Similar Frenet Curves

Abstract.

We discuss differential-geometric invariants of a Frenet curve with respect to the group of direct similarities of the Euclidean space $\mathbb{R}^n$. In terms of a spherical arc-length parameter these invariants are expressed by the Euclidean curvatures of the curve. We also prove uniqueness and existence theorems for a curve determined up to a direct similarity of $\mathbb{R}^n$. A relationship between the same invariants and the focal curvatures of a unit speed curve in $\mathbb{R}^n$ is given. All self-similar curves are completely described in any dimension.