Special curves and ruled surfaces
Cylindrical helices and Bertrand curves are studied from a new point of view: they are considered as curves on a ruled surface. It is shown that a ruled surface is the rectifying developable of a curve \(\gamma\) if and only if \(\gamma\) is the geodesic of the ruled surface which is transversal to rulings and whose Gaussian curvature vanishes along \(\pi\). As a consequence of this theorem, a new characterization of cylindrical surfaces is obtained. Another essential theorem states that a ruled surface is the principal normal surface of a space curve \(\gamma\) if and only if \(\gamma\) is the asymptotic curve of the ruled surface and has vanishing mean curvature along \(\gamma\). Applying this result, consequences on Bertrand curves and an interesting characterization of helicoids are deduced.
scientific article; zbMATH DE number 1894854