# Spacelike Sweeping Surfaces and Singularities in Minkowski 3-Space

Authors: Fatemah Mofarreh; Rashad A. Abdel-Baky; Nadia Alluhaibi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We considered the spacelike sweeping surface with rotation minimizing frames at Minkowski 3-spaceE13. We presented the new geometric invariant to demonstrate geometric properties and local singularities for this surface. Then, we derived sufficient and necessary conditions of the surface to become developable ruled surfaces. Additionally, its singularities are studied. Finally, examples are illustrated to explain the applications of the theoretical results.


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## Body

## 1. Introduction

The vision of the computer is the automatical analysis of image sequences in order to build the 3-dimensional surface form. Recently, various majors of mathematics are used for computer vision and medical imaging. Projective geometry, as an old mathematical subject, still used to characterize connections between both lines and points in several images to the same theme. In addition, differential geometry is used to characterize the shape of the curve and surface in engineering. Both Rene Thom, French mathematician, and Hassler Whitney introduced some developments in mathematical thinking and methods, especially the concept of singularity theory that contains catastrophes and bifurcations. Singularity theory now as the direct application of differential calculus is important to gain vital results in many subjects as computer vision and medical imaging (e.g., [1–7]).A canal surface is the surface that can be generated by the one-parameter set of spheres determined by a radius function and center curves: in case a radius function is a constant function, the canal surface is the envelope of the moving sphere and named the sweeping surface. Some well-known examples of the sweeping surface are circular cylinder and circular cone (radius of spheres is not constant), surface of revolution, and Dupin cycloids. More specific, the sweeping surface named the tubular surface in case the radius of the generating spheres is constant. Sweeping surfaces are very important for descriptive geometry, especially for solid and surface modeling at computer-aided design, computer-aided manufacturing (CAD/CAM), and the design of trajectory movement for robots [8–14]. It is a fact that a sweeping surface can be a developable surface. The developable surface defines the surface that can become unfolded (or developed) to the plane with the absence of any stretch or tear. As known at differential geometry, with considering sufficient differentiability, the developable surface defines the plane, conical surface, cylindrical surface, or tangent surface of the curve or the structure of one of those kinds. Therefore, the developable surface is considered as the ruled surface, such that every point at the same ruling shares the common tangent plane. The rulings are principal curvature lines which vanish normal curvature and Gaussian curvature that is vanishing at every point. As a result, the developable surface is a significant surface in (CAD/CAM) and geometric modeling as it is used for motion analysis or designing cars and ships [15–19].The essential tool to analyze the curve and surface at differential geometry is the Serret–Frenet frame that is the most used frame at Euclidean 3-space and Minkowski 3-space [16–20]. The main apparatus in the previous literatures are Serret–Frenet formulas and some linked functions on the curve as a distance-squared function in addition to the height function. Based on Serret–Frenet formulas, the singularity of those functions can be studied from the view of extrinsic differential geometry. Because the Serret–Frenet will not be defined everywhere, there is a need for a new mathematical tool to be used for analysis purposes. In [20], Bishop gave the alternative moving frame to points on the curve at Euclidean 3-space using parallel vector fields. It named rotation minimizing frame (RMF) or Bishop frame of the curves [19–21]. Analogous to Bishop frame in Euclidean 3-space, there is a similar Lorentzian frame which named Lorentzian Bishop frame, constructed along the curve at Lorentzian space, and it is the analog of the Bishop type frame as applied to Lorentzian geometry. At Lorentzian space, using Minkowski Bishop frame through the curve as a basic tool is preferred than using Serret–Frenet frame [21–23].In fact, there is no more literature review regarding singularities of sweeping surfaces relating to the Minkowski Bishop frame. Therefore, this study aims to cover some needs, where it is inspired by the study of Izumiya et al. [7] and Bishop [19]. At this study, we establish the Lorentzian Bishop frame along the unit speed spacelike curve with timelike principal normal and develop the local differential geometry of spacelike sweeping surfaces at Minkowski 3-space. Using unfolding theory at singularity theory in this study, generic singularities of this sweeping surface are classified. A new invariant relating to singularities of this sweeping surface is presented. It is founded that generic singularities of this sweeping surface are cuspidal edge and swallowtail, and these kind of singularities can be characterized by this invariant, in the same order. Afterward, we solved this problem of requiring the surface which is the spacelike sweeping surface and, at the same time, the spacelike developable surface. Some examples are introduced in order to demonstrate theoretical results.


