Bounding and Estimating the Hausdorff distance between real space algebraic curves

This is the author’s version of a work that was accepted for publication in Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Rueda S.L., Sendra J., Sendra J.R., (2014). "Bounding and Estimating the Hausdorff distance\ud between real space algebraic curves ". Computer Aided Geometric Design. vol 31 (2014)\ud 182-198; DOI 10.1016/j.cagd.2014.02.005In this paper, given two real space algebraic curves, not necessarily bounded,\ud whose Hausdor distance is nite, we provide bounds of their distance. These\ud bounds are related to the distance between the projections of the space curves onto\ud a plane (say, z = 0), and the distance between the z-coordinates of points in the\ud original curves. Using these bounds we provide an estimation method for a bound\ud of the Hausdor distance between two such curves and we check in applications that\ud the method is accurate and fas

Characterizing the finiteness of the Hausdorff distance between two algebraic curves

In this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the n-dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves C and C is finite if and only if for each infinity branch of C there exists an infinity branch of C such that the terms with positive exponent in the corresponding series are the same, and reciprocally

Sets which are not tube null and intersection properties of random measures

We show that in $\mathbb{R}^d$ there are purely unrectifiable sets of Hausdorff (and even box counting) dimension $d-1$ which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same authors and by Carbery. Our method extends also to "convex tube null sets", establishing a contrast with a theorem of Alberti, Cs\"{o}rnyei and Preiss on Lipschitz-null sets. The sets we construct are random, and the proofs depend on intersection properties of certain random fractal measures with curves.Comment: 24 pages. Referees comments incorporated. JLMS to appea

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis

Asymptotic behavior of an implicit algebraic plane curve

In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite