13,809 research outputs found
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
On the parametrization of a certain algebraic curve of genus 2
A parametrization of a certain algebraic curve of genus 2, given by a cubic equa-tion, is obtained. This curve appears in the study of Hermite-Pade´ approximants for a pair of functions with overlapping branch points on the real line. The suggested method of parametrization can be applied to other cubic curves as well
Algorithm for connectivity queries on real algebraic curves
We consider the problem of answering connectivity queries on a real algebraic
curve. The curve is given as the real trace of an algebraic curve, assumed to
be in generic position, and being defined by some rational parametrizations.
The query points are given by a zero-dimensional parametrization. We design an
algorithm which counts the number of connected components of the real curve
under study, and decides which query point lie in which connected component, in
time log-linear in , where is the maximum of the degrees and
coefficient bit-sizes of the polynomials given as input. This matches the
currently best-known bound for computing the topology of real plane curves. The
main novelty of this algorithm is the avoidance of the computation of the
complete topology of the curve.Comment: 10 pages, 2 figure
Topology of 2D and 3D Rational Curves
In this paper we present algorithms for computing the topology of planar and
space rational curves defined by a parametrization. The algorithms given here
work directly with the parametrization of the curve, and do not require to
compute or use the implicit equation of the curve (in the case of planar
curves) or of any projection (in the case of space curves). Moreover, these
algorithms have been implemented in Maple; the examples considered and the
timings obtained show good performance skills.Comment: 26 pages, 19 figure
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