Curves Whose Arcs with a Fixed Startpoint Are Similar

Abstract

The author has previously put forward a hypothesis that in n-dimensional Euclidean space, curves, any two oriented arcs of which are similar, are rectilinear. The author also proved this statement for dimensions of n = 2 and n = 3. In a space of arbitrary dimension, this hypothesis was confirmed in the class of rectifiable curves. In this study, the author provides a complete solution to this problem that is even stronger: (a) a curve in E ⁿ, where any two oriented arcs starting at an common nonfixed point are similar is rectilinear; (b) if a curve in E ⁿ has a half-tangent at its boundary point and any two of its oriented arcs emanating from this point are similar, then the curve is rectilinear; (c) if a curve in E ⁿ has a tangent at an inner point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in E² and E³ are given, that are not rectilinear, although their arcs having a common startpoint are similar, and a complete description of such curves in E² is given. Research methods are topological and set-theoretic using the apparatus of functional equations.