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.
1. Introduction
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. It is well known that Bishop introduced the Relatively Parallel Adapted Frame of regular curves embedded in Euclidean 3-space and this frame was widely applied in the area of Biology and Computer Graphics; see [1–5]. For instance, in [6], we introduced some detailed applications of Bishop Frame, so we omit them here. 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 also introduced two new spherical images and called them type-2 Bishop spherical images. We know that the properties of geometric objects are independent of the choice of the coordinate systems. But the researchers [1, 7–13] found that, when they adopted these frames, there would be some new geometric objects such as Bishop spherical images, Bishop Daroux image, and type-2 Bishop spherical images. Although 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. Actually, sometimes they are singular, for example, [7, 10]. Two questions are what about their singularities and how to recognize the types of the singularities? Thus the current study hopes to answer these questions and it is inspired by the works of Bishop [1], Yılmaz et al. [7], Pei and Sano [14] and Wang et al. [15]. On the other hand, for the reason that the vector parameterized equations of Bishop spherical images and type-2 Bishop spherical images are very complicated (see [7, 10]), it is very hard to recognize their singular points by normal way. In this paper, we will give a simple sufficient condition to describe their singular points by using the technique of singularity theory. To do this, we hope that Bishop spherical images and type-2 Bishop spherical images can be seen as the discriminants of unfolding of some functions. In this paper, adopting Bishop Frame [1] and “Type-2 Bishop Frame” [7] as the basic tools, we construct Bishop normal indicatrix height functions (denoted by , , where and is the first Bishop spherical indicatrix or the second Bishop spherical indicatrix and , , where and is the first type-2 Bishop spherical indicatrix or the second type-2 Bishop spherical indicatrix) locally around the point . These functions are the unfolding of these singularities in the local neighbourhood of and depend only on the germ that they are unfolding. We create these functions by varying a fixed point in these Bishop normal indicatrix height functions to get some families of functions. We show that these singularities are versally unfolded by the families of Bishop normal indicatrix height functions. If the singularity of and is -type () and the corresponding 2-parameter unfolding is versal, then applying the theory of singularity [14, 15], we know that discriminant set of the 2-parameter unfolding is locally diffeomorphic to a line or a cusp; thus, we finished the classification of singularities of the Bishop spherical images and type-2 Bishop spherical images, because the discriminant sets of the unfolding are precisely the Bishop spherical images and type-2 Bishop spherical images. Moreover, we see that the -singularity () of and is closely related to Bishop curvatures and type-2 Bishop curvatures. 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. Thus, we get the main results in this paper which are stated in Theorem 1. It is important that the properties of those functions related with space curve needed to be generic [15]. Once we proved that the properties of those functions were generic, we could deduce that all singularities were stable under small perturbations for our family of functions. By considering transversality, we prove that these properties given by us are generic. As applications of our main results, we give two examples.
The rest of this paper is organized as follows. Firstly, we introduce some basic concepts and the main results in the next two sections. Then, we introduce four different families of functions on that will be useful to the study of Bishop spherical images and type-2 Bishop spherical images. Afterwards, some general results on the singularity theory are used for families of function germs and the main results are proved. Finally, we give two examples to illustrate the main results and the conclusion of the work is drawn.
2. Preliminaries and Notations
In this section, we will introduce the notions of Bishop Frame, “Type-2 Bishop Frame,” Bishop spherical images, and type-2 Bishop spherical images of unit speed regular curve. Let be a regular unit speed Frenet curve in . We know that there exist accompanying three frames called Frenet frame for Frenet curve. Denote by the moving Frenet frame along the unit speed Frenet curve . Then, the Frenet formulas are given by Here, and are called curvature and torsion, respectively [16]. The Bishop Frame of the is expressed by the alternative frame equations: Here, we will call the set as Bishop Frame and and as Bishop curvatures. The relation matrix can be expressed as One can show that so that and effectively correspond to a Cartesian coordinate system for the polar coordinates and with . Here, Bishop curvatures are also defined by The orientation of the parallel transport frame includes the arbitrary choice of integration constant , which disappears from (and hence from the Frenet frame) due to the differentiation [1].
The “Type-2 Bishop Frame” of the is defined by the alternative frame equations; see [7], The relation matrix between Frenet-Serret and “Type-2 Bishop Frame" can be expressed as Here, the type-2 Bishop curvatures are defined by We will call the set “Type-2 Bishop Frame" which is properly oriented and and type-2 Bishop curvatures. One also can show that so that and also effectively correspond to a Cartesian coordinate system for the polar coordinates , with .
The following notions are the main objects in this paper. The unit sphere with center in the origin in the space is defined by Translating frame’s vector fields to the center of unit sphere, we obtain Bishop spherical images or Bishop spherical indicatrixes. The first Bishop spherical indicatrix and the second Bishop spherical indicatrix are denoted by and separately. The first type-2 Bishop spherical indicatrix and the second type-2 Bishop spherical indicatrix in [7] is denoted by and separately.
3. Cusps of Bishop Spherical Indicatrixes and Their Visualizations
The main results of this paper are in the following theorem.
