Spacelike Sweeping Surfaces and Singularities in Minkowski 3-Space

We considered the spacelike sweeping surface with rotation minimizing frames at Minkowski 3-space E1. We presented the new geometric invariant to demonstrate geometric properties and local singularities for this surface. 'en, 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.


Introduction
e 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 om, 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][2][3][4][5][6][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][9][10][11][12][13][14]. It is a fact that a sweeping surface can be a developable surface. e 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. erefore, the developable surface is considered as the ruled surface, such that every point at the same ruling shares the common tangent plane. e 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][16][17][18][19].
e 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 3space [16][17][18][19][20]. e 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][20][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][22][23].
In fact, there is no more literature review regarding singularities of sweeping surfaces relating to the Minkowski Bishop frame. erefore, 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 3space. 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.

Preliminaries
Some definitions and basic concepts are given which will be used (for instance [8,24,25]). Suppose E 3 1 is 3-dimensional Minkowski space, the 3-dimensional real vector space R 3 considers the metric where (r 1 , r 2 , r 3 ) denotes the canonical coordinates in R 3 . Any vector r of E 3 1 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. e timelike or lightlike vector at E 3 1 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. erefore, we say that a smooth map β: I ⟶ E 3 1 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 ∈ E 3 1 , the inner product is a real number 〈r, p〉 � −r 1 p 1 + r 2 p 2 + r 3 p 3 , and the vector product is given as where f 1 , f 2 , f 3 are the canonical bases of E 3 1 . For a fixed point p ∈ E 3 1 and the positive number c > 0, hyperbolic and Lorentzian (de Sitter space) spheres, in the same order, are given as We define and it is called the (open) lightcone at the vertex p. In case p � 0 and c � 1, we define LC * 0 , H 2 + , in addition to S 2 1 , in the same order.
Using Definition 1, it is observed that the Serret-Frenet frame is RMF respecting to the principal normal ξ 2 but not 2 Mathematical Problems in Engineering respecting to the tangent ζ 1 and the binormal ζ 3 . Even though the Serret-Frenet frame is not RMF respecting to ζ 1 , it is easy to derive similar RMF from it. New normal plane vectors (N 1 , N 2 ) are determined along the rotation of (ζ 2 , ζ 3 ) as with a certain Lorentzian timelike angle θ(s) ≥ 0. e set T 1 , N 1 , N 2 } is called RMF or Bishop frame. e RMF vector insure the following relations: erefore, the Bishop frame reads where Here, the Bishop curvatures are determined as κ 1 (s) � κcoshϑ and κ 2 (s) � −κsinhϑ. Comparing equations (5) and (9), it is observed that the relative velocity is One can show that As a consequence, both Serret-Frenet frame and RMF identical iff β(s) is the planar, which means τ � 0. e spacelike vector is defined as and we name it the modified Bishop Darboux vector through β(s). A Bishop spherical Darboux image e(s): erefore, we consider a new geometric invariant
Kinematically, the sweeping surface R(s, u) is generating by the moving of the profile curve x(u) through the spine curve β(s) with the orientation as introduced by F(s). e profile curve x(u) is in the 2D or 3D space that passes through the spine curve β(s) during sweeping. Interestingly, RMF allows for a simple characterization of spine curve.

Definition 3.
e surface at Minkowski 3-space E 3 1 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.

Spacelike Sweeping Surface and Its Singularities
We present the spacelike sweeping surface at Minkowski 3space E 3 1 . Consider the planar profile (cross-section) that is defined as x(u) � (0, cosh u, sinh u). Using equation (14), we obtain Using equation (9) resulted in e unit normal vector of M is e main aim of this study is given in the following theorem: e pictures of C × R, CE, and SW are shown in Figures 1-3. 3.1. Lorentzian Bishop Height Functions. We will introduce two different families of Lorentzian Bishop height functions that will be useful to study singularities of M as follows [1][2][3]: by H(s, x) � 〈β(s), x〉. It is called Lorentzian Bishop height function. e notation h x (s) � H(s, x) will be used for all fixed x ∈ S 2 1 . In addition, it is defined H: H(s, x). From here, parameter s will not be written.
We have the following proposition: Proof. Using equation (9) results in that Because of x ∈ S 2 1 , we have −a 2 1 + a 2 2 � 1. e opposite holds as well.
x > � 0, using conditions of (2), we obtain and ρ(s). by the conditions of (3), we have that

(4) Since
By using the conditions of (4), we have Using similar computation as in proof of A, we obtain B (1).

Proposition 2. Suppose β:
1 is the unit speed spacelike curve with the timelike principal normal, and is a constant vector.
In other words, N 2 is the part of circle at S 2 1 , and its center is the constant spacelike vector e 0 (s).

Unfolding of Functions by One-Variable.
We use some general results at the singularity theory for families of function germs [1][2][3]. Suppose F: (R × R r , (s 0 , x 0 )) ⟶ R is the smooth function, and f(s) � F x 0 (s, x 0 ). erefore, F  Mathematical Problems in Engineering named the r-parameter unfolding of f(s). It is said that f(s) has A k singularity at s 0 in case f (p) (s 0 ) � 0 for every 1 ≤ p ≤ k, and f (k+1) (s 0 ) ≠ 0. Additionally, f has A ⩾k singularity (k⩾1) at s 0 . Suppose the (k − 1) jet of the partial derivative zF/zx i at s 0 is j (k− 1) (zF/zx i (s, x 0 ))(s 0 ) � k−1 j�0 L ji (s − s 0 ) j (without the constant term), for i � 1, . . . , r. erefore, F(s) named the p versal unfolding in case of the k × r matrix of coefficients (L ji ) from the rank k (k ≤ r). So, we write an important set about the unfolding relative to the previous notations. e discriminant set of F is e bifurcation set of F is Similar to [1][2][3], we state the following theorem: Hence, we have the following proposition:  H(s, x, w) has the A k -singularity (k � 2, 3) at s 0 ∈ R, and H is the p-versal unfolding of h x 0 (s 0 ) Proof. 1 , and x 2 cannot be all zero. Without the loss of generality, suppose x 2 ≠ 0. en, by us, we have that erefore, the 2 jets of zH/zx i at s 0 (i � 0, 1) are as follows: let x 0 � (x 00 , x 10 , x 20 ) ∈ S 2 1 , and assume x 20 ≠ 0, then (i) In case h x 0 (s 0 ) has the A 2 singularity at s 0 , then If the matrix A has rank equal zero, then Since ‖β ′ (s 0 )‖ � ‖T 1 (s 0 )‖ � 1, we have β 2 ′ (s 0 ) ≠ 0, so that we have the contradiction as follows: en, rank (A) � 1, as well H is the p versal unfolding of h x 0 at s 0 .
(ii) In case h x 0 (s 0 ) has the A 3 singularity at s 0 ∈ R, then h x 0 ′ (s 0 ) � h x 0 ″ (s 0 ) � 0, and using Proposition 1, It is required that the 2 × 2 matrix B is a nonsingular matrix. Clearly, this matrix determinate at s 0 is x 00 x 10 x 20 Since β ′ � T 1 , we have β ′′ � κ 1 N 1 + κ 2 N 2 . Substituting these relations to the previous equality, we obtain Mathematical Problems in Engineering 7 which resulted in rank (B) � 2. (2) Using similar notations as in (1), we get We require the 2 × 3 matrix, to have the maximal rank. Using case (1) in equation (38), the second raw of G will not be zero; then, rank (G) � 2. □  , x) is It is easy to show that , and τ(s) � 1 2 .