## 2. Preliminaries

Some definitions and basic concepts are given which will be used (for instance [8, 24, 25]). Suppose E13 is 3-dimensional Minkowski space, the 3-dimensional real vector space ℝ3 considers the metric(1)<dr,dr>=dr12+dr22−dr32,where r1,r2,r3 denotes the canonical coordinates in ℝ3. Any vector r of E13 named spacelike in case <r,r≫0 or r=0, timelike in case r,r<0, and lightlike or null in case r,r=0 and r=0. The timelike or lightlike vector at E13 is named causal. Also, with the norm r=r,r, the vector r is the spacelike unit vector if r,r=1 and a timelike unit vector if r,r=−1. Therefore, we say that a smooth map β:I⟶E13 is spacelike, timelike, or lightlike, if its velocity vector β′ is spacelike, timelike, or lightlike, in the same order. Similarly, the surface is named spacelike, timelike, or lightlike if its tangent planes are spacelike, timelike, or lightlike, respectively. For any two vectors r, p∈E13, the inner product is a real number r,p=−r1p1+r2p2+r3p3, and the vector product is given as(2)r×p=f1f2−f3r1r2r3p1p2p3=r2p3−r3p2,r3p1−r1p3,−r1p2−x2p1,where f1, f2, f3 are the canonical bases of E13. For a fixed point p∈E13 and the positive number c>0, hyperbolic and Lorentzian (de Sitter space) spheres, in the same order, are given as(3)ℍ+2p,c=a∈E13|a−p,a−p=−c2,S12p,c=a∈E13|a−p,a−p=c2.We define(4)Lℂp∗=a∈E13|a−p,a−p=0−p,and it is called the (open) lightcone at the vertex p. In case p=0 and c=1, we define Lℂ0∗, ℍ+2, in addition to S12, in the same order.Supposeβ=βs is the unit speed spacelike curve with timelike principal normal in E13, and suppose κs and τs are the natural curvature and torsion of βs, in the same order. Suppose ζ1s, ζ2s, ζ3s is a Serret–Frenet frame related to the curve βs, then the Serret–Frenet formulas read(5)ζ1′ζ2′ζ3′=0κ0κ0τ0τ0ζ1ζ2ζ3=ψ×ζ1ζ2ζ3,where ψs=τζ1−κζ3 is the Darboux vector of the Serret–Frenet frame. At this study, β′s denotes the derivatives of β respecting to arc length parameter. The Serret–Frenet vector fields satisfy the following relations:(6)ζ1,ζ1=ζ3,ζ3=1,ζ2,ζ2=−1,ζ1×ζ2=ζ3,ζ1×ζ3=ζ2,ζ2×ζ3=ζ1.Definition 1.
The moving pseudoorthogonal frameξ1, ξ3, ξ3, along the nonnull space curve αs, is rotation minimizing frame (RMF) respecting to ξ1 in case its angular velocity ω insures ω,ξ1=0 or, similarly, the derivatives of ξ2 and ξ3 are both parallel to ξ1. Analogously, characterization holds when ξ2 or ξ3 is selected to be the reference direction.
Using Definition1, it is observed that the Serret–Frenet frame is RMF respecting to the principal normal ξ2 but not respecting to the tangent ζ1 and the binormal ζ3. Even though the Serret–Frenet frame is not RMF respecting to ζ1, it is easy to derive similar RMF from it. New normal plane vectors (N1,N2) are determined along the rotation of (ζ2,ζ3) as(7)T1N1N2=1000coshθsinhθ0sinhθcoshθζ1ζ2ζ3,with a certain Lorentzian timelike angle θs≥0. The set T1, N1, N2 is called RMF or Bishop frame. The RMF vector insure the following relations:(8)T1,T1=N2,N2=1,N1,N1=−1,T1×N1=N2,T1×N2=N1,N1×N2=T1.
Therefore, the Bishop frame reads(9)T1′N1′N2′=0κ1κ2κ100−κ200T1N1N2=ω˜×T1N1N2,where ω˜s=−κ2N1−κ1N2 is the Bishop Darboux vector. Here, the Bishop curvatures are determined as κ1s=κcoshϑ and κ2s=−κsinhϑ. Comparing equations (5) and (9), it is observed that the relative velocity is(10)ψs−ω˜s=τζ1.
One can show that(11)κ12−κ22=κ2,andϑ=−tanh−1κ2κ1;κ1≠0,θs=−∫s0sτds+θ0,θ0=θ0.
As a consequence, both Serret–Frenet frame and RMF identical iffβs is the planar, which means τ=0.
The spacelike vector is defined as(12)ωs=−κ2κ1N1−N2,and we name it the modified Bishop Darboux vector through βs. A Bishop spherical Darboux image es:I⟶S120,1 is defined by(13)es=ωsωs=−κ1κ12−κ22κ2κ1N1+N2.
Therefore, we consider a new geometric invariantρs=κ2κ1′−κ1κ2′.Definition 2.
The sweeping surface throughβs is the surface given as(14)M:Rs,u=βs+Fsxu=αs+x1uN1s+x2uN2s,where βs named the (at least C1-continuous) spine curve, 0≤s≤T, and s is the arc length parameter. xu is the planar profile (cross-section) curve defined as parametric presentation, xu=0x1u,x2uT, “T” refers to transposition, and u∈I⊆ℝ. The semiorthogonal matrix Fs=T1s, N1s , N2s specifies the RMF through βs.
Kinematically, the sweeping surfaceRs,u is generating by the moving of the profile curve xu through the spine curve βs with the orientation as introduced by Fs. The profile curve xu is in the 2D or 3D space that passes through the spine curve βs during sweeping. Interestingly, RMF allows for a simple characterization of spine curve.Definition 3.
The surface at Minkowski 3-spaceE13 named the timelike surface in case the induced metric at the surface is the Lorentz metric and also it named the spacelike surface in case the induced metric at the surface is a positive definite Riemannian metric, which means the normal vector on spacelike (timelike) surface is the timelike (spacelike) vector.


## 3. Spacelike Sweeping Surface and Its Singularities

We present the spacelike sweeping surface at Minkowski 3-spaceE13. Consider the planar profile (cross-section) that is defined as xu=0,coshu,sinhu. Using equation (14), we obtain(15)M:Rs,u=βs+coshuN1+sinhuN2.Using equation (9) resulted in(16)Rus,u=sinhuN1+coshuN2,Rss,u=1+κ1coshu−κ2sinhuT1..The unit normal vector ofM is(17)Ns,u=Rs×RuRs×Ru=coshuN1+sinhuN2.The main aim of this study is given in the following theorem:Theorem 1.
Supposeβ: I⟶E13 is the unit speed spacelike curve with a timelike principal normal, and κ12−κ22≠0. Then, for any fixed x∈S120,1, one has the following:A.(1)
The image of Bishop spherical Darboux is locally diffeomorphic to the line {0}×ℝ at s0 iff ρs0≠0(2)
The image of Bishop spherical Darboux is locally diffeomorphic to a cuspC×ℝ at s0 if ρs0=0, as well ρ′s0≠0B.(1)
M is locally diffeomorphic to cuspidal edge CE at s0,u0 iff x=±es0, as well ρs0≠0(2)
M is locally diffeomorphic to swallowtail SW at s0,u0 iff x=±es0, ρs0=0, as well ρ′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,x3=4v3+2uv. The pictures of C×ℝ, CE, and SW are shown in Figures 1–3.Figure 1
C×ℝ.Figure 2
CE.Figure 3
SW.