Theorem 1. Let be a regular unit speed curve. Then, one has the following.(1) Suppose that . Then, one have the following claims.(a) The first Bishop spherical image is locally diffeomorphic to a line at if . (b) The first Bishop spherical image is locally diffeomorphic to the cusp at if and . (2) Suppose that . Then, one has the following claims.(a) The second Bishop spherical image is locally diffeomorphic to a line at if . (b) The second Bishop spherical image is locally diffeomorphic to the cusp at if and . (3) Suppose that . Then, one has the following claims.(a) The first type-2 Bishop spherical image is locally diffeomorphic to a line at if . (b) The first type-2 Bishop spherical image is locally diffeomorphic to the cusp at if and .(4) Suppose that . Then, one has the following claims.(a) The second type-2 Bishop spherical image is locally diffeomorphic to a line at if . (b) The second type-2 Bishop spherical image is locally diffeomorphic to the cusp at if and , where the ordinary cusp is . The picture of cusp will be seen in Figure 1.
4. Bishop Normal Indicatrix Height Functions
In this section, we will introduce four different families of functions on 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. Let be a regular unit speed curve. Now, we define two families of smooth functions on as follows: where . We call it the first (second) Bishop normal indicatrix height function for the case . For any , we denote . We also define two families of smooth functions on as follows: where . We call it the first (second) type-2 Bishop normal indicatrix height function for the case . For any , we denote . Then we have the following proposition.
Proposition 2. Let be a regular unit speed curve. Then, one has the following.(A) Suppose that . Then, one has the following claims.(1) if and only if there are real numbers and such that and .(2) if and only if .(3) if and only if and .(4) if and only if and .(B) Suppose that . Then, one has the following claims.(1) if and only if there are real numbers and such that and .(2) if and only if .(3) if and only if and .(4) if and only if and . (C) Suppose that . Then, one has the following claims.(1) if and only if there are real numbers and such that and . (2) if and only if .(3) if and only if and .(4) if and only if and . (D) Suppose that . Then, one has the following claims.(1) if and only if there are real numbers and such that and .(2) if and only if .(3) if and only if and .(4) if and only if and .
Proof. (A) (1) If , then we have that there are real numbers and such that . Moreover, in combination with , which means , it follows that if and only if and .
(2) When , the assertion (2) follows from the fact that and . Thus, we get that if and only if .
(3) When , the assertion (3) follows from the fact that and .
(4) Under the condition that , this derivative is computed as follows: , = . Since , we get that is equivalent to condition . The assertion (4) follows.
(B) Using the same computation as the proof of (A), we can get (B).
(C) (1) Since , we have that there are real numbers and such that . By the condition that , we get . The converse direction also holds.
(2) Since , by the conditions of assertion (2) and , we have that . It follows from the fact that and that this is equivalent to condition . Therefore, we have . The converse direction also holds.
(3) Since , we have , by the conditions of assertion (3) and that this is equivalent to condition . Therefore, we have and . The converse direction also holds.
(4) Since , by the conditions of assertion (4) and , we have . This is equivalent to condition . Therefore, we have and . The converse direction also holds.
(D) Using the same computation as the proof of (C), we can get (D).
Proposition 3. Let be a regular unit speed curve. One has the following claims.(1) Suppose . Then, if and only if each of is a constant vector.(2) Suppose . Then, if and only if each of is a constant vector.(3) Suppose . Then, if and only if each of is a constant vector.(4) Suppose . Then, if and only if each of is a constant vector.
Proof. (1) Suppose . By straightforward calculations, we have
Thus, if and only if .
(2) Using the same computation as the proof of (1), we can get (2).
(3) Suppose . By straightforward calculations, we have
Thus, if and only if .
(4) Using the same computation as the proof of (3), we can get (4).
5. Versal Unfolding and Proof of the Main Result
In this section, we use some general results on the singularity theory for families of function germs. Detailed descriptions can be found in the book [16]. Let be a function germ. We call an -parameter unfolding of , where . We say that have -singularity at if for all , and . We also say that has -singularity at if for all . Let be an unfolding of and let has -singularity at . We denote the -jet of the partial derivative at by for . Then, is called a versal unfolding if the matrix of coefficients has rank , , where . We now introduce an important set concerning the unfolding. The discriminant set of is the set Then, we have the following well-known result. (cf. [16].)
Theorem 4. Let be an -parameter unfolding of which has the singularity at .
Suppose that is a versal unfolding. Then, one has the following.(a)If , then is locally diffeomorphic to .(b)If , then is locally diffeomorphic to .
By Proposition 2 and the definition of discriminant set, we have the following proposition.
Proposition 5. (1) The discriminant sets of and are, respectively,
(2) The discriminant sets of and are, respectively,
For the Bishop normal indicatrix height functions, we can consider the following propositions.
Proposition 6. (1) If has -singularity at , then is a versal unfolding of .
(2) If has -singularity at , then is a versal unfolding of .
(3) If has -singularity at , then is a versal unfolding of .
(4) If has -singularity at , then is a versal unfolding of .
