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.