Generalized normal ruled surface of a curve in the Euclidean 3-space

Abstract

In 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.

Osculating Type Ruled Surfaces with Type-2 Bishop Frame in E3

“…Moreover, many researchers have studied different curves on ruled surfaces in [13][14][15][16]. On the other hand, the special ruled surfaces with different direction vectors are called generalized rectifying ruled surfaces, generalized normal ruled surfaces, and osculating-type ruled surfaces in E 3 , as defined by Önder and Kaya [17][18][19]. Ruled surfaces with the type-2 Bishop frame were studied in [20].…”

Generalized osculating‐type ruled surfaces of singular curves

“…8 Then, generalized normal ruled surfaces, whose direction vector is a combination of N and B vectors, are introduced by Kaya and Önder. 9 Also, generalized rectifying ruled surfaces, whose direction vector is a combination of T and B vectors, were introduced by Önder. 10 Obviously, the condition that the anchor curve is regular is used in the examination of generalized ruled surfaces.…”

“…Many geometrical and topological properties of such surfaces have taken place in the literature 4–7 . In addition to these, osculating‐type ruled surfaces, which the direction vector is a combination of $$$ T $$$ and $$$ N $$$ vectors, were introduced by Kaya, Kahraman, and Önder 8 . Then, generalized normal ruled surfaces, whose direction vector is a combination of $$$ N $$$ and $$$ B $$$ vectors, are introduced by Kaya and Önder 9 .…”

“…In addition to these, osculating‐type ruled surfaces, which the direction vector is a combination of $$$ T $$$ and $$$ N $$$ vectors, were introduced by Kaya, Kahraman, and Önder 8 . Then, generalized normal ruled surfaces, whose direction vector is a combination of $$$ N $$$ and $$$ B $$$ vectors, are introduced by Kaya and Önder 9 . Also, generalized rectifying ruled surfaces, whose direction vector is a combination of $$$ T $$$ and $$$ B $$$ vectors, were introduced by Önder 10 .…”

Tangent bundle of pseudo-sphere and Non-null slant ruled surfaces

“…The same author scrutinized non-null slant ruled surfaces according to their characterizations in [17]. In [13], the author gave properties of the slant ruled surfaces in E 3 . In [28], in Euclidean 4-space, some characterizations were examined for 2-ruled surfaces.…”