SOME CURVES IN THE POLAR COORDINATE SYSTEM DEFINED BY A FUNCTION OF THE NATURAL PARAMETER

Abstract

Analytical description of curves as a function of their arc length, that is, as a function of a natural parameter, is of fundamental importance. The arc length is an internal characteristic of a curve together with its natural equation, which describes the dependence of curvature on its length. The natural equation completely defines a plane curve regardless of its location in the coordinate system. For spatial curves, in order to define them, it is additionally necessary to have a dependence of torsion on the arc length. If one dependence for a plane curve or two for a spatial curve are given, this does not mean that the curve itself can be constructed. This is due to the solution of differential equations, which, as a rule, require numerical integration methods. But if there are parametric equations with an independent variable - the arc length, then the dependences of curvature and torsion can always be found. The description of a curve as a function of a natural parameter is convenient to use in mechanics, particle motion theory, robotics, and the application of fundamental Frenet formulas in differential geometry.

A separate class of curves is curves described in the polar coordinate system. As a rule, various spirals are described in it. However, many of them cannot be described in terms of a natural parameter. These include the spirals of Archimedes, Galileo, Fermat, and others. The article studies spirals and other curves that are described in the polar coordinate system and their independent variable is a natural parameter. In the traditional description of curves in the polar system, the independent variable is the polar angle. In this study, the polar angle, like the radius vector, are functions of the arc length of the curve. To find one function, the other must be specified. This leads to solving a differential equation or, in a partial case, integrating the corresponding expression.