Distance bounds of ϵ-points on hypersurfaces
IdentifiersPermanent link (URI): http://hdl.handle.net/10017/49600
Ministerio de Educación y Ciencia
Comunidad de Madrid
Universidad de Alcalá
Pérez Díaz, S., Sendra, J. & Sendra, J.R. 2006, “Distance bounds of ϵ-points on hypersurfaces”, Theoretical Computer Science, vol. 359, no. 1-3, pp. 344-368.
MTM2005-08690-C02-01 (Ministerio de Educación y Ciencia);
CAM-UAH2005/053 (Comunidad de Madrid y Universidad de Alcalá)
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
© 2006 Elsevier
ϵ-points were introduced by the authors (see [S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2–3) (2004) 627–650 (Special issue); S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147–181; S. Pérez-Díaz, J.R. Sendra, J. Sendra, Distance properties of ϵ-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45–61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of ϵ-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an ϵ-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the ϵ-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an ϵ-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds.
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