An algorithm to parametrize approximately space curves

Abstract

This is the author’s\ud version of a work that was accepted for publication in\ud Journal of Symbolic Computation. Changes resulting from the publishing\ud process, such as peer review, editing, corrections,\ud structural formatting, and other quality control mechanisms may not be\ud reflected in this document.\ud Changes may have been made to this work since it was submitted for\ud publication.\ud A definitive version was subsequently published in Journal of Symbolic\ud Computation vol. 56 pp. 80-106 (2013).\ud DOI: 10.1016/j.jsc.2013.04.002We present an algorithm that, given a non-rational irreducible\ud real space curve, satisfying certain conditions, computes a rational\ud parametrization of a space curve near the input one. For a given\ud tolerance \epsilon > 0, the algorithm checks whether a planar projection\ud of the given space curve is \epsilon -rational and, in the affirmative\ud case, generates a planar parametrization that is lifted to a space\ud parametrization. This output rational space curve is of the same\ud degree as the input curve, both have the same structure at infinity,\ud and the Hausdorff distance between their real parts is finite.\ud Moreover, in the examples we check that the distance is small.This work has been developed, and partially supported, by the Spanish “Ministerio de Ciencia e\ud Innovación” under the Project MTM2008-04699-C03-01, and by the “Ministerio de Economía y Competitividad”\ud under the project MTM2011-25816-C02-01. All authors belong to the Research Group\ud ASYNACS (Ref. CCEE2011/R34)

Matrix Representations by Means of Interpolation

“…5 in two more examples. For the space curve of parametric degree 4 in [RSS13], we compute the 3 implicit equations of degree 4 in 0.171 sec. For the Viviani curve which has parametric degree 4 and implicit equations: x 2 1 + x 2 2 + x 2 3 = 4 a 2 , (x 1 − a) 2 + x 2 2 = a 2 , we compute 3 implicit equations of total implicit degree 4, and degree 74 in the parameter a, in 7.72 sec.…”

Characterizing the finiteness of the Hausdorff distance between two algebraic curves

“…Many authors have addressed some problems in this frame (see e.g. [2,3,[7][8][9][10], etc.) but all of them assume that the given curves are bounded or that the Hausdorff distance between them is finite.…”

“…In particular, we show how the Hausdorff distance between two algebraic curves obtained by applying approximate parametrization problems is finite (we consider examples presented in [7,8,17]). Once one has ensured that the Hausdorff distance between the two curves is finite, some additional existing methods can be applied to estimate the Hausdorff distance between two not bounded curves (see [8]), or for a chosen bounded frame of the input curves (see e.g.…”

“…In addition, the two arbitrary subsets are two real algebraic curves C and C. In this case, the Hausdorff distance between C and C is given by In general, d H (A, B) may be infinite, and some restrictions have to be imposed to guarantee its finiteness (see e.g. [7] or [8]). …”

“…[7,[14][15][16][17]). In that problem, given an affine curve C (say that it is a perturbation of a rational curve), the goal is to compute a rational proper parametrization of a rational affine curve C near C (one may state the problem also for surfaces; see [18]).…”