Proof. (1) We denote that
Under this notation, we have that
Thus, we have that
We also have that
Therefore, the -jet of at is given by
It is enough to show that the rank of the matrix is 2, where
Denote that . Then, we have
Note that is a singular point, where
This completes the proof.
(2) Using the same computation as the proof of (1), we can get (2).
(3) We denote that
Under this notation, we have
Thus, we have
We also have
Therefore, the -jet of at is given by
It is enough to show that the rank of the matrix is 2, where
Denote that . Then, we have
Note that is a singular point, where
This completes the proof.
(4) Using the same computation as the proof of (3), we can get (4).
Proof of Theorem 1. (1) By Proposition 5, the discriminant set of is
The assertion (1) of Theorem 1 follows from Proposition 6 and Theorem 4.
(2) By Proposition 5, the discriminant set of is
The assertion (2) of Theorem 1 follows from Proposition 6 and Theorem 4.
(3) By Proposition 5, the discriminant set of is
The assertion (3) of Theorem 1 follows from Proposition 6 and Theorem 4.
(4) By Proposition 5, the discriminant set of is
The assertion (4) of Theorem 1 follows from Proposition 6 and Theorem 4.
6. Generic Properties
In this section, we consider generic properties of regular curves in . The main tool is a kind of transversality theorem. Let be the space of embeddings with or equipped with Whitney -topology. Here and . We also consider the function defined by . Here . We claim that is a submersion for any , where . For any , we have . We also have the -jet extension defined by . We consider the trivialization . For any submanifold , we denote that . It is evident that both is a submersions and is an immersed submanifold of . Then is transversal to . We have the following proposition as a corollary of Lemma 6 in Wassermann [17].
Proposition 7. Let be a submanifold of . Then the set is a residual subset of . If is a closed subset, then is open.
Let be a function germ which has an -singularity at 0. It is well known that there exists a diffeomorphism germ such that . This is the classification of -singularities. For any in , we have the orbit given by the action of the Lie group of -jet diffeomorphism germs. If has an -singularity, then the codimension of the orbit is . There is another characterization of versal unfolding as follows [15].
Proposition 8. Let be an -parameter unfolding of which has an -singularity at . Then is a versal unfolding if and only if is transversal to the orbit for . Here, is the -jet extension of given by .
The generic classification theorem is given as follows.
Theorem 9. There exists an open and dense subset such that for any , the Bishop spherical images , and the type-2 Bishop spherical images , of , are locally diffeomorphic to the ordinary cusp at any singular point.
7. Examples
As applications and illustration of the main results (Theorem 1), we give two examples in this section.
Example 1. Let be a unit speed curve of defined by
with respect to a arclength parameter , where ; see Figure 2.
The curvature and torsion of this curve are, respectively, as follows:
Using the Bishop curvature equations (5) and (8), we obtain the following:
Using (3) and (7), we obtain the vector parametric equations of the Bishop spherical images and the type-2 Bishop spherical images as follows:
We see that gives some real roots , and gives some real roots . On the other hand, equation gives some real roots and gives some real roots , where . For the case that , we have that and . Hence, we have that the first Bishop spherical image is locally diffeomorphic to cusp and it has eight cusps; see Figure 3. We can also get that the second Bishop spherical image is locally diffeomorphic to cusp at and it also has eight cusps; see Figure 4. For the case that , where , we have and . Hence, we have that the first type-2 Bishop spherical image is locally diffeomorphic to cusp and it has six cusps; see Figure 5. We also can get that the second type-2 Bishop spherical image is locally diffeomorphic to cusp at , where , and it also has six cusps; see Figure 6.
Example 2. Let be a unit speed curve of defined by
with respect to a arclength parameter , where ; see Figure 7. For more details about this kind of curve, see [18].
The curvature and torsion of this curve are, respectively, as follows:
Using the Bishop curvature equations (5) and (8), we obtain
Using (3) and (7), we obtain the Bishop spherical images and the type-2 Bishop spherical images as follows:
We see that gives some real roots and gives some real roots , where is an integer. Since , we have that and . Hence, we have that and are locally diffeomorphic to a line. The pictures of Bishop spherical images are visualized in Figure 8. For the case that , we have that and . Hence, the first type-2 Bishop spherical image is locally diffeomorphic to cusp; see Figure 9. We also can get that the second type-2 Bishop spherical image is not locally diffeomorphic to cusp; see Figure 9.
8. Conclusions
In this paper, we introduce the notions of Bishop normal indicatrix height functions on a space curve embedded in Euclidean -space. We use the Bishop Frame, the type-2 Bishop Frame, and those functions to study Bishop spherical images from the singularity viewpoint. 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 of the main results, we give two examples and make them visualized by computer.
Acknowledgments
The authors would like to thank the referee for his/her valuable suggestions which improved the first version of the paper. The first author was partially supported by the Fundamental Research Funds for the Central Universities (Graduate Innovation Fund of Northeast Normal University, no. 12SSXT140), Preparatory Studies of Provincial Innovation Project of Mudanjiang Normal University, no. SY201225 and no. SY201320. The second author (Corresponding author) was partially supported by NSF of China, no. 11271063.