Similar Frenet curves. (English) Zbl 1180.53004

Summary: 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.


53A04 Curves in Euclidean and related spaces
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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