Curves whose arcs with a fixed origin are similar

Abstract

The author previously put forward the hypothesis that in the n-dimensional Euclidean space En, curves, any two oriented arcs of which are similar, are rectilinear. The same statement was proven for dimensions n = 2 and n = 3. In a space of arbitrary dimension, the hypothesis found its confirmation in the class of rectifiable curves. The work provides a complete solution to the problem, and in a stronger version:a) a curve in En, any two oriented arcs of which with a common origin (not fixed) are similar, is rectilinear;b) if a curve in En has a half-tangent at its boundary point and any two of its oriented arcs with a beginning at this point are similar, then the curve is rectilinear;c) if a curve in En has a tangent at an interior 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, in which all arcs with a common origin are similar, but they are not rectilinear, and a complete description of such curves in E2 is also given. Research methods are topological, set-theoretic, using the apparatus of functional equations.