187 research outputs found
Curves with rational chord-length parametrization
It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled
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
A partial solution to the problem of proper reparametrization for rational surfaces
Given an algebraically closed field K, and a rational parametrization P of an algebraic surface V ⊂ K3 , we consider the problem of computing a proper rational parametrization Q from P (reparametrization problem). More precisely, we present an algorithm that computes a rational parametrization Q of V such that the degree of the rational map induced by Q is less than the degree induced by P. The properness of the output parametrization Q is analyzed. In particular, if the degree of the map induced by Q is one, then Q is proper and the reparametrization problem is solved. The algorithm works if at least one of two auxiliary parametrizations defined from P is not proper
On the problem of proper reparametrization for rational curves and surfaces
A rational parametrization of an algebraic curve (resp. surface) establishes a rational correspondence of this curve (resp. surface) with the affine or projective line (resp. affine or projective plane). This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational proper parametrization trace the curve or surface once. We consider the problem of computing a proper rational parametrization from a given improper one. For the case of curves we generalize, improve and reinterpret some previous results. For surfaces, we solve the problem for some special surface's parametrizations
On threefolds covered by lines
A classification theorem is given of projective threefolds that are covered
by a two-dimensional family of lines, but not by a higher dimensional family.Comment: LaTeX 2.09, 25 page
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem
The extended L\"uroth's Theorem says that if the transcendence degree of
\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK is 1 then there exists f \in
\KK(\underline{X}) such that \KK(\mathsf{f}_1,\dots,\mathsf{f}_m) is equal
to \KK(f). In this paper we show how to compute with a probabilistic
algorithm. We also describe a probabilistic and a deterministic algorithm for
the decomposition of multivariate rational functions. The probabilistic
algorithms proposed in this paper are softly optimal when is fixed and
tends to infinity. We also give an indecomposability test based on gcd
computations and Newton's polytope. In the last section, we show that we get a
polynomial time algorithm, with a minor modification in the exponential time
decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001
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