# Time-Like Sweeping Surfaces with a Bishop Frame in the Minkowski 3-SpaceE13

Authors: Sahar H. Nazra; Rashad A. Abdel-Baky
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we consider a time-like sweeping surface and its local singularities with Bishop frame in Minkowski 3-spaceE13. Subsequently, we derive the sufficient and necessary conditions for this surface to be spacelike developable ruled surface. In particular, we mainly focus on the study for the resulting spacelike developable surface is a cylinder, cone or tangent surface. Finally, some representative curves are chosen to construct the corresponding spacelike developable surfaces.


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## Body

## 1. Introduction

Computer vision is the automatic analysis of sequences of images for the objective of recovering 3-dimensional surface form. Nowadays, different branches of mathematics, together with old and modern, have been applied to medical imaging and computer vision. Projective geometry, as an ancient mathematical subject, still used to characterize relationships between both points and lines in various images to the same subject. Differential geometry, which is even older, although it received its final newfangled look in the first half of the nineteenth century, is applied to sketch the form of curves and surfaces in engineering. A secondary revolution in mathematical notion and technique happened notion the inventive genius of the French mathematician Rene Thom. His principals slightly inspired by Whitney gave birth to what is named singularity theory, an expression which includes catastrophes and bifurcations. Nowadays, singularity theory being a direct successor of differential calculus is sure to have a large deal of advantage to say on numerous interesting outcomes in various branches such as medical imaging and computer vision (see e.g., [1–5]).Kinematically, a tubular surface is a surface traced by a one-parameter family of spheres given by the radius function and center curves: If the radius function is a constant function, the tubular surface is the envelope of a moving sphere and is named a sweeping surface. There are various modules that we are familiar with, such as rotation surface, Dupin cycloids, circular cylinder, and circular cone (radius of the spheres are not constant). One of the noticeable facts connected with the sweeping surface is that the sweeping surface can be developable surface, that is, can be developed onto a plane unaccompanied by tearing and stretching (see, e.g., [6–18]). The essential instrument to examine the curve and surface at differential geometry is the Serret–Frenet frame that is the generality used frame at Euclidean 3-space E3. The major tools in the previous literature are the Serret–Frenet formulae and somewhat connected functions on curves such as the height functions, and distance-squared functions. Depending on the Serret–Frenet formulae, the singularity of those functions can be examined from the view of extraneous differential geometry. However, the Serret–Frenet frame is not defined for all points over every curve. A novel frame is necessary for the sort of mathematical analysis that is typically done with computer graphics. Therefore, Bishop [19] defined a novel moving frame over a space curve, which could give the desired manner to ride over a space curve by manner of parallel vector fields. This particular frame is named the Bishop frame or rotation minimizing frame (RMF) of the curves. It has implementations in the branches of computer graphics and biology. For instance, it may be utilized to compute the shape of sequences of DNA using a curve defined by the Bishop frame. The Bishop frame might be also making a way to control virtual cameras in computer animations [1–5]. Lately, various fascinating characterizations of sweeping surfaces, and of developable surfaces are obtained by the Bishop frame [20–27].In this paper, we established the Lorentzian Bishop frame along a unit speed space-like curve with time-like principal normal and examined the local differential geometry of time-like sweeping surfaces in the Minkowski 3-space. Afterward, we give the necessary and sufficient conditions for the time-like sweeping surface to become a space-like developable ruled surface. We also investigated the uniqueness of such developable surface. Particularly, we fundamentally concentrated on the investigation for the resulting spacelike developable surface to be cylinder, cone or tangential surface. Lastly, examples are offered to show the spacelike developable surfaces with a common line of curvature. This study is purposed to attitude the geometric analysis of spacelike developable surfaces through the time-like sweeping surfaces.


