An application of N-Bishop frame to spherical images for direction curves

Abstract

In this paper, we first introduce [Formula: see text]-Bishop frame for a normal direction curve which is defined as an integral curve of the principal normal of a curve. We express this new frame and its properties. Afterwards, we obtain new spherical images by translating [Formula: see text]-Bishop frame vectors to the center of unit sphere [Formula: see text] in [Formula: see text]. Then, these new spherical images are called [Formula: see text]-Bishop spherical images. Additionally, we compute the Frénet–Serret equations of these new spherical images. Moreover, we show that integral curves of [Formula: see text]-Bishop spherical images of slant helices are also slant helices. Finally, we present some illustrated examples.

N-Bishop Darboux Vector of a Timelike Curve in Minkowski 3-Space

“…N-Bishop frame was introduced by Keskin et al [13]. In our paper, we worked on the N-Bishop frame in Minkowski 3-space for the a timelike curve.…”

“…Then a new Bishop frame called N-Bishop frame was introduced by Keskin. et al with using the frame methods in [13].…”

Some characterizations of the slant helices according to N-Bishop frame in Euclidean 3-space

“…in 2016, [Uzunoglu, 2016]. Keskin [Keskin, 2017]. In our study, we obtained some characterizations of the slant helices according to the N-Bishop frame were obtained.…”

Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

“…When we discuss a space curve, it is more convenient for us to use the Lorentzian Bishop frame along the curve as the basic tool than the Frenet-Serret type frame in Lorentzian space. There are several papers about Lorentzian Bishop frame [12][13][14][15][16][17].…”