Curve In N-dimensional Space Research Articles

On the optimal control function diagrams in the problem of the movement of a platform with oscillators

We consider the problem of the time-optimal movement of a rigid body moving translationally along a horizontal straight line and carrying n-linear oscillators. The only control force is applied to the platform and is limited in magnitude, there is no friction. The system is transferred from a state of rest to a specified distance with vibration damping. The evolution of optimal control functions depending on the distance of movement is investigated. A general approach to constructing a visual diagram reflecting such evolution is proposed. To do this, a geometric interpretation of the necessary optimality conditions is used as properties of an auxiliary “control” curve in n-dimensional space. Numerical examples of constructing diagrams of optimal control functions for a platform with three oscillators are given.

A simple formula for the computation of branches and asymptotes of curves and some applications

In this paper, we obtain a simple formula based on the computation of some derivatives for determining the branches and the asymptotes of curves that are defined by a parametrization. For this purpose, we use some previous results and notions presented in Blasco and Pérez-Díaz (2014a,b, 2015, 2020). From these results, we show how the generalized asymptotes of the input curve can be easily computed and we present some applications related to the ramification index and degree of the asymptote, the infinity form and the multiplicity of the infinity points. Furthermore, we show how to construct all the families of parametric curves having some given asymptotes. We develop this method for the plane case but it can be trivially adapted for dealing with rational curves in n-dimensional space. In addition, the formulaes presented can be similarly obtained for curves defined by a parametrization not necessarily rational.

Invariants of Intersection of Certain Pairs of Curves in N-Dimensional Space

Invariants of Intersection of Certain Pairs of Curves in N-Dimensional Space

Concerning acyclic continuous curves

In this paper, we propose to use the word acyclic in place of the phrase containing no simple closed curve. That is, an acyclic continuous curve is a continuous curve containing no simple closed curve. Acyclic continuous curves have been studied by Mazurkiewicz,T R. L. Wilder,? R. L. Moore,lI and the author.? As a result of Theorem 1, it follows that certain internal properties of an acyclic continuous curve which have been proved by the above authors for plane curves, are also possessed by curves in n-dimensional space. However, in the present paper, only plane point sets are considered, unless otherwise stated.