Characterization of the slant helix as successor curve of the general helix
Given a pair of unit speed curves \((\gamma,\delta)\) in Euclidean \(3\)-space \(\mathbb{E}^3\), the author calls \(\delta\) a \textit{successor curve} for \(\gamma\) if the normal vector field of \(\delta\) equals the tangent vector field of \(\gamma\). Recall that a curve is called a \textit{general helix} if its tangent vector field spans a constant angle with a fixed line in \(\mathbb{E}^3\). Likewise, a curve is called a \textit{slant helix} if its normal tangent vector field spans a constant angle with a fixed line in \(\mathbb{E}^3\).NEWLINENEWLINEIn the paper under review the author gives a description of the family of successor curves of a given curve \(\gamma\) in terms of its Frenet frame and its curvature and torsion. Furthermore, he shows that a regular \(C^2\) space curve is a general helix if and only if it is a successor curve of a plane curve (and not itself a plane curve). He also provides a characterisation of slant helices.
scientific article; zbMATH DE number 6366155