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Corollary 3.16. (i) If κ (cid:54) = 0 and a 2 (cid:54) = 0 , there are no umbilical points on the surface F ( α,q n )
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(ii) If κ = 0 or a 2 = 0 , then we get λ 1 = λ 2 = 0 and all the points of the surface F ( α,q n ) are planar and the quadratic approach of the surface is a plane
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Elements of the geometry and topology of minimal surfaces in three-dimensional space
AbstractIn this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.
Acta Universitatis Sapientiae, Mathematica – de Gruyter
Published: Aug 1, 2021
Keywords: normal ruled surface; minimal surface; helix; slant ruled surface; 53A05; 53A25
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