## 2. Preliminaries

In this section, we give some definitions and fundamental notations that we will use in this paper [6, 28, 29]. Let ℝ3=p1,p2,p3|pi∈ℝi=1,2,3∖, pi∈ℝ (i = 1, 2, 3)\} be a 3-dimensional Cartesian space. For any q=q1,q2,q3, and p=p1,p2,p3∈ℝ3, the pseudo inner product of q, and p is defined by the following:(1)<q,p>=q1p1+q2p2−q3p3.We callℝ3,<,> the Minkowski 3-space. We write E13 instead of ℝ3,<,>. We say that a non-zero vector q∈E13 is space-like, light-like or time-like 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 their vectorial product by the following:(2)q×p=ij−kq1q2q3p1p2p3=q2p3−q3p2,q3p1−q1p3,−q1p2−q2p1,where i, j, k are the canonical basis of E13.Letγ=γs be a unit speed spacelike curve with time-like principal normal in E13; by κs and τs we denote the natural curvature and torsion, respectively. Then γs is called Serret–Frenet curve if κ>0, and τ≠0. Consider the Serret–Frenet frame υ1s,υ2sυ2s, associated with curve γs, then the Serret–Frenet formulae read as follows:(3)υ1′sυ2′sυ3′s=0κ0κ0τ0τ0υ1sυ2sυ3s,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 following relations:(4)<υ1,υ1>=<υ3,υ3>=1,<υ2,υ2>=−1,υ1×υ2=υ3,υ1×υ3=υ2,υ2×υ3=υ1.The Bishop frame or rotation minimizing frame (RMF) ofγs is defined by the following alternative frame equations:(5)ξ′ξ1′ξ2′=0μ1−μ2μ100μ200ξξ1ξ2,where ωs=μ2ξ1−μ1ξ2 is the Bishop Darboux vector. Here, the Bishop curvatures are defined by μ1s=κcoshφ, and μ2s=κsinhφ. The relation matrix can be expressed as follows:(6)ξξ1ξ2=1000coshφsinhφ0sinhφcoshφυ1υ2υ3,where φs≥0 is a spacelike angle. One can show as follows:(7)μ12−μ22=κ2,φ=tanh−1μ2μ1;μ1≠0,φs=−∫s0sτds+φ0,φ0=φs0.We denote a surfaceM in E13 by the following:(8)M:ys,t=y1s,t,y2s,t,y3s,t,s,t∈D⊆ℝ2.LetN be the standard unit normal vector field on a surface M defined by N=ys×yt/ys×yt, where, yi=∂y/∂i. Then the metric (first fundamental form) I of a surface M is defined by the following:(9)I=g11ds2+2g12dsdt+g22dt2,where g11=≪ys,ys≫, g12=≪ys,yt≫,g22=≪yt,yt>. We define the second fundamental form II of M by the following:(10)II=h11ds2+2h12dsdu+h22du2,where h11=<yss,N>, h12=<yst,N>,h22=<ytt,N>. M is called a spacelike/time-like surface if the induced metric is Riemannian/Lorentzian metric on each tangent plane. This is equivalent to saying that the normal vector N is time-like/spacelike at each point of M [24, 25].A ruled surface inE13 is locally the map Dγ,x: I×ℝ⟶E13 defined by the following:(11)Dγ,xs,t=γs+txs,t∈ℝ,where γs is called the base curve, and xs the director curve. The straight lines t⟶γs+txs are called rulings. It is well known that Dγ,x is a developable surface if det γ′s,xs,x′s=0.Definition 1.
A sweeping surface alongγs is a surface defined by the following:(12)M:ys,t=γs+Asrt=γs+r1tξ1s+x2tξ2s,where γs is called the spine curve (at least C1 -continuous), 0≤s≤T, s is the arc length parameter. rt is planar profile (cross-section) curve given by parametric representation rt=0,r1t,r2tT,, the symbol ‘T’ represents transposition, with another parameter t∈I⊆ℝ. The semi orthogonal matrix As=ξs,ξ1s,ξ2s, specifies the RMF along γs.
Kinematically, the sweeping surfaceM is traced by continuous motion of the profile curve rt along the spine curve γs with the orientation as determined by As. Profile curve rt is in the 2D or 3D space which passes through the spine curve γs during sweeping. Interestingly, RMF allows for a simple characterization of spine curve.Definition 2.
A surface in the Minkowski 3-spaceE13 is named a time-like 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 (time-like) surface is a time-like (spacelike) vector.