### 3.1. Lorentzian Bishop Height Functions

We will introduce two different families of Lorentzian Bishop height functions that will be useful to study singularities ofM as follows [1–3]: H:I×S12⟶ℝ, by Hs,x=βs,x. It is called Lorentzian Bishop height function. The notation hxs=Hs,x will be used for all fixed x∈S12. In addition, it is defined H˜:I×S12×ℝ⟶ℝ, using H˜s,x,w=β,x−w. It is called extended Lorentzian Bishop height function of βs. It is denoted that h˜xs=H˜s,x. From here, parameter s will not be written.We have the following proposition:Proposition 1.
Letβ: I⟶E13 be the unit speed spacelike curve with a timelike principal normal, with κ12−κ22≠0. Then,A.(1)
hx′s=0 iff x=a1N1+a2N2, and −a12+a22=1.(2)
hx′s=hx″s=0 iff x=±es(3)
hx′s=hx″s=hx″′s=0 iff x=±es, and ρs=0(4)
hx′s=hx″s=hx″′s=hx4s=0 iff x=±es, and ρs=ρ′s=0(5)
hx′s=hx″s=hx″′s=hx4s=hx5s=0 iff x=±es, and ρs=ρ′s=ρ″s=0B.(1)
h˜xs=0 iff there is β,x=w(2)
h˜xs=h˜x′s=0 if there are a1, a2∈ℝ, that is, x=coshuN1+sinhuN2, and β,x=w(3)
h˜xs=h˜x′s=h˜x″s=h˜x″s=0 iff x=±es, β,x=w, and ρs=0(4)
h˜xs=h˜x′s=h˜x″s=h˜x″s=h˜x″′s=0 iff x=±es, β,x=w, and ρs=ρ′s=0(5)
h˜xs=h˜x′s=h˜x″s=h˜x″s=h˜x″′s=h˜x4s=0 iff x=±es, β,x=w and ρs=ρ′s=ρ′′s=0(6)
h˜xs=h˜x′s=h˜x′′s=h˜x′′s=h˜x″′s=h˜x4s=h˜x4s=0 iff x=±es, β,x=w, and ρs=ρ′s=ρ″s=ρ″′s=0Proof.
Using equation (9) results in that T1′2≠0 iff κ12−κ22≠0.A.(1)
Becausehx′s=<T1,x> and T1, N1, N2 is RMF through βs, then there are a1, a2∈ℝ such that x=a1N1+a2N2. Because of x∈S12, we have −a12+a22=1. The opposite holds as well.(2)
Becausehx″s=<T1′,x>=<κ1N1+κ2N2,x>=0, we have that −a1κ1+a2κ2=0. It follows from the fact −a12+a22=1 that a1=∓κ2/κ12−κ22 and a2=±κ1/κ12−κ22. Therefore, we obtain(18)x=∓κ1κ12−κ22κ2κ1N1+N2s=±es.Therefore,hx′s=hx″s=0 iff x=±es.(3)
Becausehx″′s=<T1″,x>=<κ12−κ22T1+κ1′N1+κ2′N2,x>=0, using conditions of (2), we obtain(19)±κ1κ12−κ22κ2κ1′−κ1κ2′κ1s=±κ1κ12−κ22ρκ1s=0.Thus,hx′s=hx″s=hx″′s=0 iff x=±es, and ρs.(4)
Since(20)hx4s=<T1‴,x>=<3κ1κ1′−κ2κ2′T1+κ1″+κ1κ12−κ22N1+κ2″+κ2κ12−κ22N2,x>=0by the conditions of (3), we have that(21)±κ1κ12−κ22κ2κ1′−κ1κ2′′κ1s=±1κ12−κ22ρ′s=0.Thus,hx′s=hx″s=hx″′s=hx4s=0 iff x=±es, and ρs=ρ′s=0.(5)
Sincehx5s=<T14,x>0, we have(22)<κ12−κ222+4κ2κ2″−κ1κ1″+3κ1′2−κ2′2T1+κ1″′+5κ1κ1′κ1−κ2′κ2+κ2′κ12−κ1′κ22N1+κ2″′+5κ2κ2′κ2−κ1′κ1+κ1′κ22−κ2′κ12N2,x>=0,.By using the conditions of (4), we have(23)±1κ12−κ22κ1κ2‴−κ2κ1‴+κ2κ1′−κ1κ2′κ12−κ22κ1=0.Therefore,hx′s=hx′′s=hx″′s=hx4s=hx5=0 iff x=±es, as well ρs=ρ′s=ρ″s=0.BUsing similar computation as in proof of A, we obtain B (1).Proposition 2.
Supposeβ: I⟶E13 is the unit speed spacelike curve with the timelike principal normal, and κ12−κ22>0. Then, we have ρs=0 iff(24)es=−κ1κ12−κ22κ2κ1N1+N2,is a constant vector.Proof.
Letκ12−κ22≠0. Using simple calculations, we have that(25)e′s=ρsκ12−κ223κ1N1+κ2N2.
Thus,e′s=0 iff ρs=κ2κ1′−κ1κ2′=0.Proposition 3.
Supposeβ: I⟶E13 is the unit speed spacelike curve with the timelike principal normal, and κ12−κ22>0. Then, it is stated that(a)
β is the B-slant helix iff κ2/κ1 is a constant(b)
N2 is the part of circle at S12, and its center is the spacelike constant vector e0.Proof.
(a) Lets haveρs=κ2κ1′−κ1κ2′=0. Hence, we can write(26)κ2κ1′=κ1κ2′−κ2κ1′κ12=−ρsκ12=0.This means thatκ2/κ1 = constant, such that, β is the B-slant helix.(b)
Suppose thatκ12−κ22>0. Since(27)e,N2=−κ1κ12−κ22κ2κ1N1+N2,N2=−11−κ22/κ12=constant.
In other words,N2 is the part of circle at S12, and its center is the constant spacelike vector e0s.


### 3.2. Unfolding of Functions by One-Variable

We use some general results at the singularity theory for families of function germs [1–3]. Suppose F:ℝ×ℝr,s0,x0⟶ℝ is the smooth function, and fs=Fx0s,x0. Therefore, F named the r-parameter unfolding of fs. It is said that fs has Ak singularity at s0 in case fps0=0 for every 1≤p≤k, and fk+1s0≠0. Additionally, f has A⩾k singularity (k⩾1) at s0. Suppose the k−1 jet of the partial derivative ∂F/∂xi at s0 is jk−1∂F/∂xis,x0s0=∑j=0k−1Ljis−s0j (without the constant term), for i=1,…,r. Therefore, Fs named the p versal unfolding in case of the k×r matrix of coefficients Lji from the rank kk≤r. So, we write an important set about the unfolding relative to the previous notations. The discriminant set of F is(28)DF=x∈ℝr|there existsswithFs,x=∂F∂ss,x=0ats,x.The bifurcation set ofF is(29)BF=x∈ℝr|there existsswith∂F∂ss,x=∂2F∂s2s,x=0ats,x.Similar to [1–3], we state the following theorem:Theorem 2.
SupposeF:ℝ×ℝr,s0,x0⟶ℝ is the r-parameter unfolding of fs, with Ak singularity at s0.
ConsideringF is the p versal unfolding,(a)
In casek=1, DF is locally diffeomorphic to 0×ℝr−1, and BF=∅;(b)
In casek=2, DF is locally diffeomorphic to C×ℝr−2, and BF is locally diffeomorphic to 0×ℝr−1;(c)
In casek=3, DF is locally diffeomorphic to SW×ℝr−3, and BF is locally diffeomorphic to C×ℝr−2.
Hence, we have the following proposition:Proposition 4.
Supposeβ: I⟶E13 is the unit speed spacelike curve with the timelike principal normal, and κ12−κ22≠0.(1)
In casehxs=Hs,x has the Ak singularity (k=2,3 at s0∈ℝ, and H is the p versal unfolding of hx0s0(2)
In caseh˜xs=H˜s,x,w has the Ak-singularity (k=2, 3) at s0∈ℝ, and H˜ is the p–versal unfolding of h˜x0s0Proof.
(1)
Sincex=x0,x1,x2∈S12, −x02+x12+x22=1,x0,x1, and x2 cannot be all zero. Without the loss of generality, suppose x2≠0. Then, by x2=1+x02−x12, we have(30)Hs,x=−x0β0s+x1β1s+1+x02−x12β2s.Thus, we have that(31)∂H∂x0=−β0s+x0β2s1+x02−x12,∂H∂x1=β1s−x1β2s1+x12−x22,∂2H∂s∂x0=−β0′s+x0β2′s1+x02−x12,∂2H∂s∂x1=β1′s−x1β2′s1+x12−x22.Therefore, the 2 jets of∂H/∂xi at s0 (i = 0, 1) are as follows: let x0=x00,x10,x20∈S12, and assume x20≠0, then(32)j1∂H∂x0s,x0=−β0′s+x00β2′sx20s−s0,j1∂H∂x1s,x0=β1′s−x10β2′sx20s−s0,(33)j2∂H∂x0s,x0=−β0′s+x00β2sx20s−s0+12−β0′′+x00β2′′sx20s−s02,j2∂H∂x1s,x0=β1′s−x10β2′sx20s−s012β1′′s−x10β2′′sx20s−s02.(i)
In casehx0s0 has the A2 singularity at s0, then hx0′s0=0. So, the 2−1×2 matrix of coefficients Lji is(34)A=−β0′s+x00β2′sx20β1′s−x10β2′sx20.If the matrixA has rank equal zero, then(35)β0′s=x00β2′sx20,β1′s=x10β2′sx20.Sinceβ′s0=T1s0=1, we have β2′s0≠0, so that we have the contradiction as follows:(36)0=β0′s0,β1′s0,β2′s0,x00,x10,x20=−β0′s0x00+β1′s0x10+β2′s0x20=−x002β2′s0x20+x102β2′s0x20+β2′s0x20=β2′s0x20−x002+x102+x202=β2′s0x20≠0.Then, rankA=1, as well H is the p versal unfolding of hx0 at s0.(ii)
In casehx0s0 has the A3 singularity at s0∈ℝ, then hx0′s0=hx0″s0=0, and using Proposition 1,(37)es0=κ1κ12−κ22κ2κ1N1+N2,whereκ12−κ22>0, ρ′s0=0, and ρ′′s0≠0. So the 3−1×2 matrix of the coefficients Lji is(38)B=L11L12L21L22=−β0′s+x00β2sx20β1′s−x10β2′sx20−β0′′+x00β2″sx20β1′′s−x10β2″sx20.It is required that the2×2 matrix B is a nonsingular matrix. Clearly, this matrix determinate at s0 is(39)detB=1x20−β0′β1′β2′−β0″β1″β2″x00x10x20=1x20β′×β″,e0=∓κ1x20κ12−κ22β′×β″,κ2κ1N1+N2.Sinceβ′=T1, we have β′′=κ1N1+κ2N2. Substituting these relations to the previous equality, we obtain(40)detB=∓κ12−κ22x20≠0,which resulted in rankB=2.(2)
Using similar notations as in (1), we get(41)H˜s,x,x2=−x0β0s+x1β1s+1+x02−x12β2s−x2.
We require the2×3 matrix,(42)G=−β0′s+x00β2sx20β1′s−x10β2′sx20−1−β0′′+x00β2′′sx20β1′′s−x10β2′′sx200,to have the maximal rank. Using case (1) in equation (38), the second raw of G will not be zero; then, rank G=2.Proof of  Theorem 1.
(1) Using Proposition1, the bifurcation set of Hs,x is(43)BH=κ1κ12−κ22κ2κ1N1+N2s∈ℝs∈ℝ.
The assertion (1) of Theorem1 follows from Propositions 1 and 4 and Theorem 2. The discriminant set of H˜s,x is written as(44)DH˜=x0=β+coshuN1+sinhuN2|s∈ℝ.
The assertion (1) of Theorem1 follows from Propositions 1 and 4 and Theorem 2.Example 1.
Consider this spacelike helix:(45)αs=32sinhs,s2,32coshs,−1≤s≤1.
It is easy to show that(46)ζ1s=32sinhs,12,32coshs,ζ2s=sinhs,0,coshs,ζ3s=12coshs,−32,12sinhs,κs=32,andτs=12..
Takingθ0=0, we obtain θs=−1/2s. By the use of equation (9), we have(47)κ1s=coshs2,andκ2s=−sinhs2.
Hence, the geometric invariant is(48)ρs=12.
Therefore, the transformation matrix is written as(49)T1N1N2=1000coshs2−sinhs20−sinhs2coshs2ζ1ζ2ζ3.
From this, we have(50)N1=N11N12N13=coshs2sinhs−12sinhs2coshs32sinhs2coshs2coshs−12sinhs2sinhs,N2=N21N22N23=12coshs2coshs−sinhs2sinhs−32coshs212coshs2sinhs−sinhs2coshs.
Hence, the Bishop spherical Darboux image (Figure4)(51)es=sinhs2N11N12N13−coshs2N21N22N23.
The spacelike sweeping surface (Figure5) is(52)M:Rs,u=32sinhs,s2,32coshs+coshuN11N12N13+sinhuN21N22N23.Figure 4
Bishop spherical Darboux image has a cusp ass = 0.Figure 5
Spacelike sweeping surface with spacelike helix singularity curve.


