On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-Space E 31

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 in E1.


Introduction
Kinematically, a sweeping surface is a surface traced by a oneparameter 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][2][3][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][6][7][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 relation-ship 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][12][13][14][15][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][19][20][21][22][23].
In this paper, the classification of singularity type of timelike sweeping surfaces is studied with the RMF in E 3 1 . 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.
A ruled surface in E 3 1 is locally the map D ðγ,xÞ : where αðsÞ is called the directrix curve and xðsÞ the director curve. The straight lines t ⟶ αðsÞ + txðsÞ are called rulings. It is well known that D ðγ,xÞ is a developable surface iff detðα ′ ðsÞ, xðsÞ, x ′ ðsÞÞ = 0.

Definition 1.
A surface in the Minkowski 3-space E 3 1 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.

Timelike Sweeping Surfaces and Singularities
In this section, the classification of singularity type of timelike sweeping surfaces is studied with the RMF in E 3 1 . Let γ = γðsÞ 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 γðsÞ as follows: where rðtÞ = ð0, r 1 ðtÞ, r 2 ðtÞÞ T is called planar profile (cross section); "T" represents transposition, with another parameter t ∈ I ⊆ ℝ. The semiorthogonal matrix ðsÞ = fξðsÞ, ξ 1 ðsÞ, ξ 2 ðsÞg specifies the RMF along γðsÞ. We will utilize "dot" to indicate the derivative with respect to the arc length parameter of the profile curve rðtÞ.
The tangent vectors and the unit normal vector to the surface, respectively, are From Equation (3.3), it follows that nðs, tÞ is contained in the normal plane of the spine curve γðsÞ, since it is orthogonal to ξ. Thus, the normal of the profile curve rðtÞ and the surface normal are identical. Through this work, we will assume that the profile curve rðtÞ is a unit speed timelike curve, that is, r :2 1 − r :2 2 = −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 (1) M is locally diffeomorphic to cuspidal edge CE at ðs 0 , t 0 Þ iff x = ±gðs 0 Þ and σðs 0 Þ ≠ 0 (2) M is locally diffeomorphic to swallowtail SW at ðs 0 , t 0 Þ iff x = ±gðs 0 Þ, σðs 0 Þ ≠ 0, and σ ′ ðs 0 Þ = 0 The graphs of C × ℝ, CE, and SW are seen in Figures 1-3. 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 of M as follows: H : I × S 2 1 ⟶ ℝ, by Hðs, xÞ = <γðsÞ, x > . We call it the Lorentzian Bishop height function. We use the notation h x ðsÞ = Hðs, xÞ for any fixed x ∈ S 2 1 . We also defineH : We call it the extended Lorentzian Bishop height function of γðsÞ. We denote that h x ðsÞ =Hðs, xÞ. From now on, we shall often not write the parameter s. Then, we have the following proposition.
(a) γ is a B-slant helix iff μ 2 /μ 1 is constant (b) ξ 2 is a part of circle on ℍ 2 + whose center is the timelike constant vector g 0 Proof.
This means that ξ 2 is a part of circle on ℍ 2 + whose center is the constant timelike vector g 0 ðsÞ+.☐

Unfolding of Functions by One Variable.
In this subsection, we use some general results on the singularity theory for families of function germs. Let F : ðℝ × ℝ r , ðs 0 , x 0 ÞÞ ⟶ ℝ be a smooth function and fðsÞ = F x 0 ðs, x 0 Þ. Then, F is called an r-parameter unfolding of fðsÞ. We say that fðsÞ has A k -singularity at s 0 if f ðpÞ ðs 0 Þ = 0 for all 1 ≤ p ≤ k and f ðk+1Þ ðs 0 Þ ≠ 0. We also say that f has A k -singularity (k1) at s 0 . Let the ðk − 1Þ-jet of the partial derivative ∂F/∂x i at s 0 be j ðk−1Þ ðð∂F/∂x i Þðs, x 0 ÞÞðs 0 Þ = Σ k−1 j=0 L ji ðs − s 0 Þ j (without the constant term), for i = 1, ⋯, r. Then, FðsÞ is called a p-versal unfolding if the k × r matrix of coefficients ðL ji Þ has rank ðk ≤ rÞ. So, we write important set about the unfolding relative to the above notations. The discriminant set of F is the set The bifurcation set of F is the set We can also give the following theorem [12,13].
Theorem 6. Let F : ðℝ × ℝ r , ðs 0 , x 0 ÞÞ ⟶ ℝ be an r -parameter unfolding of fðsÞ, which has the A k singularity at s 0 . Suppose that F is a p-versal unfolding.
(a) If k = 1, then D F is locally diffeomorphic to f0g × ℝ r−1 and B F = ∅ (b) If k = 2, then D F is locally diffeomorphic to C × ℝ r−2 , and B F is locally diffeomorphic to f0g × ℝ r−1 (c) If k = 3, then D F is locally diffeomorphic to SW × ℝ r−3 , and B F is locally diffeomorphic to C × ℝ r−2 Hence, we have the following fundamental proposition.
So the ð2 − 1Þ × 2 matrix of coefficients ðL ji Þ is Suppose that the rank of the matrix A is zero, then we have Since kγ ′ ðs 0 Þk = kξðs 0 Þk = 1, we have γ 2 ′ ðs 0 Þ ≠ 0 so that we have the contradiction as follows: Therefore, rank ðAÞ = 1, and H is the (p) versal unfolding of h x 0 at s 0 .
(2) Under the same notations as in (4), we havẽ We require the 2 × 3 matrix Abstract and Applied Analysis to have the maximal rank. By case (4) in Equation (3.14), the second row of G does not vanish, so rank ðGÞ = 2+.☐ Proof of Theorem 1 (see (4)). By Proposition 3, the bifurcation set of Hðs, xÞ is The timelike Bishop spherical Darboux image is shown in Figure 4 The timelike sweeping surface family is By choosing r 1 ðtÞ = cosh t and r 2 ðtÞ = sinh t, then we immediately have a timelike sweeping surface (see Figure 5).

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 curve rðtÞ degenerates into a is locally diffeomorphic to the cuspidal edge; see Figure 6.
We can obtain the singular locus of M as follows: is locally diffeomorphic to swallowtail; see Figure 7. Also, the singular locus of M ⊥ is

Conclusion
In this paper, we introduced the notion of timelike sweeping surfaces with rotation minimizing frames in Minkowski 3space E 3 1 . 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.

Data Availability
All of the data are available within the paper.

Conflicts of Interest
The authors declare that they have no conflict of interest.