The Ruled Surfaces According to Bishop Frame in Minkowski 3-Space

Abstract

We investigate the ruled surfaces generated by a straight line in Bishop frame moving along a spacelike curve in Minkowski 3-space. We obtain the distribution parameters, mean curvatures. We give some results and theorems related to be developable and minimal of them. Furthermore, we show that, if the base curve of the ruled surface is also an asymtotic curve and striction line, then the ruled surface is developable.

Curves and surfaces of spacelike curves according to Bishop frame and their singularities

“…We know that the properties of geometric objects are independent of the choice of the coordinate systems. But, the researchers found that, when they adopted this frame, there will be some new geometric objects such as parallel slant curves, Bishop spherical images, Bishop Darboux images, etc.. After that, many regular curves and surfaces related to the Bishop frame have been treated in the Euclidean space [5, 6, 13-17, 22, 24], Minkowski space [4,21,23,26], dual space [11] and Heisenberg group Heis 3 [12]. The current study hopes to research those singular curves and surfaces associate to the Bishop frame instead of regular ones.…”

Legendrian dualities between spherical indicatrixes of curves and surfaces according to Bishop frame

“…Inspired by the work of Bishop, in [19], the authors introduced a new version of Bishop frame using a common vector field as binormal vector field of a regular curve and called this frame as "Type-2 Bishop frame". There are many applications of Bishop frames in differential geometry such as [1, 3,8,9,10,11,12,17,19,20]. Up to now, different types of surfaces and curves such as ruled surfaces [9,20], tubular surfaces [8], special Bishop motion and Bishop Darboux rotation axis of the space curve [3] and B-canal surfaces in terms of biharmonic B-slant helices in Heisenberg group Heis 3 [10] have been studied according to Bishop frames.…”

“…There are many applications of Bishop frames in differential geometry such as [1, 3,8,9,10,11,12,17,19,20]. Up to now, different types of surfaces and curves such as ruled surfaces [9,20], tubular surfaces [8], special Bishop motion and Bishop Darboux rotation axis of the space curve [3] and B-canal surfaces in terms of biharmonic B-slant helices in Heisenberg group Heis 3 [10] have been studied according to Bishop frames. Inspired by the work of Bishop, in [9], the authors introduced the ruled surface with Bishop frame N 2 (s) as its directrix, i.e., ϕ(s, t) = γ(s) + uN 2 (s), which they also called N 2 (s)-ruled surface with the Bishop frame.…”

“…They studied the shape operator and the fundamental forms of the N 2 (s)-ruled surface. In [20], N. Yüksel studied the ruled surfaces generated by a straight line in Bishop frame moving along a spacelike curve in Minkowski 3-space. He obtained the distribution parameters, mean curvatures and gave some results and theorems related to the developable and minimal of them.…”

“…Two questions are: what about their singularities and how to recognize the types of their singularities? Thus the current study hopes to answer these questions and it is inspired by the works of Bishop [1], Kiliçouǧlu and Hacisalihoǧlu [9], N. Yüksel [20], Pei and Sano [15]. On the other hand, for the reason that the vector parameterized equations of the ruled surfaces with Bishop frame N i (s) as their directrix are very complicated (see Ex.…”