Introduction

Abstract

One of the initial chapters of differential geometry is the theory of curves. In a classical form the theory of curves introduces such notions as the length of a curve, which is a function of its arc, the notion of a tangent and osculatory plane of a curve, and defines certain numerical characteristics, such as curvature and torsion as functions of a curve point. It should be remarked that differential geometry commonly studies only the curves obeying certain conditions of regularity. These conditions are imposed by the requirement that the apparatus of differential calculus be applied, but they are hardly justified in a geometrical sense. Moreover, the classical differential-geometrical method often does not work, even in the cases when we deal with regular curves. For instance, a plane curve y = x3 is regular, and analytical, too. At the same time, viewing it as a spatial curve, we see that the differential-geometrical theory of spatial curves cannot be applied to the curve in question, since at the point x = 0 the first two derivates of the radius-vector of the curve turn to zero.KeywordsPlane CurveConvex SurfaceDifferential CalculusIntegral GeometryPolygonal LineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.