the Numerical Condition of Algebraic Curves and Surfaces. 1: Implicit Equations. Computer Aided Geometric Design. Vol. 5 pp. 215–252.
Marotta, V., (2003). Resultants and Neighborhoods of a Polynomial. Symbolic and Numerical Scientific Computation, SNSC’01, Hagemberg, Austria. Lectures Notes in Computer Science 2630. Springer Verlag.
Mignotte. M., (1992). Mathematics for Computer Algebra. Springer-Verlag, New York, Inc.
Pan V.Y. (2001). Univariate polynomials: nearly optimal algorithms for factorization and rootfinding. ISSAC 2001, London, Ontario, Canada. ACM Press New York, NY, USA. pp. 253–267.
Pérez-Díaz, S., Sendra, J., Sendra, J.R., Parametrization of Approximate Algebraic Curves by Lines. Theoretical Computer Science Vol.315/2-3. pp.627–650.
Pérez-Díaz, S., Sendra, J., Sendra, J.R., Parametrization of Approximate Algebraic Surfaces by Lines. Preprint.
Sasaki, T., Terui, A., (2002). A Formula for Separating Small Roots of a Polynomial. ACM, SIGSAM Bulletim, Vol.36, N.3.
Winkler, J.R., (2001). Condition Numbers of a Nearly Singular Simple Root of a Polynomial. Applied Numerical Mathematics, 38,3; pp.275–285.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pérez-Díaz, S., Sendra, J., Sendra, J. (2005). Distance Properties of ∈-Points on Algebraic Curves. In: Computational Methods for Algebraic Spline Surfaces. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27157-0_4
Download citation
DOI: https://doi.org/10.1007/3-540-27157-0_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23274-2
Online ISBN: 978-3-540-27157-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)