### 3.3. Singularities of Developable Surfaces

It is known that the developable surface is a special case of ruled surface. This kind of surfaces is used extensively in manufacture of automobile body parts, airplane wings, as well some ship parts. As a result, we analyze the case that the profile curvex degenerates to the line. Therefore, we have the next spacelike developable surface:(53)M:Qs,u=βs+uN2s,u∈ℝ.Moreover, we have the following timelike developable surface:(54)M⊥:Q⊥s,u=βs+uN1s,u∈ℝ.Clearly,Qs,0=αs (respectively, Q⊥s,0=αs), 0≤s≤L, such that, the surface M (respectively, M⊥) interpolates the curve αs. We also have(55)M:Qs×Qu=1−uκ2N1s,as well(56)M⊥:Qs⊥×Qu⊥=1+uκ1N1s.Therefore, we obtainM (respectively, M⊥) that is nonsingular at s0,u0 if and only if 1−u0κ2s0≠0 (respectively 1+u0κ1s0≠0). Hence, we can classify the singularities of developable surface M by using κ2.Theorem 3.
SupposeM is the spacelike developable given as in equation (53). Therefore, we have(1)
M is locally diffeomorphic to cuspidal edge at s0,u0 iff κ2s0=0 and κ2′s0≠0(2)
M is locally diffeomorphic to swallowtail at s0,u0 iff κ2s0≠0 and κ2′s0/κ22s0≠0Proof.
If there is a parameters0, that is, κ2s0=0, and u0′=κ2′s0/κ22s0≠0 (κ2′s0≠0), M is locally diffeomorphic to the cuspidal edge at s0,u0. As a result, assertion (1) holds. In addition, in case there is a parameter s0, that is, u0=1/κ2s0≠0, u0′=κ2′s0/κ22s0=0, and 1/κ2s0″≠0, and M is locally diffeomorphic to swallowtail at s0,u0, assertion (2) holds.Example 2.
By making using of Example1, we have(1)
Ifs0=0, then κ2s0=0 and κ2′s0≠0. The timelike developable surface(57)M:Qs,u=32sinhs,s2,32coshs+u12coshs2coshs−sinhs2sinhs−32coshs212coshs2sinhs−sinhs2coshs,whereu∈ℝ is locally diffeomorphic to the cuspidal edge (Figure 6).(2)
In cases0=0, κ1s0≠0 and κ2′s0=0. The spacelike developable surface is(58)M⊥:Q⊥s,u=32sinhs,s2,32coshs+ucoshs2sinhs−12sinhs2coshs32sinhs2coshs2coshs−12sinhs2sinhs,where u∈ℝ is locally diffeomorphic to swallowtail (Figure 7).Figure 6
The timelike developable surface.Figure 7
The spacelike developable surface.


