Topology of 2D and 3D rational curves

Self Cite

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.

Section: Introductionmentioning

confidence: 99%

Detecting symmetries of rational plane and space curves

Alcázar

Hermoso

Muntingh

2014

Self Cite

“…As we mentioned in the introduction, the topology graph G of a parametric space curve can be computed by the method in [27].…”

Section: Subdivision Algorithmmentioning

confidence: 99%

“…1. Compute the certified vertex list V with all character points as vertices with the method in [27]. The parameters and the left and right Frenet frames are recorded.…”

Section: Subdivision Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Certified approximation of parametric space curves with cubic B-spline curves

Shen

Yuan

Gao

2012

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given precision and preserve the geometric features of the given curve such as the topology, singular points, etc. The approximated curve is divided into segments called quasi-cubic Bézier curve segments which have properties similar to a cubic rational Bézier curve. And the approximate curve is naturally constructed as the associated cubic rational Bézier curve of the control tetrahedron of a quasi-cubic curve. A novel optimization method is proposed to select proper weights in the cubic rational Bézier curve to approximate the given curve. The error of the approximation is controlled by the size of its tetrahedron, which converges to zero by subdividing the curve segments. As an application, approximate implicit equations of the approximated curves can be computed. Experiments show that the method can approximate space curves of high degrees with high precision and very few cubic Bézier curve segments.

“…Plotting and correct visualization of algebraic curves, both in the case when they are implicitly defined by means of a polynomial f (x, y) = 0, or by a parametrization ϕ(t) = (x(t), y(t)) with x(t), y(t) being rational, have received a great deal of attention in the literature on scientific computation (see for example [1], [4], [6], [7], [8], [9], [11]). With this paper, we want to initiate a similar study for curves which are written in polar coordinates.…”

Section: Introductionmentioning

confidence: 99%

On the shape of curves that are rational in polar coordinates

Alcázar

Diaz–Toca

2012

Self Cite

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.