# Cusps of Bishop Spherical Indicatrixes and Their Visualizations

Authors: Haiming Liu; Donghe Pei
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Hi:I×S2→ℝ, Hi(s,v)=〈Ni(s),v〉, where i=1,2 and Ni(s) is the first Bishop spherical indicatrix or the second Bishop spherical indicatrix and Hj:I×S2→ℝ, Hj(s,v)=〈ζj(s),v〉, where j=3,4 and ζj(s) is the first type-2 Bishop spherical indicatrix or the second type-2 Bishop spherical indicatrix) locally around the point (s0,v0). These functions are the unfolding of these singularities in the local neighbourhood of (s0,v0) and depend only on the germ that they are unfolding. We create these functions by varying a fixed point v 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 hiv=Hi(s,v) and hjv=Hj(s,v) is Ak-type (k=1,2) 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 Ak-singularity (k=1,2) of hiv and hjv 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 γ(s) 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 onγ 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. Letγ=γ(s) be a regular unit speed Frenet curve in 𝔼3. We know that there exist accompanying three frames called Frenet frame for Frenet curve. Denote by (T(s),N(s),B(s)) the moving Frenet frame along the unit speed Frenet curve γ(s). Then, the Frenet formulas are given by
(1)(T′(s)N′(s)B′(s))=(0k(s)0-k(s)0τ(s)0-τ(s)0)(T(s)N(s)B(s)).
Here, k(s) and τ(s) are called curvature and torsion, respectively [16]. The Bishop Frame of the γ(s) is expressed by the alternative frame equations:
(2)(T′(s)N1′(s)N2′(s))=(0k1(s)k2(s)-k1(s)00-k2(s)00)(T(s)N1(s)N2(s)).
Here, we will call the set (T(s),N1(s),N2(s)) as Bishop Frame and k1(s)=〈T′(s),N1(s)〉 and k2(s)=〈T′(s),N2(s)〉 as Bishop curvatures. The relation matrix can be expressed as
(3)(T(s)N(s)B(s))=(1000cosθ(s)sinθ(s)0-sinθ(s)cosθ(s))(T(s)N1(s)N2(s)).
One can show that
(4)k(s)=k12(s)+k22(s),θ(s)=arctan(k2(s)k1(s)),wherek1(s)≠0,τ(s)=dθ(s)ds,
so that k1(s) and k2(s) effectively correspond to a Cartesian coordinate system for the polar coordinates k(s) and θ(s) with θ=∫‍τ(s)ds. Here, Bishop curvatures are also defined by
(5)k1(s)=k(s)cosθ(s),k2(s)=k(s)sinθ(s).
The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ0, which disappears from τ (and hence from the Frenet frame) due to the differentiation [1].The “Type-2 Bishop Frame” of theγ(s) is defined by the alternative frame equations; see [7],
(6)(ζ1′(s)ζ2′(s)B′(s))=(00-ϵ1(s)00-ϵ2(s)ϵ1(s)ϵ2(s)0)(ζ1(s)ζ2(s)B(s)).
The relation matrix between Frenet-Serret and “Type-2 Bishop Frame" can be expressed as
(7)(T(s)N(s)B(s))=(sinθ(s)-cosθ(s)0cosθ(s)sinθ(s)0001)(ζ1(s)ζ2(s)B(s)).
Here, the type-2 Bishop curvatures are defined by
(8)ϵ1(s)=-τ(s)cosθ(s),ϵ2(s)=-τ(s)sinθ(s).
We will call the set (ζ1(s),ζ2(s),B(s)) “Type-2 Bishop Frame" which is properly oriented and ϵ1(s)=〈B′(s),ζ1(s)〉 and ϵ2(s)=〈B′(s),ζ2(s)〉 type-2 Bishop curvatures. One also can show that
(9)κ(s)=dθ(s)ds,
so that ϵ1(s) and ϵ2(s) also effectively correspond to a Cartesian coordinate system for the polar coordinates τ(s), θ(s) with θ=∫κ(s)ds.The following notions are the main objects in this paper. The unit sphere with center in the origin in the space𝔼3 is defined by
(10)S2={x∈𝔼3∣〈x,x〉=1}.
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 ℱℬ𝒩(s)=N1(s) and 𝒮ℬ𝒩(s)=N2(s) separately. The first type-2 Bishop spherical indicatrix and the second type-2 Bishop spherical indicatrix in [7] is denoted by ℱ𝒩ℬ𝒩(s)=ζ1(s) and 𝒮𝒩ℬ𝒩(s)=ζ2(s) separately.


## 3. Cusps of Bishop Spherical Indicatrixes and Their Visualizations

The main results of this paper are in the following theorem.Theorem 1.
Letγ:I→𝔼3 be a regular unit speed curve. Then, one has the following.(1)
Suppose thatk2(s)≠0.                                  Then, one have the following claims.
(a)
The first Bishop spherical imageN1(s)is locally diffeomorphic to a line{0}×ℝats0ifk1(s0)≠0.(b)
The first Bishop spherical imageN1(s)is locally diffeomorphic to the cuspCats0ifk1(s0)=0andk1′(s0)≠0.(2)
Suppose thatk1(s)≠0.Then, one has the following claims.
(a)
The second Bishop spherical imageN2(s)is locally diffeomorphic to a line{0}×ℝats0ifk2(s0)≠0.(b)
The second Bishop spherical imageN2(s)is locally diffeomorphic to the cuspCats0ifk2(s0)=0andk2′(s0)≠0.(3)
Suppose thatϵ2(s)≠0.                                    Then, one has the following claims.
(a)
The first type-2 Bishop spherical imageζ1(s)is locally diffeomorphic to a line{0}×ℝats0ifϵ1(s0)≠0.(b)
The first type-2 Bishop spherical imageζ1(s)is locally diffeomorphic to the cuspCats0ifϵ1(s0)=0andϵ1′(s0)≠0.(4)
Suppose thatϵ1(s)≠0.                                     Then, one has the following claims.
(a)
The second type-2 Bishop spherical imageζ2(s)is locally diffeomorphic to a line{0}×ℝats0ifϵ2(s0)≠0.(b)
The second type-2 Bishop spherical imageζ2(s)is locally diffeomorphic to the cuspCats0ifϵ2(s0)=0andϵ2′(s0)≠0, where the ordinary cusp isC={(x1,x2)∣x12=x23}.                                                 The picture of cusp will be seen in Figure1.Figure 1
Cusp.


## 4. Bishop Normal Indicatrix Height Functions

In this section, we will introduce four different families of functions onγ 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 γ:I→𝔼3 be a regular unit speed curve. Now, we define two families of smooth functions on I as follows:
(11)Hi:I×S2⟶ℝbyHi(s,v)=〈Ni(s),v〉,
where i=1,2. We call it the first (second) Bishop normal indicatrix height function for the case i=1(i=2). For any v∈S2, we denote hiv(s)=Hi(s,v). We also define two families of smooth functions on I as follows:
(12)Hj:I×S2⟶ℝbyHj(s,v)=〈ζj(s),v〉,
where j=3,4. We call it the first (second) type-2 Bishop normal indicatrix height function for the case j=3(j=4). For any v∈S2, we denote hjv(s)=Hj(s,v). Then we have the following proposition.Proposition 2.
Letγ:I→𝔼3 be a regular unit speed curve. Then, one has the following.(A)
Suppose thatk1(s)≠0.                                    Then, one has the following claims.
(1)
h1v(s)=0if and only if there are real numbersλandμsuch thatv=λT(s)+μN2(s)andλ2+μ2=1.(2)
h1v(s)=h1v′(s)=0if and only ifv=∓N2(s).(3)
h1v(s)=h1v′(s)=h1v′′(s)=0if and only ifv=∓N2(s)andk2(s)=0.(4)
h1v(s)=h1v′(s)=h1v′′(s)=h1v(3)(s)=0if and only ifv=∓N2(s)andk2(s)=k2′(s)=0.(B)
Suppose thatk2(s)≠0.                                  Then, one has the following claims.
(1)
h2v(s)=0if and only if there are real numbersλandμsuch thatv=λT(s)+μN1(s)andλ2+μ2=1.(2)
h2v(s)=h2v′(s)=0if and only ifv=∓N1(s).(3)
h2v(s)=h2v′(s)=h2v′′(s)=0if and only ifv=∓N1(s)andk1(s)=0.(4)
h2v(s)=h2v′(s)=h2v′′(s)=h2v(3)(s)=0if and only ifv=∓N1(s)andk1(s)=k1′(s)=0.(C)
Suppose thatϵ1(s)≠0.                                     Then, one has the following claims.
(1)
h3v(s)=0if and only if there are real numbersλandμsuch thatv=λζ2(s)+μB(s)andλ2+μ2=1.(2)
h3v(s)=h3v′(s)=0if and only ifv=∓ζ2(s).(3)
h3v(s)=h3v′(s)=h3v′′(s)=0if and only ifv=∓ζ2(s)andϵ2(s)=0.(4)
h3v(s)=h3v′(s)=h3v′′(s)=h3v(3)(s)=0if and only ifv=∓ζ2(s)andϵ2(s)=ϵ2′(s)=0.(D)
Suppose thatϵ2(s)≠0.                                   Then, one has the following claims.
(1)
h4v(s)=0if and only if there are real numbersλandμsuch thatv=λζ1(s)+μB(s)andλ2+μ2=1.(2)
h4v(s)=h4v′(s)=0if and only ifv=∓ζ1(s).(3)
h4v(s)=h4v′(s)=h4v′′(s)=0if and only ifv=∓ζ1(s)andϵ1(s)=0.(4)
h4v(s)=h4v′(s)=h4v′′(s)=h4v(3)(s)=0if and only ifv=∓ζ1(s)andϵ1(s)=ϵ1′(s)=0.Proof.
(A) (1) Ifh1v(s)=〈N1(s),v〉=0, then we have that there are real numbers λ and μ such that v=λT(s)+μN2(s). Moreover, in combination with v∈S2, which means λ2+μ2=1, it follows that h1v(s)=0 if and only if v=λT(s)+μN2(s) and λ2+μ2=1.
(2) Whenh1v(s)=0, the assertion (2) follows from the fact that h1v′(s)=〈k1(s)T(s),v〉=k1(s)λ and k1(s)≠0. Thus, we get that h1v(s)=h1v′(s)=0 if and only if v=∓N2(s).
(3) Whenh1v(s)=h1v′(s)=0, the assertion (3) follows from the fact that h1v′′(s)=〈k1′(s)T(s)+k12(s)N1(s)+k1(s)k2(s)N2(s),∓N2(s)〉=∓k1(s)k2(s) and k1(s)≠0.
(4) Under the condition thath1v(s)=h1v′(s)=h1v′′(s)=0, this derivative is computed as follows: h1v(3)(s)=〈(k1′′(s)+k13(s)+k1(s)k22(s))T(s)+3k1(s)k1′(s)N1(s)+(2k1′(s)k2(s)+k1(s)k2′(s))N2(s), ∓N2(s)〉 = 2k1′(s)k2(s)+k1(s)k2′(s)=k1(s)k2′(s). Since k1(s)≠0, we get that h1v(3)(s)=0 is equivalent to condition k2′(s)=0. The assertion (4) follows.
(B) Using the same computation as the proof of (A), we can get (B).
(C) (1) Sinceh3v(s)=〈ζ1(s),v〉=0, we have that there are real numbers λ and μ such that v=λB(s)+μζ2(s). By the condition that v∈S2, we get λ2+μ2=1. The converse direction also holds.
(2) Sinceh3v′(s)=〈-ϵ1(s)B(s),v〉=0, by the conditions of assertion (2) and ϵ1(s)≠0, we have that ϵ1(s)λ=0. It follows from the fact that ϵ1(s)λ=0 and ϵ1(s)≠0 that this is equivalent to condition λ=0. Therefore, we have v=±ζ2(s). The converse direction also holds.
(3) Sinceh3v′′(s)=〈-ϵ1′(s)B(s)+ϵ12(s)ζ1(s)+ϵ1ϵ2(s)ζ2(s),v〉=0, we have ϵ1(s)ϵ2(s)=0, by the conditions of assertion (3) and ϵ1(s)≠0 that this is equivalent to condition ϵ2(s)=0. Therefore, we have v=±ζ2(s) and ϵ2(s)=0. The converse direction also holds.
