Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves

In this paper we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions to the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve

An algorithm to parametrize approximately space curves

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)

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

Improved limits on short-wavelength gravitational waves from the cosmic microwave background

The cosmic microwave background (CMB) is affected by the total radiation density around the time of decoupling. At that epoch, neutrinos comprised a significant fraction of the radiative energy, but there could also be a contribution from primordial gravitational waves with frequencies greater than ~ 10^-15 Hz. If this cosmological gravitational wave background (CGWB) were produced under adiabatic initial conditions, its effects on the CMB and matter power spectrum would mimic massless non-interacting neutrinos. However, with homogenous initial conditions, as one might expect from certain models of inflation, pre big-bang models, phase transitions and other scenarios, the effect on the CMB would be distinct. We present updated observational bounds for both initial conditions using the latest CMB data at small scales from the South Pole Telescope (SPT) in combination with Wilkinson Microwave Anisotropy Probe (WMAP), current measurements of the baryon acoustic oscillations, and the Hubble parameter. With the inclusion of the data from SPT the adiabatic bound on the CGWB density is improved by a factor of 1.7 to 10^6 Omega_gw < 8.7 at the 95% confidence level (C.L.), with weak evidence in favor of an additional radiation component consistent with previous analyses. The constraint can be converted into an upper limit on the tension of horizon-sized cosmic strings that could generate this gravitational wave component, with Gmu < 2 10^-7 at 95% C.L., for string tension Gmu. The homogeneous bound improves by a factor of 3.5 to 10^6 Omega_gw < 1.0 at 95% C.L., with no evidence for such a component from current data.Comment: 5 pages, 3 figure

An Algebraic Analysis of Conchoids to Algebraic Curves

We study the conchoid to an algebraic affine plane curve C from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside C, the notion of conchoid involves a point A in the affine plane (the focus) and a nonzero field element d (the distance).We introduce the formal definition of conchoid by means of incidence diagrams.We prove that the conchoid is a 1-dimensional algebraic set having atmost two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to C, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve