Curves whose arcs with a fixed origin are similar

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.