(4) Sinceh3v(3)(s)=〈(ϵ1′(s)ϵ1(s))ζ1(s)+ϵ1(s)ϵ2′(s)ζ2(s)-(ϵ13(s)+ϵ1(s)ϵ2(s)2-ϵ1′′(s))B(s),v〉=0, by the conditions of assertion (4) and ϵ1(s)≠0, we have ϵ1(s)ϵ2′(s)=0. This is equivalent to condition ϵ2′(s)=0. Therefore, we have v=±ζ2(s) and ϵ2(s)=ϵ2′(s)=0. The converse direction also holds.
(D) Using the same computation as the proof of (C), we can get (D).Proposition 3.
Letγ:I→𝔼3 be a regular unit speed curve. One has the following claims.(1)
Supposek1(s)≠0. Then,k2(s)=0if and only if each ofv=∓N2(s)is a constant vector.(2)
Supposek2(s)≠0.                                   Then,k1(s)=0if and only if each ofv=∓N1(s)is a constant vector.(3)
Supposeϵ1(s)≠0.                                   Then,ϵ2(s)=0if and only if each ofv=∓ζ2(s)is a constant vector.(4)
Supposeϵ2(s)≠0.                                   Then,ϵ1(s)=0if and only if each ofv=∓ζ1(s)is a constant vector.Proof.
(1) Supposek1(s)≠0. By straightforward calculations, we have
(13)v′(s)=±k2(s)T(s).
Thus, v′(s)≡0 if and only if k2(s)=0.
(2) Using the same computation as the proof of (1), we can get (2).
(3) Supposeϵ1(s)≠0. By straightforward calculations, we have
(14)v′(s)=±ϵ2(s)B(s).
Thus, v′(s)≡0 if and only if ϵ2(s)=0.
(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 F:(ℝ×ℝr,(s0,x0))→ℝ be a function germ. We call F an r-parameter unfolding of f(s), where f(s)=F(s,x0). We say that f(s) have Ak-singularity at s0 if f(p)(s0)=0 for all 1≤p≤k, and f(k+1)(s0)≠0. We also say that f(s) has A≥k-singularity at s0 if f(p)(s0)=0 for all 1≤p≤k. Let F be an unfolding of f(s) and let f(s) has Ak-singularity (k≥1) at s0. We denote the (k-1)-jet of the partial derivative ∂F/∂xi at s0 by j(k-1)((∂F/∂xi)(x,s0))(s0)=∑j=1k-1‍ajisj for i=1,…,r. Then, F is called a versal unfolding if the k×r matrix of coefficients (a0i,aji) has rank k, (k≤r), where a0i=(∂F/∂xi)(x0,s0). We now introduce an important set concerning the unfolding. The discriminant set of F is the set
(15)𝔇F={x∈ℝr∣F(s,x)=∂F∂s(s,x)=0}.
Then, we have the following well-known result. (cf. [16].)Theorem 4.
LetF:(ℝ×ℝr,(s0,x0))→ℝ be an r-parameter unfolding of f(s) which has the Ak singularity at s0.
Suppose thatF is a (p) versal unfolding. Then, one has the following.(a)
Ifk=1, then 𝔇F is locally diffeomorphic to 0×ℝr-1.(b)
Ifk=2, then 𝔇F is locally diffeomorphic to C×ℝr-2.By Proposition2 and the definition of discriminant set, we have the following proposition.Proposition 5.
(1) The discriminant sets ofH1 and H2 are, respectively,
(16)𝔇H1={v=∓N2(s)∣s∈I},𝔇H2={v=∓N1(s)∣s∈I}.
(2) The discriminant sets ofH3 and H4 are, respectively,
(17)𝔇H3={v=∓ζ2(s)∣s∈I},𝔇H4={v=∓ζ1(s)∣s∈I}.For the Bishop normal indicatrix height functions, we can consider the following propositions.Proposition 6.
(1) Ifh1v0(s) has Ak-singularity (k=1,2) at s0, then H1(s,v) is a (p) versal unfolding of h1v0(s).
