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 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

Feasible, Robust and Reliable Automation and Control for Autonomous Systems

The Special Issue book focuses on highlighting current research and developments in the automation and control field for autonomous systems as well as showcasing state-of-the-art control strategy approaches for autonomous platforms. The book is co-edited by distinguished international control system experts currently based in Sweden, the United States of America, and the United Kingdom, with contributions from reputable researchers from China, Austria, France, the United States of America, Poland, and Hungary, among many others. The editors believe the ten articles published within this Special Issue will be highly appealing to control-systems-related researchers in applications typified in the fields of ground, aerial, maritime vehicles, and robotics as well as industrial audiences

Detecting Cusps and Inflection Points in Curves

In many applications it is desirable to analyze parametric curves for undesirable features like cusps and inflection points. Previously known algorithms to analyze such features are limited to cubics and in many cases are for planar curves only. We present a general purpose method to detect cusps in polynomial or rational space curves of arbitrary degree. If a curve has no cusp in its defining interval, it has a regular parametrization and our algorithm computes that. In particular, we show that if a curve has a proper parametrization then the necessary and sufficient condition for the existence of cusps is given by the vanishing of the first derivative vector. We present a simple algorithm to compute the proper parametrization of a polynomial curve and reduce the problem of detecting cusps in a rational curve to that of a polynomial curve. Finally, we use the regular parametrizations to analyze for inflection points