Cusps of Bishop Spherical Indicatrixes and Their Visualizations

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.


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][2][3][4][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][8][9][10][11][12][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   :  ×  2 → R,   (, k) = ⟨N  (), k⟩, where  = 1, 2 and N  () is the first Bishop spherical indicatrix or the second Bishop spherical indicatrix and   :  ×  2 → R,   (, k) = ⟨  (), k⟩, where  = 3, 4 and   () is the first type-2 Bishop spherical indicatrix or the second type-2 Bishop spherical indicatrix) locally around the point ( 0 , k 0 ).These functions are the unfolding of these singularities in the local neighbourhood of ( 0 , k 0 ) and depend only on the germ that they are unfolding.We create these functions by varying a fixed point k in these Bishop normal indicatrix height

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 E 3 .We know that there exist accompanying three frames called Frenet frame for Frenet curve.Denote by (T(), N(), B()) 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 (T(), N 1 (), N 2 ()) as Bishop Frame and  1 () = ⟨T  (), N 1 ()⟩ and  2 () = ⟨T  (), N 2 ()⟩ as Bishop curvatures.The relation matrix can be expressed as ) . ( One can show that so that  1 () and  2 () 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  0 , 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], ) .(6) The relation matrix between Frenet-Serret and "Type-2 Bishop Frame" can be expressed as ) .
The following notions are the main objects in this paper.The unit sphere with center in the origin in the space E 3 is defined by

Cusps of Bishop Spherical Indicatrixes and Their Visualizations
The main results of this paper are in the following theorem.
Theorem 1.Let  :  → E 3 be a regular unit speed curve.
Then, one has the following.
The picture of cusp will be seen in Figure 1.

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  :  → E 3 be a regular unit speed curve.Now, we define two families of smooth functions on  as follows: where  = 1, 2. We call it the first (second) Bishop normal indicatrix height function for the case  = 1 ( = 2).For any k ∈  2 , we denote ℎ V () =   (, k).We also define two families of smooth functions on  as follows: where  = 3, 4. We call it the first (second) type-2 Bishop normal indicatrix height function for the case  = 3 ( = 4).
Proposition 2. Let  :  → E 3 be a regular unit speed curve.
Then, one has the following.
(1) ℎ 2V () = 0 if and only if there are real numbers  and  such that k = T()+N 1 () and Then, one has the following claims.
(D) Using the same computation as the proof of (C), we can get (D).Proposition 3. Let  :  → E 3 be a regular unit speed curve.One has the following claims.
(2) Using the same computation as the proof of (1), we can get (2).
(4) Using the same computation as the proof of (3), we can get (4).
By Proposition 2 and the definition of discriminant set, we have the following proposition.

Proposition 5. (1)
The discriminant sets of  1 and  2 are, respectively, (2) The discriminant sets of  3 and  4 are, respectively, For the Bishop normal indicatrix height functions, we can consider the following propositions.
Proof.(1) We denote that N 1 () = ( 11 () ,  12 () ,  13 ()) , Under this notation, we have that Thus, we have that We also have that Therefore, the 2-jet of ( 1 /V  )(, V) ( = 1, 2) at  0 is given by It is enough to show that the rank of the matrix  is 2, where Denote that 6 Mathematical Problems in Engineering Note that k ∈ D  1 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  1 () = ( 11 () ,  12 () ,  13 ()) , Under this notation, we have Thus, we have We also have Therefore, the 2-jet of ( 3 /V  )(, V) ( = 1, 2) at  0 is given by It is enough to show that the rank of the matrix  is 2, where ) . (31) Note that k ∈ D  3 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 The assertion (1) of Theorem 1 follows from Proposition 6 and Theorem 4.
(3) By Proposition 5, the discriminant set D  4 of  4 is The assertion (3) of Theorem 1 follows from Proposition 6 and Theorem 4.
(4) By Proposition 5, the discriminant set D  3 of  3 is The assertion (4) of Theorem 1 follows from Proposition 6 and Theorem 4.

Generic Properties
In this section, we consider generic properties of regular curves in E 3 .The main tool is a kind of transversality theorem.Let Emb  (, E 3 ) be the space of embeddings  :  → E 3 with   () ̸ = 0 or   () ̸ = 0 equipped with Whitney  ∞ -topology.Here  = 1, 2 and  = 3, 4. We also consider the function Here  = 1, 2, 3, 4. We claim that H k is a submersion for any k ∈  2 , where ℎ V (u) = H  (u, k).For any  ∈ Emb  (, E 3 ), we have   = H  ∘(×  2 ).We also have the ℓ-jet extension We have the following proposition as a corollary of Lemma 6 in Wassermann [17].

Proposition 7.
Let  be a submanifold of  ℓ (1, 1).Then the set Let  : (R, 0) → (R, 0) be a function germ which has an   -singularity at 0. It is well known that there exists a diffeomorphism germ  : (R, 0) → (R, 0) such that  ∘  = ± +1 .This is the classification of   -singularities.For any  =   (0) in  ℓ (1, 1), 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  : (R × R  , 0) → (R, 0) be an parameter unfolding of  : (R, 0) → (R, 0) which has an   -singularity at 0. Then  is a versal unfolding if and only if   1  is transversal to the orbit   ( j (0)) for  ≥  + 1.Here, The generic classification theorem is given as follows.

Examples
As applications and illustration of the main results (Theorem 1), we give two examples in this section.

Conclusions
In this paper, we introduce the notions of Bishop normal indicatrix height functions on a space curve embedded in Euclidean 3-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.

Theorem 9 .
There exists an open and dense subset O ⊂ Emb  (, E 3 ) such that for any  ∈ O, the Bishop spherical images N  (),  = 1, 2 and the type-2 Bishop spherical images   (),  = 3, 4 of (), are locally diffeomorphic to the ordinary cusp at any singular point.

Example 2 .
Let () be a unit speed curve of E 3 defined by

Figure 3 :
Figure 3: The first Bishop spherical image which has eight cusps.

Figure 4 :
Figure 4: The second Bishop spherical image which has eight cusps.

Figure 5 :
Figure 5: The first type-2 Bishop spherical image which has six cusps.

Figure 6 :
Figure 6: The second type-2 Bishop spherical image which has six cusps.

Figure 8 :
Figure 8: (a) The first Bishop spherical image.(b) The second Bishop spherical image.