Mannheim offsets of ruled surfaces. (English) Zbl 1184.53008
Summary: In recent works [Math. Pract. Theory 37, No. 1, 141–143 (2007; Zbl 1141.53300); J. Geom. 88, No. 1–2, 120–126 (2008; Zbl 1138.53007)], H. Liu and F. Wang studied Mannheim partner curves in the three dimensional space. In this paper, we extend the theory of Mannheim curves to ruled surfaces and define two ruled surfaces which are offset in the sense of Mannheim. It is shown that, every developable ruled surface has a Mannheim offset if and only if an equation is satisfied between the geodesic curvature and the arc-length of spherical indicatrix of it. Moreover, we obtain that the Mannheim offset of a developable ruled surface has constant distance from it. Finally, examples are given.
References:
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| [5] | H. Liu and F. Wang, “Mannheim partner curves in 3-space,” Journal of Geometry, vol. 88, no. 1-2, pp. 120-126, 2008. · Zbl 1138.53007 · doi:10.1007/s00022-007-1949-0 |
| [6] | F. Wang and H. Liu, “Mannheim partner curves in 3-Euclidean space,” Mathematics in Practice and Theory, vol. 37, no. 1, pp. 141-143, 2007. · Zbl 1141.53300 |
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