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

In this paper, we first introduce N-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 N-Bishop frame vectors to the center of unit sphere S-2 in E-3. Then, these new spherical images are called N-Bishop spherical images. Additionally, we compute the Frenet-Serret equations of these new spherical images. Moreover, we show that integral curves of N-Bishop spherical images of slant helices are also slant helices. Finally, we present some illustrated examples.