Curves whose arcs with a fixed origin are similar

I. V. Polikanova

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2023), pp. 26-40

Résumé

The author previously put forward the hypothesis that in the $n$-dimensional Euclidean space $E^n$, 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 $E^n$, any two oriented arcs of which with a common origin (not fixed) 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 with a beginning at this point are similar, then the curve is rectilinear; c) if a curve in $E^n$ 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 $E^2$ and $E^3$ 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 $E^2$ is also given. Research methods are topological, set-theoretic, using the apparatus of functional equations.

Détail

Keywords: straight line, curve with similar arcs, criterion for the straightness of a curve, logarithmic spiral, concho-spiral, functional equation, functional exponential Cauchy equation.
Mots-clés : tangent cone