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Special curves and ruled surfaces. (English) Zbl 1035.53024
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.

53A25Differential line geometry
53A05Surfaces in Euclidean space
53A04Curves in Euclidean space
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