(2) Ifh2v0(s) has Ak-singularity (k=1,2) at s0, then H2(s,v) is a (p) versal unfolding of h2v0(s).
(3) Ifh3v0(s) has Ak-singularity (k=1,2) at s0, then H3(s,v) is a (p) versal unfolding of h3v0(s).
(4) Ifh4v0(s) has Ak-singularity (k=1,2) at s0, then H4(s,v) is a (p) versal unfolding of h4v0(s).Proof.
(1) We denote that(18)N1(s)=(n11(s),n12(s),n13(s)),v=(v1,v2,±1-v12-v22).
Under this notation, we have that
(19)H1(s,v)=n11(s)v1+n12(s)v2±n13(s)1-v12-v22.
Thus, we have that
(20)∂H1∂vi=n1i(s)∓vin13(s)1-v12-v22,i=1,2.
We also have that
(21)∂∂s∂H1∂vi=n1i′(s)∓vin13′(s)1-v12-v22,i=1,2,∂2∂s2∂H1∂vi=n1i′′(s)∓vin13′′(s)1-v12-v22,i=1,2.
Therefore, the 2-jet of (∂H1/∂vi)(s,v)(i=1,2) at s0 is given by
(22)j2(∂H1∂vi(s,v0))(s0)=∂∂s∂H1∂vi(s-s0)+12∂2∂s2∂H1∂vi(s-s0)2=a1i(s-s0)+12a2i(s-s0)2.
It is enough to show that the rank of the matrix A is 2, where
(23)A=(n11′(s)∓v1n13′(s)1-v12-v22n12′(s)∓v2n13′(s)1-v12-v22n11′′(s)∓v1n13′′(s)1-v12-v22n12′′(s)∓v2n13′′(s)1-v12-v22).
Denote that v3=±1-v12-v22. Then, we have
(24)detA=(n12′′n11′-n12′n11′′)-v1v3(n12′′n13′-n12′n13′′)-v2v3(n13′′n11′-n13′n11′′)=1v3〈v,N1′(s)∧N1′′(s)〉=1v3〈1v3v,-k1(s)T(s)∧-k1′(s)T(s)-k12(s)N1(s)-k1(s)k2(s)N2(s)1v3〉=1v3〈v,k13(s)N2(s)-k12(s)k2(s)N1(s)〉=1v3〈∓N2(s),k13(s)N2(s)-k12(s)k2(s)N1(s)〉=∓1v3k13(s)≠0.
Note that v∈𝔇H1 is a singular point, where
(25)v=∓N2(s).
This completes the proof.
(2) Using the same computation as the proof of (1), we can get (2).
(3) We denote that(26)ζ1(s)=(ζ11(s),ζ12(s),ζ13(s)),v=(v1,v2,±1-v12-v22).
Under this notation, we have
(27)H3(s,v)=ζ11(s)v1+ζ12(s)v2±ζ13(s)1-v12-v22.
Thus, we have
(28)∂H3∂vi=ζ1i(s)∓viζ13(s)1-v12-v22,i=1,2.
We also have
(29)∂∂s∂H3∂vi=ζ1i′(s)∓viζ13′(s)1-v12-v22,i=1,2,∂2∂s2∂H3∂vi=ζ1i′′(s)∓viζ13′′(s)1-v12-v22,i=1,2.
Therefore, the 2-jet of (∂H3/∂vi)(s,v)(i=1,2) at s0 is given by
(30)j2(∂H3∂vi(s,v0))(s0)=∂∂s∂H3∂vi(s-s0)+12∂2∂s2∂H3∂vi(s-s0)2=a1i(s-s0)+12a2i(s-s0)2.
It is enough to show that the rank of the matrix B is 2, where
(31)B=(ζ11′(s)∓v1ζ13′(s)1-v12-v22ζ12′(s)∓v2ζ13′(s)1-v12-v22ζ11′′(s)∓v1ζ13′′(s)1-v12-v22ζ12′′(s)∓v2ζ13′′(s)1-v12-v22).
Denote that v3=±1-v12-v22. Then, we have
(32)detB=(ζ12′′ζ11′-ζ12′ζ11′′)-v1v3(ζ12′′ζ13′-ζ12′ζ13′′)-v2v3(ζ13′′ζ11′-ζ13′ζ11′′)=1v3〈v,ζ1′(s)∧ζ1′′(s)〉=1v3〈(-ϵ1′(s)B(s)-ϵ12(s)ζ1(s)-(ϵ1(s)ϵ2(s))ζ2(s))v,(-ϵ1(s)B(s))∧(-ϵ1′(s)B(s)-ϵ12(s)ζ1(s)-(ϵ1(s)ϵ2(s))ζ2(s)-ϵ1′(s)B(s)-ϵ12(s)ζ1(s))(-ϵ1′(s)B(s)-ϵ12(s)ζ1(s)-(ϵ1(s)ϵ2(s))ζ2(s))〉=1v3〈v,ϵ13(s)ζ2(s)-ϵ13(s)ϵ2(s)ζ1(s)〉=1v3〈∓ζ2(s),ϵ13(s)ζ2(s)-ϵ13(s)ϵ2(s)ζ1(s)〉=∓1v3ϵ13(s)≠0.
Note that v∈𝔇H3 is a singular point, where
(33)v=∓ζ2(s).
This completes the proof.(4) Using the same computation as the proof of (3), we can get (4).Proof of Theorem1.
(1) By Proposition5, the discriminant set 𝔇H2 of H2 is
(34)𝔇H2={v=∓N1(s)∣s∈I}.
The assertion (1) of Theorem 1 follows from Proposition 6 and Theorem 4.
(2) By Proposition5, the discriminant set 𝔇H1 of H1 is
(35)𝔇H1={v=∓N2(s)∣s∈I}.
The assertion (2) of Theorem 1 follows from Proposition 6 and Theorem 4.
(3) By Proposition5, the discriminant set 𝔇H4 of H4 is
(36)𝔇H4={v=∓ζ1(s)∣s∈I}.
The assertion (3) of Theorem 1 follows from Proposition 6 and Theorem 4.
(4) By Proposition5, the discriminant set 𝔇H3 of H3 is
(37)𝔇H3={v=∓ζ2(s)∣s∈I}.
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 in𝔼3. The main tool is a kind of transversality theorem. Let EmbB(I,𝔼3) be the space of embeddings γ:I→𝔼3 with ki(s)≠0 or ϵj(s)≠0 equipped with Whitney C∞-topology. Here i=1,2 and j=3,4. We also consider the function ℋk:𝔼3×S2→ℝ defined by ℋk(u,v)=〈u,v〉. Here k=1,2,3,4. We claim that ℋkv is a submersion for any v∈S2, where hkv(u)=ℋk(u,v). For any γ∈EmbB(I,𝔼3), we have Hk=ℋk∘(γ×idS2). We also have the ℓ-jet extension j1ℓHk:I×S2→Jℓ(I,ℝ) defined by j1ℓHk(s,v)=jℓhkv(s). We consider the trivialization Jℓ(I,ℝ)≡I×ℝ×Jℓ(1,1). For any submanifold Q⊂Jℓ(1,1), we denote that Q~=I×{0}×Q. It is evident that both j1ℓHk is a submersions and Q~ is an immersed submanifold of jℓ(I,ℝ). Then j1ℓHk is transversal to Q~. We have the following proposition as a corollary of Lemma 6 in Wassermann [17].Proposition 7.
LetQ be a submanifold of Jℓ(1,1). Then the set
(38)TQ={γ∈EmbB(I,𝔼3)∣jℓHkistransversaltoQ~}
is a residual subset of Emb(I,𝔼3). If Q is a closed subset, then TQ is open.Letf:(ℝ,0)→(ℝ,0) be a function germ which has an Ak-singularity at 0. It is well known that there exists a diffeomorphism germ ϕ:(ℝ,0)→(ℝ,0) such that f∘ϕ=±sk+1. This is the classification of Ak-singularities. For any z=jlf(0) in Jℓ(1,1), we have the orbit Ll(z) given by the action of the Lie group of l-jet diffeomorphism germs. If f has an Ak-singularity, then the codimension of the orbit is k. There is another characterization of versal unfolding as follows [15].Proposition 8.