## 3.1. Lorentzian Bishop Height Functions

We will introduce two different families of Lorentzian Bishop height functions that will be useful to study singularities ofM as follows [1–3]: H:I×S12⟶ℝ, by Hs,x=βs,x. It is called Lorentzian Bishop height function. The notation hxs=Hs,x will be used for all fixed x∈S12. In addition, it is defined H˜:I×S12×ℝ⟶ℝ, using H˜s,x,w=β,x−w. It is called extended Lorentzian Bishop height function of βs. It is denoted that h˜xs=H˜s,x. From here, parameter s will not be written.We have the following proposition:Proposition 1.
Letβ: I⟶E13 be the unit speed spacelike curve with a timelike principal normal, with κ12−κ22≠0. Then,A.(1)
hx′s=0 iff x=a1N1+a2N2, and −a12+a22=1.(2)
hx′s=hx″s=0 iff x=±es(3)
hx′s=hx″s=hx″′s=0 iff x=±es, and ρs=0(4)
hx′s=hx″s=hx″′s=hx4s=0 iff x=±es, and ρs=ρ′s=0(5)
hx′s=hx″s=hx″′s=hx4s=hx5s=0 iff x=±es, and ρs=ρ′s=ρ″s=0B.(1)
h˜xs=0 iff there is β,x=w(2)
h˜xs=h˜x′s=0 if there are a1, a2∈ℝ, that is, x=coshuN1+sinhuN2, and β,x=w(3)
h˜xs=h˜x′s=h˜x″s=h˜x″s=0 iff x=±es, β,x=w, and ρs=0(4)
h˜xs=h˜x′s=h˜x″s=h˜x″s=h˜x″′s=0 iff x=±es, β,x=w, and ρs=ρ′s=0(5)
h˜xs=h˜x′s=h˜x″s=h˜x″s=h˜x″′s=h˜x4s=0 iff x=±es, β,x=w and ρs=ρ′s=ρ′′s=0(6)
h˜xs=h˜x′s=h˜x′′s=h˜x′′s=h˜x″′s=h˜x4s=h˜x4s=0 iff x=±es, β,x=w, and ρs=ρ′s=ρ″s=ρ″′s=0Proof.
Using equation (9) results in that T1′2≠0 iff κ12−κ22≠0.A.(1)
Becausehx′s=<T1,x> and T1, N1, N2 is RMF through βs, then there are a1, a2∈ℝ such that x=a1N1+a2N2. Because of x∈S12, we have −a12+a22=1. The opposite holds as well.(2)
Becausehx″s=<T1′,x>=<κ1N1+κ2N2,x>=0, we have that −a1κ1+a2κ2=0. It follows from the fact −a12+a22=1 that a1=∓κ2/κ12−κ22 and a2=±κ1/κ12−κ22. Therefore, we obtain(18)x=∓κ1κ12−κ22κ2κ1N1+N2s=±es.Therefore,hx′s=hx″s=0 iff x=±es.(3)
Becausehx″′s=<T1″,x>=<κ12−κ22T1+κ1′N1+κ2′N2,x>=0, using conditions of (2), we obtain(19)±κ1κ12−κ22κ2κ1′−κ1κ2′κ1s=±κ1κ12−κ22ρκ1s=0.Thus,hx′s=hx″s=hx″′s=0 iff x=±es, and ρs.(4)
Since(20)hx4s=<T1‴,x>=<3κ1κ1′−κ2κ2′T1+κ1″+κ1κ12−κ22N1+κ2″+κ2κ12−κ22N2,x>=0by the conditions of (3), we have that(21)±κ1κ12−κ22κ2κ1′−κ1κ2′′κ1s=±1κ12−κ22ρ′s=0.Thus,hx′s=hx″s=hx″′s=hx4s=0 iff x=±es, and ρs=ρ′s=0.(5)
Sincehx5s=<T14,x>0, we have(22)<κ12−κ222+4κ2κ2″−κ1κ1″+3κ1′2−κ2′2T1+κ1″′+5κ1κ1′κ1−κ2′κ2+κ2′κ12−κ1′κ22N1+κ2″′+5κ2κ2′κ2−κ1′κ1+κ1′κ22−κ2′κ12N2,x>=0,.By using the conditions of (4), we have(23)±1κ12−κ22κ1κ2‴−κ2κ1‴+κ2κ1′−κ1κ2′κ12−κ22κ1=0.Therefore,hx′s=hx′′s=hx″′s=hx4s=hx5=0 iff x=±es, as well ρs=ρ′s=ρ″s=0.BUsing similar computation as in proof of A, we obtain B (1).Proposition 2.
Supposeβ: I⟶E13 is the unit speed spacelike curve with the timelike principal normal, and κ12−κ22>0. Then, we have ρs=0 iff(24)es=−κ1κ12−κ22κ2κ1N1+N2,is a constant vector.Proof.
Letκ12−κ22≠0. Using simple calculations, we have that(25)e′s=ρsκ12−κ223κ1N1+κ2N2.
Thus,e′s=0 iff ρs=κ2κ1′−κ1κ2′=0.Proposition 3.
Supposeβ: I⟶E13 is the unit speed spacelike curve with the timelike principal normal, and κ12−κ22>0. Then, it is stated that(a)
β is the B-slant helix iff κ2/κ1 is a constant(b)
N2 is the part of circle at S12, and its center is the spacelike constant vector e0.Proof.
(a) Lets haveρs=κ2κ1′−κ1κ2′=0. Hence, we can write(26)κ2κ1′=κ1κ2′−κ2κ1′κ12=−ρsκ12=0.This means thatκ2/κ1 = constant, such that, β is the B-slant helix.(b)
Suppose thatκ12−κ22>0. Since(27)e,N2=−κ1κ12−κ22κ2κ1N1+N2,N2=−11−κ22/κ12=constant.
In other words,N2 is the part of circle at S12, and its center is the constant spacelike vector e0s.


## 3.2. Unfolding of Functions by One-Variable

We use some general results at the singularity theory for families of function germs [1–3]. Suppose F:ℝ×ℝr,s0,x0⟶ℝ is the smooth function, and fs=Fx0s,x0. Therefore, F named the r-parameter unfolding of fs. It is said that fs has Ak singularity at s0 in case fps0=0 for every 1≤p≤k, and fk+1s0≠0. Additionally, f has A⩾k singularity (k⩾1) at s0. Suppose the k−1 jet of the partial derivative ∂F/∂xi at s0 is jk−1∂F/∂xis,x0s0=∑j=0k−1Ljis−s0j (without the constant term), for i=1,…,r. Therefore, Fs named the p versal unfolding in case of the k×r matrix of coefficients Lji from the rank kk≤r. So, we write an important set about the unfolding relative to the previous notations. The discriminant set of F is(28)DF=x∈ℝr|there existsswithFs,x=∂F∂ss,x=0ats,x.The bifurcation set ofF is(29)BF=x∈ℝr|there existsswith∂F∂ss,x=∂2F∂s2s,x=0ats,x.Similar to [1–3], we state the following theorem:Theorem 2.
SupposeF:ℝ×ℝr,s0,x0⟶ℝ is the r-parameter unfolding of fs, with Ak singularity at s0.
ConsideringF is the p versal unfolding,(a)
In casek=1, DF is locally diffeomorphic to 0×ℝr−1, and BF=∅;(b)
In casek=2, DF is locally diffeomorphic to C×ℝr−2, and BF is locally diffeomorphic to 0×ℝr−1;(c)
In casek=3, DF is locally diffeomorphic to SW×ℝr−3, and BF is locally diffeomorphic to C×ℝr−2.
Hence, we have the following proposition:Proposition 4.
Supposeβ: I⟶E13 is the unit speed spacelike curve with the timelike principal normal, and κ12−κ22≠0.(1)
In casehxs=Hs,x has the Ak singularity (k=2,3 at s0∈ℝ, and H is the p versal unfolding of hx0s0(2)
In caseh˜xs=H˜s,x,w has the Ak-singularity (k=2, 3) at s0∈ℝ, and H˜ is the p–versal unfolding of h˜x0s0Proof.
(1)
Sincex=x0,x1,x2∈S12, −x02+x12+x22=1,x0,x1, and x2 cannot be all zero. Without the loss of generality, suppose x2≠0. Then, by x2=1+x02−x12, we have(30)Hs,x=−x0β0s+x1β1s+1+x02−x12β2s.Thus, we have that(31)∂H∂x0=−β0s+x0β2s1+x02−x12,∂H∂x1=β1s−x1β2s1+x12−x22,∂2H∂s∂x0=−β0′s+x0β2′s1+x02−x12,∂2H∂s∂x1=β1′s−x1β2′s1+x12−x22.Therefore, the 2 jets of∂H/∂xi at s0 (i = 0, 1) are as follows: let x0=x00,x10,x20∈S12, and assume x20≠0, then(32)j1∂H∂x0s,x0=−β0′s+x00β2′sx20s−s0,j1∂H∂x1s,x0=β1′s−x10β2′sx20s−s0,(33)j2∂H∂x0s,x0=−β0′s+x00β2sx20s−s0+12−β0′′+x00β2′′sx20s−s02,j2∂H∂x1s,x0=β1′s−x10β2′sx20s−s012β1′′s−x10β2′′sx20s−s02.(i)
In casehx0s0 has the A2 singularity at s0, then hx0′s0=0. So, the 2−1×2 matrix of coefficients Lji is(34)A=−β0′s+x00β2′sx20β1′s−x10β2′sx20.If the matrixA has rank equal zero, then(35)β0′s=x00β2′sx20,β1′s=x10β2′sx20.Sinceβ′s0=T1s0=1, we have β2′s0≠0, so that we have the contradiction as follows:(36)0=β0′s0,β1′s0,β2′s0,x00,x10,x20=−β0′s0x00+β1′s0x10+β2′s0x20=−x002β2′s0x20+x102β2′s0x20+β2′s0x20=β2′s0x20−x002+x102+x202=β2′s0x20≠0.Then, rankA=1, as well H is the p versal unfolding of hx0 at s0.(ii)
In casehx0s0 has the A3 singularity at s0∈ℝ, then hx0′s0=hx0″s0=0, and using Proposition 1,(37)es0=κ1κ12−κ22κ2κ1N1+N2,whereκ12−κ22>0, ρ′s0=0, and ρ′′s0≠0. So the 3−1×2 matrix of the coefficients Lji is(38)B=L11L12L21L22=−β0′s+x00β2sx20β1′s−x10β2′sx20−β0′′+x00β2″sx20β1′′s−x10β2″sx20.It is required that the2×2 matrix B is a nonsingular matrix. Clearly, this matrix determinate at s0 is(39)detB=1x20−β0′β1′β2′−β0″β1″β2″x00x10x20=1x20β′×β″,e0=∓κ1x20κ12−κ22β′×β″,κ2κ1N1+N2.Sinceβ′=T1, we have β′′=κ1N1+κ2N2. Substituting these relations to the previous equality, we obtain(40)detB=∓κ12−κ22x20≠0,which resulted in rankB=2.(2)
Using similar notations as in (1), we get(41)H˜s,x,x2=−x0β0s+x1β1s+1+x02−x12β2s−x2.
We require the2×3 matrix,(42)G=−β0′s+x00β2sx20β1′s−x10β2′sx20−1−β0′′+x00β2′′sx20β1′′s−x10β2′′sx200,to have the maximal rank. Using case (1) in equation (38), the second raw of G will not be zero; then, rank G=2.Proof of  Theorem 1.
(1) Using Proposition1, the bifurcation set of Hs,x is(43)BH=κ1κ12−κ22κ2κ1N1+N2s∈ℝs∈ℝ.
The assertion (1) of Theorem1 follows from Propositions 1 and 4 and Theorem 2. The discriminant set of H˜s,x is written as(44)DH˜=x0=β+coshuN1+sinhuN2|s∈ℝ.
The assertion (1) of Theorem1 follows from Propositions 1 and 4 and Theorem 2.Example 1.
Consider this spacelike helix:(45)αs=32sinhs,s2,32coshs,−1≤s≤1.
It is easy to show that(46)ζ1s=32sinhs,12,32coshs,ζ2s=sinhs,0,coshs,ζ3s=12coshs,−32,12sinhs,κs=32,andτs=12..
Takingθ0=0, we obtain θs=−1/2s. By the use of equation (9), we have(47)κ1s=coshs2,andκ2s=−sinhs2.
Hence, the geometric invariant is(48)ρs=12.
Therefore, the transformation matrix is written as(49)T1N1N2=1000coshs2−sinhs20−sinhs2coshs2ζ1ζ2ζ3.
From this, we have(50)N1=N11N12N13=coshs2sinhs−12sinhs2coshs32sinhs2coshs2coshs−12sinhs2sinhs,N2=N21N22N23=12coshs2coshs−sinhs2sinhs−32coshs212coshs2sinhs−sinhs2coshs.
Hence, the Bishop spherical Darboux image (Figure4)(51)es=sinhs2N11N12N13−coshs2N21N22N23.
The spacelike sweeping surface (Figure5) is(52)M:Rs,u=32sinhs,s2,32coshs+coshuN11N12N13+sinhuN21N22N23.Figure 4
Bishop spherical Darboux image has a cusp ass = 0.Figure 5
Spacelike sweeping surface with spacelike helix singularity curve.


## 3.3. Singularities of Developable Surfaces

It is known that the developable surface is a special case of ruled surface. This kind of surfaces is used extensively in manufacture of automobile body parts, airplane wings, as well some ship parts. As a result, we analyze the case that the profile curvex degenerates to the line. Therefore, we have the next spacelike developable surface:(53)M:Qs,u=βs+uN2s,u∈ℝ.Moreover, we have the following timelike developable surface:(54)M⊥:Q⊥s,u=βs+uN1s,u∈ℝ.Clearly,Qs,0=αs (respectively, Q⊥s,0=αs), 0≤s≤L, such that, the surface M (respectively, M⊥) interpolates the curve αs. We also have(55)M:Qs×Qu=1−uκ2N1s,as well(56)M⊥:Qs⊥×Qu⊥=1+uκ1N1s.Therefore, we obtainM (respectively, M⊥) that is nonsingular at s0,u0 if and only if 1−u0κ2s0≠0 (respectively 1+u0κ1s0≠0). Hence, we can classify the singularities of developable surface M by using κ2.Theorem 3.
SupposeM is the spacelike developable given as in equation (53). Therefore, we have(1)
M is locally diffeomorphic to cuspidal edge at s0,u0 iff κ2s0=0 and κ2′s0≠0(2)
M is locally diffeomorphic to swallowtail at s0,u0 iff κ2s0≠0 and κ2′s0/κ22s0≠0Proof.
If there is a parameters0, that is, κ2s0=0, and u0′=κ2′s0/κ22s0≠0 (κ2′s0≠0), M is locally diffeomorphic to the cuspidal edge at s0,u0. As a result, assertion (1) holds. In addition, in case there is a parameter s0, that is, u0=1/κ2s0≠0, u0′=κ2′s0/κ22s0=0, and 1/κ2s0″≠0, and M is locally diffeomorphic to swallowtail at s0,u0, assertion (2) holds.Example 2.
By making using of Example1, we have(1)
Ifs0=0, then κ2s0=0 and κ2′s0≠0. The timelike developable surface(57)M:Qs,u=32sinhs,s2,32coshs+u12coshs2coshs−sinhs2sinhs−32coshs212coshs2sinhs−sinhs2coshs,whereu∈ℝ is locally diffeomorphic to the cuspidal edge (Figure 6).(2)
In cases0=0, κ1s0≠0 and κ2′s0=0. The spacelike developable surface is(58)M⊥:Q⊥s,u=32sinhs,s2,32coshs+ucoshs2sinhs−12sinhs2coshs32sinhs2coshs2coshs−12sinhs2sinhs,where u∈ℝ is locally diffeomorphic to swallowtail (Figure 7).Figure 6
The timelike developable surface.Figure 7
The spacelike developable surface.


## 4. Conclusion

In this study, we introduced the notion of spacelike sweeping surfaces with rotation minimizing frames at Minkowski 3-spaceE13. Using the singularity, we classified the generic properties and present a new geometric invariant related to the singularities of this spacelike sweeping surface. It leads to 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, in the same order. Subsequently, we derived the sufficient and necessary conditions for the surface to be developable ruled surfaces. At the end, some examples are introduced in order to clarify the theoretical results.



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