Abstract
The main result of this paper is using Bishop Frame and “Type-2 Bishop Frame” to study the cusps of Bishop spherical images and type-2 Bishop spherical images which are deeply related to a space curve and to make them visualized by computer. We find that the singular points of the Bishop spherical images and type-2 Bishop spherical images correspond to the point where Bishop curvatures and type-2 Bishop curvatures vanished and their derivatives are not equal to zero. As applications and illustration of the main results, two examples are given.
Highlights
This paper is written as one of the research projects on visualization of singularities of submanifolds generated by regular curves embedded in Euclidean 3-space by using Bishop Frame
The main results of this paper are in the following theorem
We will introduce four different families of functions on γ that will be useful to study the singular points of the Bishop spherical images and type-2 Bishop spherical images of unit speed regular curve
Summary
This paper is written as one of the research projects on visualization of singularities of submanifolds generated by regular curves embedded in Euclidean 3-space by using Bishop Frame. Inspired by the work of Bishop, in [7], 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 “Type-2 Bishop Frame.” They introduced two new spherical images and called them type-2 Bishop spherical images. Bishop spherical images and type-2 Bishop spherical images have been well studied from the standpoint of differential geometry when they are regular spherical curves, there are little papers on their singularities. Sometimes they are singular, for example, [7, 10]. We give two examples to illustrate the main results and the conclusion of the work is drawn
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