On the numerical condition of algebraic curves and surfaces 1. Implicit equations
Computer Aided Geometric Design
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Parametrization of approximate algebraic curves by lines
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Parametrization of approximate algebraic surfaces by lines
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Parametrization of ε-rational curves: extended abstract
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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.