Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 203-212.

Special Curves and Ruled Surfaces Dedicated to Professor Koichi Ogiue on his sixtieth birthday Shyuichi Izumiya Nobuko Takeuchi Department of Mathematics, Hokkaido University Sapporo 060-0810, Japan e-mail: izumiya@math.sci.hokudai.ac.jp Department of Mathematics, Tokyo Gakugei University Koganei, Tokyo 184-8501, Japan e-mail: nobuko@u-gakugei.ac.jp Abstract. We study cylindrical helices and Bertrand curves as curves on ruled urfaces. Some results in this paper clarify that the cylindrical helix is related to aussian curvature and the Bertrand curve is related to mean curvature of ruled urfaces. All arguments in this paper are elementary and classical. There are some rticles which investigate curves on ruled surfaces [1, 7]. However, the main results re not obtained in these articles and do shade new light on an old subject. SC2000: 58C27, 53A25, 53A05 eywords: cylindrical helix, Bertrand curve, ruled surfaces 1. Introduction In [5] we have studied singularities of the rectifying developable (surface) of a space curve. We observed that the rectifying developable along a curve Ω is non-singular if and only if Ω is a cylindrical helix. In this case the rectifying developable is a cylindrical surface. The notion of cylindrical helices is a generalization of the notion of circular helices. On the other hand, the notion of Bertrand curves is another generalization of the notion of circular helices. These two curves have been classically studied as special curves in Euclidean space.

In this paper we study these curves from the view point of geometry of curves on ruled surfaces. The principal normal surface of a space curve Ω is defined to be a ruled surface

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag