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

Computing the topology of a planar or space hyperelliptic curve

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.Comment: 34 pages, lot of figure

Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.Comment: 25 page

PTOPO: A Maple package for the topology of parametric curves

International audiencePTOPO is a MAPLE package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature

On the shape of curves that are rational in polar coordinates

In this paper we provide a computational approach to the shape of curves which are rational in polar coordinates, i.e. which are defined by means of a parametrization (r(t),θ(t)) where both r(t), θ(t) are rational functions. Our study includes theoretical aspects on the shape of these curves, and algorithmic results which eventually lead to an algorithm for plotting the “interesting parts” of the curve, i.e. the parts showing the main geometrical features

Detecting Symmetries of Rational Plane and Space Curves

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.Comment: 19 page

Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”

This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only “by values”. This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points

On the Geometry and the Topology of Parametric Curves

International audienceWe consider the problem of computing the topology and describing the geometry of a parametric curve in R. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in implicit space. When the parametrization involves polynomials of degree at most and maximum bitsize of coefficients , then the worst case bit complexity of PTOPO is O (6 + 5 + 4 (2 +) + 3 (2 + 3) + 3 2). This bound matches the current record bound O (6 + 5) for the problem of computing the topology of a planar algebraic curve given in implicit form. For planar and space curves, if = max{ , }, the complexity of PTOPO becomes O (6), which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of 10. However, visualizing the curve on top of the abstract graph construction, increases the bound to O (7). We have implemented PTOPO in maple for the case of planar curves. Our experiments illustrate its practical nature

Computing the topology of a plane or space hyperelliptic curve

International audienceWe present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a Maple implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed