# On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-SpaceE13

Authors: Nadia Alluhaibi; Rashad A. Abdel-Baky; Monia Naghi
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

The Bishop frame or rotation minimizing frame (RMF) is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of computer graphics, engineering, and biology. The main aim of this paper is using the RMF for classification of singularity type of timelike sweeping surface and Bishop spherical Darboux image which is mightily concerning a unit speed spacelike curve with timelike binormal vector inE13.


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## Body

## 1. Introduction

Kinematically, a sweeping surface is a surface traced by a one-parameter family of spheres with centers on a regular space curve, its directrix or spine. If the radii of the spheres are fixed, the sweeping surface is called tubular. There are several examples that we are familiar with, such as circular cylinder (spine is a line, the axis of the cylinder), right circular cone (spine is a line (the axis), radii of the spheres not constant), torus (directrix is a circle), and rotation surface (spine is a line). This visualization is a popularization of the classical notation of a partner of a planar curve [1–4]. One of the noteworthy facts linked with the sweeping surface is that the sweeping surface can be developable surface, that is, can be developed onto a plane without tearing and stretching. Therefore, sweeping surfaces have great usefulness in considerable product design which uses leather, paper, and sheet metal as materials (see, e.g., [5–8]). The developable surface can be represented using the Serret–Frenet frame of space curves from the viewpoint of singularity theory. In [9], Izumiya and Takeuchi defined the rectifying developable surfaces of space curves, where they proved that a regular curve is a geodesic of its rectifying developable surface and revealed the relationship between singularities of the rectifying developable surface and geometric invariants. Ishikawa investigated the relationship between the singularities of tangent developable surfaces and some types of space curves [10]. He also gave a classification of tangent developable surfaces by using the local topological property. There are several works about the singularity theory of developable ruled surfaces by using the Serret–Frenet frame of space curves, for example, [11–16]. However, the Serret–Frenet frame is undefined wherever the curvature vanishes, such as at points of inflection or along straight sections of the curve. A new frame is needed for the kind of mathematical analysis that is typically done with computer graphics. Therefore, Bishop [17] introduced the rotation minimizing frame (RMF) or Bishop frame, which could provide the desired means to ride along a space curve with vanished second 1derivative. After that, many research works linked to the RMF have been treated in the Euclidean space and Minkowski space [18–23].In this paper, the classification of singularity type of timelike sweeping surfaces is studied with the RMF inE13. We present a new invariant related to the singularities of these sweeping surfaces. It is demonstrated that the generic singularities of this sweeping surface are cuspidal edge and swallowtail, and the types of these singularities can be characterized by this invariant, respectively. Afterwards, we have solved the problem of requiring the surface that is timelike sweeping surface and at the same time spacelike/timelike developable surface. Two examples are presented to explain the theoretical results.


## 2. Preliminaries

In this section, we give some definitions and basic concepts that we will use in this paper (see, for instance, [1, 8, 24]). Let ℝ3=p1,p2,p3∣,pi∈ℝi=1,2,3 be a 3-dimensional Cartesian space. For any q=q1,q2,q3 and p=p1,p2,p3∈ℝ3, the pseudoscalar product of q and p is defined by
(1)<q,p>=q1p1+q2p2−q3p3.We callℝ3,<,> Minkowski 3-space. We write E13 instead of ℝ3,<,>. We say that a nonzero vector q∈E13 is spacelike, lightlike, or timelike if <q,q>>0, <q,q>=0, or <q,q><0, respectively. The norm of the vector q∈E13 is defined to be q=<q,q>. For any two vectors q, p∈E13, we define the cross product by
(2)q×p=ij−kq1q2q3p1p2p3=q2p3−q3p2,q3p1−q1p3,−q1p2−q2p1,where i, j, and k are the canonical basis of E13. The hyperbolic and Lorentzian unit spheres, respectively, are
(3)ℍ+2=q∈E13∣q2=q12+q22−q32=−1,S12=q∈E13∣q2=q12+q22−q32=1.Letγ=γs be a unit speed spacelike curve with timelike binormal normal in E13; by κs and τs, we denote the natural curvature and torsion, respectively. Consider the Serret–Frenet frame {υ1s,υ2s, υ2s} associated with curve γs, then the Serret–Frenet formulae read
(4)υ1′sυ2′sυ3′s=0κ0−κ0τ0τ0υ1sυ2sυ3s,where υ1ss=γ′s, υ2s=γ′′s/γ′′s, and υ3s=−υ1s×υ2s are called the unit tangent vector, the principal normal vector, and the binormal vector, respectively. Here, “prime” denotes the derivative with respect to the parameter s. The Serret–Frenet vector fields satisfy the relations
(5)<υ1,υ1>=<υ2,υ2>=1,<υ3,υ3>=−1,υ1×υ2=−υ3,υ1×υ3=−υ2,υ2×υ3=υ1.The Bishop frame or rotation minimizing frame (RMF) ofγs is defined by the alternative frame equations
(6)ξ′ξ1′ξ2′=0μ1−μ2−μ100−μ200ξξ1ξ2,where ωs=−μ2ξ1+μ1ξ2 is RMF Darboux vector. Here, the Bishop curvatures are defined by μ1s=κcoshφ, μ2s=κsinhφ. The relation matrix can be expressed as
(7)ξξ1ξ2=1000coshφsinhφ0sinhφcoshφυ1υ2υ3,where φs≥0 is a hyperbolic angle. One can show that
(8)μ12−μ22=κ2,φ=tanh−1μ2μ1;μ1≠0,φs=−τs0sds+φ0,φ0=φs0.We define a Bishop spherical Darboux imagegs:I⟶ℍ+2 as
(9)gs=−μ2ξ1+μ1ξ2μ12−μ22.Then, we define a new geometric invariantσs=μ2μ1′−μ1μ2′.A ruled surface inE13 is locally the map Dγ,x: I×ℝ⟶E13 defined by
(10)Dγ,xs,t=αs+txs,t∈ℝ,where αs is called the directrix curve and xs the director curve. The straight lines t⟶αs+txs are called rulings. It is well known that Dγ,x is a developable surface iff detα′s,xs,x′s=0.Definition 1.
A surface in the Minkowski 3-spaceE13 is called a timelike surface if the induced metric on the surface is a Lorentz metric and is called a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on spacelike (timelike) surface is a timelike (spacelike) vector.


## 3. Timelike Sweeping Surfaces and Singularities

In this section, the classification of singularity type of timelike sweeping surfaces is studied with the RMF inE13. Let γ=γs be a unit speed spacelike curve with timelike binormal as defined on the RMF frame. Then, we can give the parametric form of sweeping surface given by the spine curve γs as follows:
(11)M:ys,t=γs+Rsrt=γs+r1tξ1s+r2tξ2s,where rt=0,r1t,r2tT is called planar profile (cross section); “T” represents transposition, with another parameter t∈I⊆ℝ. The semiorthogonal matrix s=ξs,ξ1s,ξ2s specifies the RMF along γs. We will utilize “dot” to indicate the derivative with respect to the arc length parameter of the profile curve rt.The tangent vectors and the unit normal vector to the surface, respectively, are(12)yts,t=r.1ξ1+r.2ξ2,yss,t=1−μ1r1−μ2r2ξ,ns,t=yt×ysyt×ys=r.2ξ1+r.1ξ2.From Equation (3.3), it follows thatns,t is contained in the normal plane of the spine curve γs, since it is orthogonal to ξ. Thus, the normal of the profile curve rt and the surface normal are identical. Through this work, we will assume that the profile curve rt is a unit speed timelike curve, that is, r.12−r.22=−1. Thus, M is a timelike sweeping surface. From now on, we shall often not write the parameter s explicitly in our formulae.Our aim of this work is the following theorem.Theorem 2.
For the timelike sweeping surface Equation (3.1), withμ12−μ22≠0, one has the following
(A)(1)
gs is locally diffeomorphic to a timelike line {0}×ℝ at s0 iff σs0≠0(2)
gs is locally diffeomorphic to the cusp C×ℝ at s0 iff σs0≠0 and σ′s0=0
(B)(1)
M is locally diffeomorphic to cuspidal edge CE at s0,t0 iff x=±gs0 and σs0≠0(2)
M is locally diffeomorphic to swallowtail SW at s0,t0 iff x=±gs0, σs0≠0, and σ′s0=0
Here,C×ℝ=x1,x2∣x12=x23×ℝ, CE=x1,x2,x3∣x1=u,x2=v2,x3=v3, and SW=x1,x2,x3∣x1=u,x2=3v2+uv2,3=4v3+2uv. The graphs of C×ℝ, CE, and SW are seen in Figures 1–3.Figure 1
C×ℝ.Figure 2
CE.Figure 3
SW.


### 3.1. Lorentzian Bishop Height Functions

Now, we will define two different families of Lorentzian Bishop height functions that will be useful to study the singularities ofM as follows: H:I×S12⟶ℝ, by Hs,x=<γs,x>. We call it the Lorentzian Bishop height function. We use the notation hxs=Hs,x for any fixed x∈S12. We also define H~:I×S12×ℝ⟶ℝ, by H~s,x,w=<γ,x>−w. We call it the extended Lorentzian Bishop height function of γs. We denote that h~xs=H~s,x. From now on, we shall often not write the parameter s. Then, we have the following proposition.Proposition 3.
Letγ=γs be a unit speed spacelike curve with timelike binormal normal and μ12−μ22≠0. Then, the following hold:
(A)(1)
hx′s=0 iff x=a1ξ1+a2ξ2 and a12−a22=−1(2)
hx′s=hx′′s=0 iff x=±gs(3)
hx′s=hx′′s=hx′′′s=0 iff x=±gs and σs=0(4)
hx′s=hx′′s=hx′′′s=hx4s=0 iff x=±gs and σs=σ′s=0(5)
hx′s=hx′′s=hx′′′s=hx4s=hx5s=0 iff x=±gs and σs=σ′s=σ′′s=0
(B)(1)
h~xs=0 iff there exist <γ,x>=w(2)
h~xs=h~x′s=0 iff there exist t∈ℝ such that x=sinhtξ1+coshtξ2 and <γ,x>=w(3)
h~xs=h~x′s=h~x′′s=h~x′′s=0 iff x=±gs, <γ,x>=w, and σs=0(4)
h~xs=h~x′s=h~x′′s=h~x′′s=h~x′′′s=0 iff x=±gs, <γ,x>=w, and σs=σ′s=0(5)
h~xs=h~x′s=h~x′′s=h~x′′s=h~x′′′s=h~x4s=0 iff x=±gs, <γ,x>=w, and σs=σ′s=σ′′s=0(6)
h~xs=h~x′s=h~x′′s=h~x′′s=h~x′′′s=h~x4s=h~x5s=0 iff x=±gs, <γ,x>=w, and σs=σ′s=σ′′s=σ′′′s=0Proof.
According to Equation (2.2), we haveξ′2≠0 iff μ12−μ22≠0.
(A)(1)
Sincehx′s=<ξ,x> and {ξ,ξ1, ξ2} is RMF along γs, then there exist a1, a2∈ℝ such that x=a1ξ1+a2ξ2. By the condition that x∈ℍ+2, we get a12−a22=−1. The converse direction also holds(2)
Sincehx′′s=<ξ′,x>=<μ1ξ1−μ2ξ2,x>=0, we have a1μ1+a2μ2=0. It follows from the fact a12−a22=−1 that a1=±μ2/μ12−μ22 and a2=∓μ1/μ12−μ22. Therefore, we have(13)x=∓1μ12−μ22−μ2ξ1+μ1ξ2s=gs.
Therefore,hx′s=hx′′s=0 iff x=±gs.
(3)
Sincehx′′′s=<ξ′′,x>=<−μ12+μ22ξ+μ1′ξ1−μ2′ξ2,x>=0, by the conditions of (2), we have(14)∓1μ12−μ22−μ2μ1′+μ1μ2′s=±σμ12−μ22s=0.
Thus,hx′s=hx′′s=hx′′′s=0 iff x=±gs and σs=0.
(4)
Since(15)hx4s=<ξ′′′,x>=<−3μ1μ1′−μ2μ2′ξ+μ1′′+μ1−μ12+μ22ξ1−μ2′′+μ2−μ12+μ22ξ2,x>=0,by the conditions of (7), we have
(16)∓1μ12−μ22μ2μ1′−μ1μ2′′s=±σ′μ22−μ12s=0.
Thus,hx′s=hx′′s=hx′′s=hx4s=0 iff x=±gs and σs=σ′s=0.
(5)
Sincehx5s=<ξ4,x>0, we have(17)<μ12−μ222−4μ1μ1′′−μ2μ2′′−3μ1′2−μ2′2ξ+μ1′′′−5μ1μ1′μ1−μ2′μ2+μ1′−μ12+μ22ξ1−μ2′′′−5μ2μ1′μ1−μ2′μ2+μ2′−μ12+μ22ξ2,x>=0.
By using the conditions of (8), we have
(18)±1μ12−μ22μ1μ2′′′−μ2μ1′′′+μ1μ2′−μ2μ1′−μ12+μ22s=0.
Therefore,hx′s=hx′′s=hx′′′s=hx4s=hx5=0 iff x=±gs and σs=σ′s=σ′′s=0.
(B)
Using the same computation as the proof of (A), we can get (B) (4) +.☐Proposition 4.
Letγ=γs be a unit speed spacelike curve with timelike binormal and μ12−μ22≠0. Then, we have σs=0 iff gs is a constant timelike vector.Proof.
By simple calculations, we have(19)g′s=σsμ12−μ223−μ1ξ1+μ2ξ2.
Thus,g′s=0 iff σs=μ2μ1′−μ1μ2′=0+.☐Proposition 5.
Letγ: I⟶E13 be a unit speed spacelike curve with timelike binormal and μ12−μ22≠0. Then, we have the following.
(a)
γ is a B-slant helix iff μ2/μ1 is constant(b)
ξ2 is a part of circle on ℍ+2 whose center is the timelike constant vector g0Proof.
(a)
Suppose thatσs=μ1μ2′−μ2μ1′=0. Hence, we can write(20)μ2μ1′=μ1μ2′−μ2μ1′μ12=−σsμ12=0.
Thus,μ2/μ1=constant, that is, γ is a B-slant helix.
(b)
Since(21)<g,ξ2>=μ1μ12−μ22<−μ2μ1ξ1+ξ2,ξ2>=−11−μ22/μ12=const.
This means thatξ2 is a part of circle on ℍ+2 whose center is the constant timelike vector g0s+.☐


### 3.2. Unfolding of Functions by One Variable

In this subsection, we use some general results on the singularity theory for families of function germs. LetF:ℝ×ℝr,s0,x0⟶ℝ be a smooth function and fs=Fx0s,x0. Then, Fis called an r-parameter unfolding of fs. We say that fs has Ak-singularity at s0 if fps0=0 for all 1≤p≤k and fk+1s0≠0. We also say that f has Ak-singularity (k1) at s0. Let the k−1-jet of the partial derivative ∂F/∂xi at s0 be jk−1∂F/∂xis,x0s0=Σj=0k−1Ljis−s0j (without the constant term), for i=1,⋯,r. Then, Fs is called a p-versal unfolding if the k×r matrix of coefficients Lji has rank k≤r. So, we write important set about the unfolding relative to the above notations. The discriminant set of F is the set
(22)DF=x∈ℝr∣thereexistsswithFs,x=∂F∂ss,x=0ats,x.The bifurcation set ofF is the set
(23)BF=x∈ℝr∣thereexistsswith∂F∂ss,x=∂2F∂s2s,x=0ats,x.We can also give the following theorem [12, 13].Theorem 6.
LetF:ℝ×ℝr,s0,x0⟶ℝ be an r-parameter unfolding of fs, which has the Ak singularity at s0.
Suppose thatF is a p-versal unfolding.
(a)
Ifk=1, then DF is locally diffeomorphic to 0×ℝr−1 and BF=∅(b)
Ifk=2, then DF is locally diffeomorphic to C×ℝr−2, and BF is locally diffeomorphic to 0×ℝr−1(c)
Ifk=3, then DF is locally diffeomorphic to SW×ℝr−3, and BF is locally diffeomorphic to C×ℝr−2
Hence, we have the following fundamental proposition.Proposition 7.
Letγ: I⟶E13 be a unit speed spacelike curve with timelike binormal and μ12−μ22≠0. (1) If hxs=Hs,x has an Ak-singularity k=2,3 at s0∈ℝ, then H is a p-versal unfolding of hx0s0. (2) If h~xs=H~s,x,w has an Ak-singularity k=2,3 at s0∈ℝ, then H~ is a p-versal unfolding of h~x0s0.Proof.
(see (4)).
Sincex=x0,x1,x2∈ℍ+2, x02+x12−x22=−1, x0, x1, and x2 cannot be all zero. Without loss of generality, we may assume that x2≠0. Then, by x2=1+x02+x12, we have
(24)Hs,x=x0γ0s+x1γ1s−1+x02+x12γ2s.
Thus, we have(25)∂H∂x0=γ0s−x0γ2s1+x02−x12,∂H∂x1=γ1s−x1γ2s1+x02+x12,∂2H∂s∂x0=γ0′s−x0γ2′s−1+x02−x12,∂2H∂s∂x1=γ1′s−x1γ2′s1+x02−x12.
Therefore, the 2-jets of∂H/∂xi at s0 (i=0,1) are as follows. Let x0=x00,x10,x20∈ℍ+2, and assume that x20≠0, then
(26)j1∂H∂x0s,x0=γ0′s−x0γ2′s1+x02−x12s−s0,j1∂H∂x1s,x0=γ1′s−x1γ2′s1+x02−x12s−s0,j2∂H∂x0s,x0=γ0′s−x00γ2′sx20s−s0+12γ0′′−x00γ2′′sx20s−s02,j2∂H∂x1s,x0=γ1′s−x10γ2′sx20s−s0,12γ1′′s−x10γ2′′sx20s−s02.(i)
Ifhx0s0 has the A2-singularity at s0, then hx0′s0=0. So the 2−1×2 matrix of coefficients Lji is(27)A=γ0′s−x00γ2′sx20γ1′s−x10γ2′sx20.
Suppose that the rank of the matrixA is zero, then we have
(28)γ0′s=x00γ2′sx20,γ1′s=x10γ2′sx20.
Sinceγ′s0=ξs0=1, we have γ2′s0≠0 so that we have the contradiction as follows:
(29)0=<γ0′s0,γ1′s0,γ2′s0,x00,x10,x20>=γ0′s0x00+γ1′s0x10−γ2′s0x20=x002γ2′s0x20+x102γ2′s0x20−γ2′s0x20=γ2′s0x20x002+x102−x202=γ2′s0x20≠0.
Therefore, rankA=1, and H is the (p) versal unfolding of hx0 at s0.
(ii)
Ifhx0s0 has the A3-singularity at s0∈ℝ, then hx0′s0=hx0′′s0=0, and by Proposition 3.(30)gs0=μ1μ12−μ22−μ2μ1ξ1+ξ2,where μ12−μ22>0, σ′s0=0, and σ′′s0≠0. So, the 3−1×2 matrix of the coefficients Lji is
(31)B=L11L12L21L22=γ0′s+x00γ2′sx20γ1′s−x10γ2′sx20γ0′′+x00γ2′′sx20γ1′′s−x10γ2′′sx20.
For the purpose, we also require the2×2 matrix B to be nonsingular, which always does. In fact, the determinate of this matrix at s0 is
(32)detB=1x20−γ0′γ1′γ2′−γ0′′γ1′′γ2′′x00x10x20=1x20<γ′×γ′′,g0>=∓1x20μ12−μ22<γ′×γ′′,−μ2ξ1+μ1ξ2>.
Sinceγ′=ξ, we have γ′′=μ1ξ1−μ2ξ2. Substituting these relations to the above equality, we have
(33)detB=±μ12−μ22x20≠0.
This means that rankB=2.
(2) Under the same notations as in (4), we have
(34)H~s,x,x2=x0γ0s+x1γ1s−1+x02+x12γ2s−x2.
We require the2×3 matrix
(35)G=γ0′s−x00γ2sx20γ1′s−x10γ2′sx20−1γ0′′−x00γ2′′sx20γ1′′s−x10γ2′′sx200,to have the maximal rank. By case (4) in Equation (3.14), the second row of G does not vanish, so rank G=2+.☐Proof of Theorem 1 (see (4)).
By Proposition3, the bifurcation set of Hs,x is
(36)BH=1μ12−μ22−μ2ξ1+μ1ξ2s∈ℝs∈ℝ.The assertion (4) of Theorem 2 follows from Propositions 3 and 7 and Theorem 6. The discriminant set of H~s,x is given as follows:
(37)DH~=x0=γ+sinhtξ1+coshtξ2∣s∈ℝ.The assertion (4) of Theorem 2 follows from Propositions 3 and 7 and Theorem 6+.Example 1.
Given the spacelike helix(38)γs=coshs,2s,sinhs,−3≤s≤3.
It is easy to show that(39)υ1s=sinhs,2,coshs,υ2s=coshs,0,sinhs,υ3s=2sinhs,1,2coshs,κs=1,andτs=2.
Then,φs=−2s+φ0. If we choose φ0=0, for example, we have
(40)μ1s=cosh2s,μ2s=−sinh2s.
We can calculate the geometric invariant(41)σs=−2,σ′s=0.
We also have(42)ξ1=ξ11ξ12ξ13=cosh2scoshs−2sinh2ssinhs−sinh2scosh2ssinhs−2sinh2scoshs,ξ2=ξ21ξ22ξ23=−sinh2scoshs+2cosh2ssinhscosh2s−sinh2ssinhs+2cosh2scoshs.
The timelike Bishop spherical Darboux image is shown in Figure4)
(43)gs=sinh2sξ11ξ12ξ13+cosh2sξ21ξ22ξ23.
The timelike sweeping surface family is(44)ys,t=coshs,2s,sinhs+r1tξ11ξ12ξ13+r2tξ21ξ22ξ23.
By choosingr1t=cosht and r2t=sinht, then we immediately have a timelike sweeping surface (see Figure 5).Figure 4
Bishop spherical Darboux image has a cusp ass=0.Figure 5
Timelike sweeping surface with spacelike helix singularity curve.


### 3.3. Developable Surfaces

Developable surfaces can be briefly introduced as special cases of ruled surfaces. Such surfaces are widely used, for example, in the manufacture of automobile body parts, airplane wings, and ship hulls. Therefore, we analyze the case that the profile curvert degenerates into a timelike line. Then, we have the following timelike developable surface
(45)M:Ds,t=γs+tξ2s,t∈ℝ.We also have the following spacelike developable surface(46)M⊥:D⊥s,t=γs+tξ1s,t∈ℝ.It is clear thatDs,0=γs (resp. D⊥s,01=γs), 0≤s≤L, that is, the surface M (resp. M⊥) interpolate the curve γs. Also, we have
(47)∂D∂s×∂D∂t=−1−tμ2ξ1s.∂D⊥∂s×∂D⊥∂t=−1−tμ1ξ2s.Thus, we have thatM (resp. M⊥) is nonsingular at s0,t0 if and only if 1−t0μ2s0≠0 (resp. 1−t0μ1s0≠0). We designate μs to represent μis (i=1,2), and based on Theorem 3.3 in [23], we can give the following corollary.Corollary 8.
For the developable ruled surfacesDs,t and D⊥s,t, we have the following:
(1)
D (resp. D⊥) is locally diffeomorphic to the cuspidal edge CE C2,3×ℝ at s0,t0 iff μs0=0 and μ′s0≠0(2)
D (resp. D⊥) is locally diffeomorphic to swallowtail SW at s0,t0 iff μs0≠0 and μ′s0=0Example 2.
By making using of Example1, we have the following:
(1)
Ifs0=0, then μ2s0=0 and μ2′s0≠0. The timelike developable surface(48)Ds,t=coshs,2s,sinhs+t−sinh2scoshs+2cosh2ssinhscosh2s−sinh2ssinhs+2cosh2scoshs,t∈ℝis locally diffeomorphic to the cuspidal edge; see Figure 6. We can obtain the singular locus of M as follows:
(49)Ss=2coshs−2coth2ssinhs2s−coth2s2sinhs−2coth2scoshs.(2)
Ifs0=0, then μ1s0≠0 and μ1′s0=0. The spacelike developable surface(50)D⊥s,t=coshs,2s,sinhs+tcosh2scoshs−2sinh2ssinhs−sinh2scosh2ssinhs−2sinh2scoshs,t∈ℝis locally diffeomorphic to swallowtail; see Figure 7. Also, the singular locus of M⊥ is
(51)S⊥s=2coshs−2tanh2ssinhs2s−tanh2s2sinhs−2tanh2scoshs.Figure 6
Timelike developable surface with its singular curve.Figure 7
Spacelike developable surface with its singular curve.


## 3.1. Lorentzian Bishop Height Functions

Now, we will define two different families of Lorentzian Bishop height functions that will be useful to study the singularities ofM as follows: H:I×S12⟶ℝ, by Hs,x=<γs,x>. We call it the Lorentzian Bishop height function. We use the notation hxs=Hs,x for any fixed x∈S12. We also define H~:I×S12×ℝ⟶ℝ, by H~s,x,w=<γ,x>−w. We call it the extended Lorentzian Bishop height function of γs. We denote that h~xs=H~s,x. From now on, we shall often not write the parameter s. Then, we have the following proposition.Proposition 3.
Letγ=γs be a unit speed spacelike curve with timelike binormal normal and μ12−μ22≠0. Then, the following hold:
(A)(1)
hx′s=0 iff x=a1ξ1+a2ξ2 and a12−a22=−1(2)
hx′s=hx′′s=0 iff x=±gs(3)
hx′s=hx′′s=hx′′′s=0 iff x=±gs and σs=0(4)
hx′s=hx′′s=hx′′′s=hx4s=0 iff x=±gs and σs=σ′s=0(5)
hx′s=hx′′s=hx′′′s=hx4s=hx5s=0 iff x=±gs and σs=σ′s=σ′′s=0
(B)(1)
h~xs=0 iff there exist <γ,x>=w(2)
h~xs=h~x′s=0 iff there exist t∈ℝ such that x=sinhtξ1+coshtξ2 and <γ,x>=w(3)
h~xs=h~x′s=h~x′′s=h~x′′s=0 iff x=±gs, <γ,x>=w, and σs=0(4)
h~xs=h~x′s=h~x′′s=h~x′′s=h~x′′′s=0 iff x=±gs, <γ,x>=w, and σs=σ′s=0(5)
h~xs=h~x′s=h~x′′s=h~x′′s=h~x′′′s=h~x4s=0 iff x=±gs, <γ,x>=w, and σs=σ′s=σ′′s=0(6)
h~xs=h~x′s=h~x′′s=h~x′′s=h~x′′′s=h~x4s=h~x5s=0 iff x=±gs, <γ,x>=w, and σs=σ′s=σ′′s=σ′′′s=0Proof.
According to Equation (2.2), we haveξ′2≠0 iff μ12−μ22≠0.
(A)(1)
Sincehx′s=<ξ,x> and {ξ,ξ1, ξ2} is RMF along γs, then there exist a1, a2∈ℝ such that x=a1ξ1+a2ξ2. By the condition that x∈ℍ+2, we get a12−a22=−1. The converse direction also holds(2)
Sincehx′′s=<ξ′,x>=<μ1ξ1−μ2ξ2,x>=0, we have a1μ1+a2μ2=0. It follows from the fact a12−a22=−1 that a1=±μ2/μ12−μ22 and a2=∓μ1/μ12−μ22. Therefore, we have(13)x=∓1μ12−μ22−μ2ξ1+μ1ξ2s=gs.
Therefore,hx′s=hx′′s=0 iff x=±gs.
(3)
Sincehx′′′s=<ξ′′,x>=<−μ12+μ22ξ+μ1′ξ1−μ2′ξ2,x>=0, by the conditions of (2), we have(14)∓1μ12−μ22−μ2μ1′+μ1μ2′s=±σμ12−μ22s=0.
Thus,hx′s=hx′′s=hx′′′s=0 iff x=±gs and σs=0.
(4)
Since(15)hx4s=<ξ′′′,x>=<−3μ1μ1′−μ2μ2′ξ+μ1′′+μ1−μ12+μ22ξ1−μ2′′+μ2−μ12+μ22ξ2,x>=0,by the conditions of (7), we have
(16)∓1μ12−μ22μ2μ1′−μ1μ2′′s=±σ′μ22−μ12s=0.
Thus,hx′s=hx′′s=hx′′s=hx4s=0 iff x=±gs and σs=σ′s=0.
(5)
Sincehx5s=<ξ4,x>0, we have(17)<μ12−μ222−4μ1μ1′′−μ2μ2′′−3μ1′2−μ2′2ξ+μ1′′′−5μ1μ1′μ1−μ2′μ2+μ1′−μ12+μ22ξ1−μ2′′′−5μ2μ1′μ1−μ2′μ2+μ2′−μ12+μ22ξ2,x>=0.
By using the conditions of (8), we have
(18)±1μ12−μ22μ1μ2′′′−μ2μ1′′′+μ1μ2′−μ2μ1′−μ12+μ22s=0.
Therefore,hx′s=hx′′s=hx′′′s=hx4s=hx5=0 iff x=±gs and σs=σ′s=σ′′s=0.
(B)
Using the same computation as the proof of (A), we can get (B) (4) +.☐Proposition 4.
Letγ=γs be a unit speed spacelike curve with timelike binormal and μ12−μ22≠0. Then, we have σs=0 iff gs is a constant timelike vector.Proof.
By simple calculations, we have(19)g′s=σsμ12−μ223−μ1ξ1+μ2ξ2.
Thus,g′s=0 iff σs=μ2μ1′−μ1μ2′=0+.☐Proposition 5.
Letγ: I⟶E13 be a unit speed spacelike curve with timelike binormal and μ12−μ22≠0. Then, we have the following.
(a)
γ is a B-slant helix iff μ2/μ1 is constant(b)
ξ2 is a part of circle on ℍ+2 whose center is the timelike constant vector g0Proof.
(a)
Suppose thatσs=μ1μ2′−μ2μ1′=0. Hence, we can write(20)μ2μ1′=μ1μ2′−μ2μ1′μ12=−σsμ12=0.
Thus,μ2/μ1=constant, that is, γ is a B-slant helix.
(b)
Since(21)<g,ξ2>=μ1μ12−μ22<−μ2μ1ξ1+ξ2,ξ2>=−11−μ22/μ12=const.
This means thatξ2 is a part of circle on ℍ+2 whose center is the constant timelike vector g0s+.☐


## 3.2. Unfolding of Functions by One Variable

In this subsection, we use some general results on the singularity theory for families of function germs. LetF:ℝ×ℝr,s0,x0⟶ℝ be a smooth function and fs=Fx0s,x0. Then, Fis called an r-parameter unfolding of fs. We say that fs has Ak-singularity at s0 if fps0=0 for all 1≤p≤k and fk+1s0≠0. We also say that f has Ak-singularity (k1) at s0. Let the k−1-jet of the partial derivative ∂F/∂xi at s0 be jk−1∂F/∂xis,x0s0=Σj=0k−1Ljis−s0j (without the constant term), for i=1,⋯,r. Then, Fs is called a p-versal unfolding if the k×r matrix of coefficients Lji has rank k≤r. So, we write important set about the unfolding relative to the above notations. The discriminant set of F is the set
(22)DF=x∈ℝr∣thereexistsswithFs,x=∂F∂ss,x=0ats,x.The bifurcation set ofF is the set
(23)BF=x∈ℝr∣thereexistsswith∂F∂ss,x=∂2F∂s2s,x=0ats,x.We can also give the following theorem [12, 13].Theorem 6.
LetF:ℝ×ℝr,s0,x0⟶ℝ be an r-parameter unfolding of fs, which has the Ak singularity at s0.
Suppose thatF is a p-versal unfolding.
(a)
Ifk=1, then DF is locally diffeomorphic to 0×ℝr−1 and BF=∅(b)
Ifk=2, then DF is locally diffeomorphic to C×ℝr−2, and BF is locally diffeomorphic to 0×ℝr−1(c)
Ifk=3, then DF is locally diffeomorphic to SW×ℝr−3, and BF is locally diffeomorphic to C×ℝr−2
Hence, we have the following fundamental proposition.Proposition 7.
Letγ: I⟶E13 be a unit speed spacelike curve with timelike binormal and μ12−μ22≠0. (1) If hxs=Hs,x has an Ak-singularity k=2,3 at s0∈ℝ, then H is a p-versal unfolding of hx0s0. (2) If h~xs=H~s,x,w has an Ak-singularity k=2,3 at s0∈ℝ, then H~ is a p-versal unfolding of h~x0s0.Proof.
(see (4)).
Sincex=x0,x1,x2∈ℍ+2, x02+x12−x22=−1, x0, x1, and x2 cannot be all zero. Without loss of generality, we may assume that x2≠0. Then, by x2=1+x02+x12, we have
(24)Hs,x=x0γ0s+x1γ1s−1+x02+x12γ2s.
Thus, we have(25)∂H∂x0=γ0s−x0γ2s1+x02−x12,∂H∂x1=γ1s−x1γ2s1+x02+x12,∂2H∂s∂x0=γ0′s−x0γ2′s−1+x02−x12,∂2H∂s∂x1=γ1′s−x1γ2′s1+x02−x12.
Therefore, the 2-jets of∂H/∂xi at s0 (i=0,1) are as follows. Let x0=x00,x10,x20∈ℍ+2, and assume that x20≠0, then
(26)j1∂H∂x0s,x0=γ0′s−x0γ2′s1+x02−x12s−s0,j1∂H∂x1s,x0=γ1′s−x1γ2′s1+x02−x12s−s0,j2∂H∂x0s,x0=γ0′s−x00γ2′sx20s−s0+12γ0′′−x00γ2′′sx20s−s02,j2∂H∂x1s,x0=γ1′s−x10γ2′sx20s−s0,12γ1′′s−x10γ2′′sx20s−s02.(i)
Ifhx0s0 has the A2-singularity at s0, then hx0′s0=0. So the 2−1×2 matrix of coefficients Lji is(27)A=γ0′s−x00γ2′sx20γ1′s−x10γ2′sx20.
Suppose that the rank of the matrixA is zero, then we have
(28)γ0′s=x00γ2′sx20,γ1′s=x10γ2′sx20.
Sinceγ′s0=ξs0=1, we have γ2′s0≠0 so that we have the contradiction as follows:
(29)0=<γ0′s0,γ1′s0,γ2′s0,x00,x10,x20>=γ0′s0x00+γ1′s0x10−γ2′s0x20=x002γ2′s0x20+x102γ2′s0x20−γ2′s0x20=γ2′s0x20x002+x102−x202=γ2′s0x20≠0.
Therefore, rankA=1, and H is the (p) versal unfolding of hx0 at s0.
(ii)
Ifhx0s0 has the A3-singularity at s0∈ℝ, then hx0′s0=hx0′′s0=0, and by Proposition 3.(30)gs0=μ1μ12−μ22−μ2μ1ξ1+ξ2,where μ12−μ22>0, σ′s0=0, and σ′′s0≠0. So, the 3−1×2 matrix of the coefficients Lji is
(31)B=L11L12L21L22=γ0′s+x00γ2′sx20γ1′s−x10γ2′sx20γ0′′+x00γ2′′sx20γ1′′s−x10γ2′′sx20.
For the purpose, we also require the2×2 matrix B to be nonsingular, which always does. In fact, the determinate of this matrix at s0 is
(32)detB=1x20−γ0′γ1′γ2′−γ0′′γ1′′γ2′′x00x10x20=1x20<γ′×γ′′,g0>=∓1x20μ12−μ22<γ′×γ′′,−μ2ξ1+μ1ξ2>.
Sinceγ′=ξ, we have γ′′=μ1ξ1−μ2ξ2. Substituting these relations to the above equality, we have
(33)detB=±μ12−μ22x20≠0.
This means that rankB=2.
(2) Under the same notations as in (4), we have
(34)H~s,x,x2=x0γ0s+x1γ1s−1+x02+x12γ2s−x2.
We require the2×3 matrix
(35)G=γ0′s−x00γ2sx20γ1′s−x10γ2′sx20−1γ0′′−x00γ2′′sx20γ1′′s−x10γ2′′sx200,to have the maximal rank. By case (4) in Equation (3.14), the second row of G does not vanish, so rank G=2+.☐Proof of Theorem 1 (see (4)).
By Proposition3, the bifurcation set of Hs,x is
(36)BH=1μ12−μ22−μ2ξ1+μ1ξ2s∈ℝs∈ℝ.The assertion (4) of Theorem 2 follows from Propositions 3 and 7 and Theorem 6. The discriminant set of H~s,x is given as follows:
(37)DH~=x0=γ+sinhtξ1+coshtξ2∣s∈ℝ.The assertion (4) of Theorem 2 follows from Propositions 3 and 7 and Theorem 6+.Example 1.
Given the spacelike helix(38)γs=coshs,2s,sinhs,−3≤s≤3.
It is easy to show that(39)υ1s=sinhs,2,coshs,υ2s=coshs,0,sinhs,υ3s=2sinhs,1,2coshs,κs=1,andτs=2.
Then,φs=−2s+φ0. If we choose φ0=0, for example, we have
(40)μ1s=cosh2s,μ2s=−sinh2s.
We can calculate the geometric invariant(41)σs=−2,σ′s=0.
We also have(42)ξ1=ξ11ξ12ξ13=cosh2scoshs−2sinh2ssinhs−sinh2scosh2ssinhs−2sinh2scoshs,ξ2=ξ21ξ22ξ23=−sinh2scoshs+2cosh2ssinhscosh2s−sinh2ssinhs+2cosh2scoshs.
The timelike Bishop spherical Darboux image is shown in Figure4)
(43)gs=sinh2sξ11ξ12ξ13+cosh2sξ21ξ22ξ23.
The timelike sweeping surface family is(44)ys,t=coshs,2s,sinhs+r1tξ11ξ12ξ13+r2tξ21ξ22ξ23.
By choosingr1t=cosht and r2t=sinht, then we immediately have a timelike sweeping surface (see Figure 5).Figure 4
Bishop spherical Darboux image has a cusp ass=0.Figure 5
Timelike sweeping surface with spacelike helix singularity curve.