LetF:(ℝ×ℝr,0)→(ℝ,0) be an r-parameter unfolding of f:(ℝ,0)→(ℝ,0) which has an Ak-singularity at 0. Then F is a versal unfolding if and only if j1lF is transversal to the orbit Ll(jlf(0)~) for l≥k+1. Here, j1l(ℝ×ℝr,0)→Jℓ(ℝ,ℝ) is the l-jet extension of F given by j1lF(s,x)=jlFx(s).The generic classification theorem is given as follows.Theorem 9.
There exists an open and dense subset𝒪⊂EmbB(I,𝔼3) such that for any γ∈𝒪, the Bishop spherical images Ni(s), i=1,2 and the type-2 Bishop spherical images ζj(s), j=3,4 of γ(s), are locally diffeomorphic to the ordinary cusp at any singular point.


## 7. Examples

As applications and illustration of the main results (Theorem1), we give two examples in this section.Example 1.
Letγ(s) be a unit speed curve of 𝔼3 defined by
(39)γ(s)=(3cos15s,3sin15s,45s)
with respect to a arclength parameter s, where s∈[0,50π]; see Figure 2.
The curvature and torsion of this curve are, respectively, as follows:(40)k(s)=325,τ(s)=425.
Using the Bishop curvature equations (5) and (8), we obtain the following:
(41)k1(s)=325cos(425s),k2(s)=325sin(425s),ϵ1(s)=-425cos(325s),ϵ2(s)=-425sin(325s).
Using (3) and (7), we obtain the vector parametric equations of the Bishop spherical images N1(s),N2(s) and the type-2 Bishop spherical images ζ1(s),ζ2(s) as follows:
(42)N1(s)=(-cos(425s)cos(15s)-45sin(425s)sin(15s),-cos(425s)sin(15s)+45sin(425s)cos(15s),-35sin(425s)),N2(s)=(-sin(425s)cos(15s)+45cos(425s)sin(15s),-sin(425s)sin(15s)-45cos(425s)cos(15s),35cos(425s)),ζ1(s)=(-35sin(325s)sin(15s)-cos(325s)cos(15s),35sin(325s)cos(15s)-cos(325s)sin(15s),45sin(325s)),ζ2(s)=(35cos(325s)sin(15s)-sin(325s)cos(15s),-35cos(325s)cos(15s)-sin(325s)sin(15s),-45cos(325s)).
We see that k1(s)=0 gives some real roots s=(25/4)kπ+(25/8)π, and k1′(s)=0 gives some real roots s=(25/4)kπ. On the other hand, equation k2(s)=0 gives some real roots s=(25/4)kπ and k2′(s)=0 gives some real roots s=(25/4)kπ+(25/8)π, where k=0,1,…,7. For the case that s=(25/4)kπ+(25/8)π, we have that k1(s)=0 and k1′(s)≠0. Hence, we have that the first Bishop spherical image N1(s) is locally diffeomorphic to cusp and it has eight cusps; see Figure 3. We can also get that the second Bishop spherical image N2(s) is locally diffeomorphic to cusp at s=(25/4)kπ and it also has eight cusps; see Figure 4. For the case that s=(25/3)kπ+(25/6)π, where k=0,…,5, we have ϵ1(s)=0 and ϵ1′(s)≠0. Hence, we have that the first type-2 Bishop spherical image ζ1(s) 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 ζ2(s) is locally diffeomorphic to cusp at s=(25/3)kπ, where k=0,…,5, and it also has six cusps; see Figure 6.Figure 2
Curve.Figure 3
The first Bishop spherical image which has eight cusps.Figure 4
The second Bishop spherical image which has eight cusps.Figure 5
The first type-2 Bishop spherical image which has six cusps.Figure 6
The second type-2 Bishop spherical image which has six cusps.Example 2.
Letγ(s) be a unit speed curve of 𝔼3 defined by
(43)γ(s)=(25sin2s-140sin8s,-25cos2s+140cos8s,415sin3s)
with respect to a arclength parameter s, where s∈[0.34π,0.66π]; see Figure 7. For more details about this kind of curve, see [18].
The curvature and torsion of this curve are, respectively, as follows:(44)k(s)=-4sin3s,τ(s)=4cos3s.
