Izumiya, Shyuichi; Takeuchi, Nobuko Special curves and ruled surfaces. (English) Zbl 1035.53024 Beitr. Algebra Geom. 44, No. 1, 203-212 (2003). 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. Reviewer: Jozsef Szilasi (Debrecen) Cited in 24 Documents MSC: 53A25 Differential line geometry 53A05 Surfaces in Euclidean and related spaces 53A04 Curves in Euclidean and related spaces Keywords:cylindrical helix; Bertrand curve; ruled surfaces Citations:Zbl 0153.50803; Zbl 0770.53002; Zbl 0326.53001; Zbl 0227.53007; Zbl 0973.58023; Zbl 0809.53004; Zbl 0077.15401; Zbl 0082.36704 PDFBibTeX XMLCite \textit{S. Izumiya} and \textit{N. Takeuchi}, Beitr. Algebra Geom. 44, No. 1, 203--212 (2003; Zbl 1035.53024) Full Text: EuDML EMIS