Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 44, No. 1, pp. 203-212 (2003)

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Special Curves and Ruled Surfaces

Shyuichi Izumiya and Nobuko Takeuchi

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan, e-mail:; Department of Mathematics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan, e-mail:

Abstract: We study cylindrical helices and Bertrand curves as curves on ruled surfaces. Some results in this paper clarify that the cylindrical helix is related to Gaussian curvature and the Bertrand curve is related to mean curvature of ruled surfaces. All arguments in this paper are elementary and classical. There are some articles which investigate curves on ruled surfaces [B], [K]. However, the main results are not obtained in these articles and do shade new light on an old subject.

[B] Brauner, H.: Neuere Untersuchungen über windschiefe Flächen. Jahresber. DMV 1967, 61--85.

[K] Koch, R.: Regelflächen in euklidischen Räumen. In: O.Giering und J. Hoschek, Geometrie und ihre Anwendungen, München 1994, 71--106.

Keywords: cylindrical helix, Bertrand curve, ruled surfaces

Classification (MSC2000): 58C27, 53A25, 53A05

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Electronic version published on: 3 Apr 2003. This page was last modified: 4 May 2006.

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