## 3.3. Developable Surfaces

Developable surfaces can be briefly introduced as special cases of ruled surfaces. Such surfaces are widely used, for example, in the manufacture of automobile body parts, airplane wings, and ship hulls. Therefore, we analyze the case that the profile curvert degenerates into a timelike line. Then, we have the following timelike developable surface
(45)M:Ds,t=γs+tξ2s,t∈ℝ.We also have the following spacelike developable surface(46)M⊥:D⊥s,t=γs+tξ1s,t∈ℝ.It is clear thatDs,0=γs (resp. D⊥s,01=γs), 0≤s≤L, that is, the surface M (resp. M⊥) interpolate the curve γs. Also, we have
(47)∂D∂s×∂D∂t=−1−tμ2ξ1s.∂D⊥∂s×∂D⊥∂t=−1−tμ1ξ2s.Thus, we have thatM (resp. M⊥) is nonsingular at s0,t0 if and only if 1−t0μ2s0≠0 (resp. 1−t0μ1s0≠0). We designate μs to represent μis (i=1,2), and based on Theorem 3.3 in [23], we can give the following corollary.Corollary 8.
For the developable ruled surfacesDs,t and D⊥s,t, we have the following:
(1)
D (resp. D⊥) is locally diffeomorphic to the cuspidal edge CE C2,3×ℝ at s0,t0 iff μs0=0 and μ′s0≠0(2)
D (resp. D⊥) is locally diffeomorphic to swallowtail SW at s0,t0 iff μs0≠0 and μ′s0=0Example 2.
By making using of Example1, we have the following:
(1)
Ifs0=0, then μ2s0=0 and μ2′s0≠0. The timelike developable surface(48)Ds,t=coshs,2s,sinhs+t−sinh2scoshs+2cosh2ssinhscosh2s−sinh2ssinhs+2cosh2scoshs,t∈ℝis locally diffeomorphic to the cuspidal edge; see Figure 6. We can obtain the singular locus of M as follows:
(49)Ss=2coshs−2coth2ssinhs2s−coth2s2sinhs−2coth2scoshs.(2)
Ifs0=0, then μ1s0≠0 and μ1′s0=0. The spacelike developable surface(50)D⊥s,t=coshs,2s,sinhs+tcosh2scoshs−2sinh2ssinhs−sinh2scosh2ssinhs−2sinh2scoshs,t∈ℝis locally diffeomorphic to swallowtail; see Figure 7. Also, the singular locus of M⊥ is
(51)S⊥s=2coshs−2tanh2ssinhs2s−tanh2s2sinhs−2tanh2scoshs.Figure 6
Timelike developable surface with its singular curve.Figure 7
Spacelike developable surface with its singular curve.


## 4. Conclusion

In this paper, we introduced the notion of timelike sweeping surfaces with rotation minimizing frames in Minkowski 3-spaceE13. By applying singularity, we classified the generic properties and present a new geometric invariant related to the singularities of this timelike sweeping surface. It leads to the fact that the generic singularities of this sweeping surface are cuspidal edge and swallowtail, and the types of these singularities can be characterized by this geometric invariant, respectively. Finally, some examples are presented to explain the theoretical results.



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