Using the Bishop curvature equations (5) and (8), we obtain
(45)k1(s)=-4sin(3s)cos(43sin3s),k2(s)=-4sin(3s)sin(43sin3s),ϵ1(s)=-4cos(3s)cos(43cos3s),ϵ2(s)=-4cos(3s)sin(43cos3s).
Using (3) and (7), we obtain the Bishop spherical images N1(s),N2(s) and the type-2 Bishop spherical images ζ1(s),ζ2(s) as follows:
(46)N1(s)=1sin2s(4cos2s-1)2×(24125sin(43sin3s)cos3s(cos8s-cos2s)(sin8s-sin2s)×(25cos(43sin3s)-325sin(43sin3s)sin3s)+24125sin(43sin3s)cos3s(cos8s-cos2s)),25cos(43sin3s)(cos2s-cos8s)-sin(43sin3s)×(825cos3s(sin8s-sin2s)+(1225cos2s-325cos8s)sin3s),-35cos(43sin3s)sin3s-sin(43sin3s)×((825cos2s-225cos8s)(cos2s-cos8s)+(825sin2s-225sin8s)(sin2s-sin8s))),N2(s)=1sin2s(4cos2s-1)2×(25(sin8s-sin2s)sin(43sin3s)+cos(43sin3s)(325sin8s-1225sin2s)sin3s-825cos3s(cos2s-cos8s),25(cos2s-cos8s)sin(43sin3s)+cos(43sin3s)×(825cos3s(sin8s-sin2s)+(1225cos2s-325cos8s)sin3s),-35sin(43sin3s)sin3s+cos(43sin3s)×((825cos2s-225cos8s)(cos2s-cos8s)-(825sin2s-225sin8s)(sin8s-sin2s))).ζ1(s)=(35cos((4/3)cos3s)sin3ssin2(s)(4cos2(s)-1)2sin(43cos(3s))(45cos(2s)-15cos(8s))+14×cos((4/3)cos(3s))(-(8/5)sin2s+(8/5)sin8s)sin2(s)(4cos2(s)-1)2,sin(43cos(3s))(45sin(2s)-15sin(8s))+14×cos((4/3)cos(3s))((8/5)cos2s-(8/5)cos8s)sin2(s)(4cos2(s)-1)2,45sin(43cos3s)cos3s-35cos((4/3)cos3s)sin3ssin2(s)(4cos2(s)-1)2),ζ2(s)=(35sin((4/3)cos3s)sin3ssin2(s)(4cos2(s)-1)2-cos(43cos(3s))(45cos(2s)-15cos(8s))+14×sin((4/3)cos(3s))(-(8/5)sin2s+(8/5)sin8s)sin2(s)(4cos2(s)-1)2,-cos(43cos(3s))(45sin(2s)-15sin(8s))+14×sin((4/3)cos(3s))((8/5)cos2s-(8/5)cos8s)sin2(s)(4cos2(s)-1)2,-45cos(43cos3s)cos3s-35sin((4/3)cos3s)sin3ssin2(s)(4cos2(s)-1)2).
We see that k1(s)=0 gives some real roots s=(1/3)kπ and k1′(s)=0 gives some real roots s=(1/3)kπ+(π/6), where k is an integer. Since s∈[0.34π,0.66π], we have that k1(s)≠0 and k2(s)≠0. Hence, we have that N1(s) and N2(s) are locally diffeomorphic to a line. The pictures of Bishop spherical images are visualized in Figure 8. For the case that s=(1/2)π, we have that ϵ1(s)=0 and ϵ1′(s)≠0. Hence, the first type-2 Bishop spherical image ζ1(s) is locally diffeomorphic to cusp; see Figure 9. We also can get that the second type-2 Bishop spherical image ζ2(s) is not locally diffeomorphic to cusp; see Figure 9.Figure 7
Curve.Figure 8
(a) The first Bishop spherical image. (b) The second Bishop spherical image.Figure 9
(a) The first type-2 Bishop spherical image. (b) The second type-2 Bishop spherical image.


## 8. Conclusions

In this paper, we introduce the notions of Bishop normal indicatrix height functions on a space curve embedded in Euclidean3-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.



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