Surfaces of osculating circles in Euclidean space (Q6148913)

The authors investigate surfaces in~\(\mathbb{R}^3\) that are the union of osculating circles of some regular space curve~\(C\). If \(\alpha\colon I\subset\mathbb{R}\to C\) is an arc-length parametrization with Frenet frame \(\{T,N\}\) and curvature \(\kappa\), then its surface of osculating circles~\(\mathcal{O}(\alpha)\) is parametrized as follows: \[ I\times\mathbb{R}\to\mathbb{R}^3,\quad (s,u)\mapsto \alpha(s)+\frac{1}{\kappa(s)}\bigl(\sin(u) T(s)+(1-\cos u)N(s)\bigr). \] It is shown that umbilical points in \(\mathcal{O}(\alpha)\) lie on parametric curves for which the variable \(s\) is fixed. These parametric curves form lines of curvature. The article contributes the following classification results: \begin{itemize} \item The surface~\(\mathcal{O}(\alpha)\) is a canal surface if and only if \(\alpha\) has constant curvature. \item If \(\mathcal{O}(\alpha)\) is a Weingarten surface, then it is contained in either a plane or sphere, or the curve~\(\alpha\) is a Salkowski curve. Moreover, the surface~\(\mathcal{O}(\alpha)\) is a non-umbilical Weingarten surface if and only if it is a non-umbilical canal surface. \item If \(\mathcal{O}(\alpha)\) has constant Gauss curvature or constant mean curvature, then it is contained in either a plane or sphere. \end{itemize}