## 3. Time-Like Sweeping Surface and Singularities

In this section, we define a time-like sweeping surface in Minkowski 3-spaceE13. Consider the planar profile curve given by rt=0,r1t,r2t. By using (12), it follows the following:(13)M:ys,t=γs+r1tξ1+r2tξ2.Parametrizing a sweeping surface in this way excludes sweeping surfaces with stationary tangent vectorυ1, whose geometric ownerships are less interesting and to a certain scope easier to explore. We will use “dot” to indicate the derivative with respect to the arc length parameter of the profile curve rt . Without loss of generality, we can assume the profile curve rt is a unit speed time-like curve, i that is, r12.−r22.=−1.From equation (13), it follows that(14)yts,t=r1.ξ1+r2.ξ2,yss,t=1+μ1r1+μ2r2ξ.The unit normalN is found by the following:(15)Ns,t=yt×ysyt×ys=r2.ξ1+r1.ξ2.Note thatNs,t2=1 means that M is a time-like surface. (15) shows that the surface normal N 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.Proposition 1.
Consider a pointy in the normal plane of the spine curve γs. The tangent vector yss,t is consistently parallel to the tangent vector ξ of the spine curve γs.For the time-like sweeping surfaceM, the elements of the first fundamental form are as follows:(16)g11=<ys,ys>=1+μ1r1+μ2r22,g12=<ys,yt>=0,g22=<yt,yt>=−1..A direct computation gives the following:(17)yss=r1μ1′+r2μ2′ξ+1+μ1r1+μ2r2μ1ξ1−μ2ξ2,yst=r˙1μ1+r˙2μ2ξ,ytt=r¨1ζ1+r¨2ζ2..Also, we have the following:(18)h11=−1+r1μ1+r2μ2μ1r˙2+μ2r˙1,h12=0,h22=−r˙2r¨1+r˙1r¨2..(18) shows that the parametric curves s-and t are lines of curvature g12=h12=0. Therefore, the iso-parametric curve is as follows:(19)πt:ζt:=yt,s0=γs0+r1tζ1s0+r2tζ2s0.is a planar unit speed time-like line of curvature. (19) locate a set of one-parameter family of time-like planes in E13. The time-like unit tangent vector to ζu is as follows:(20)tζt=r˙1tζ1s0+r˙2tζ2s0.The unit principal normal vector ofζt is given by the following:(21)nζt=ξs0×tζu=r˙2ξ1+r˙1ξ2=Ns0,t.Hence, the surface normalNs0,t is identical to the principal normal nζt, that is, the curve ζt is a time-like planar geodesic line of curvature on yt,s0. Surfaces whose parametric curves are lines of curvature have different applications in geometric design [2–7]. Hence, one has to define the offset surfaces yft,s=yt,s+fNs,t of a given time-like surface yt,s at a definite distance f. Therefore, the offsetting operation for time-like sweeping surface can be reduced to the offsetting of planar profile curve, which is completely easier to deal with.Hence, we can state the following proposition:Proposition 2.
Consider a time-like sweeping surfaceM represented by equation (13), and rft be the planar offset of the profile rt at distance f. Then the offset surface yft,s is again a time-like sweeping surface, traced by the spine curve γs and profile curve rft.


### 3.1. Singularity

We now study singularity as follows: It can be seen thatyt,s has singularities if(22)yt×ys=1+μ1r1+μ2r2=0.Via (5), the following can be obtained as follows:(23)ρs+r1tcoshφs+r2tsinhφs=0,where ρ=κ−1s is the radius of curvature of γs. By ρs, we have r1=−ρcoshφ, and r2=ρsinhφ. Hence, from (13), the singular locus is as follows:(24)Cs=γs+ρs−coshφξ1+sinhφξ2.In view of the kinematical generation ofM, the singular points occur between the profile curve r=rt and the curvature axis (revolution axis):(25)Lt=r1,r2|ρ0+r1coshφ0+r2sinhφ0=0.Thus, the time-like sweeping surface has a 2nd-order contact with the time-like surface of revolution that is generated by rotating the profile curver=rt around Lt.Hence, we obtain the following corollary:Corollary 1.
The time-like sweeping surfaceM represented by (13), has no singular points if the condition(26)1+μ1r1+μ2r2≠0,is satisfied for all s, and t.


#### 3.1.1. Convexity and Parabolic Curves

Conditions that guarantee the convexity or curves which produce parabolic points of a surface are desired in several implementations (such as manufacturing of sculptured surfaces, or layered manufacturing). Therefore, we will investigate in what conditions the parametric curves are parabolic curves as in the following: The Gaussian curvature ofM at a regular point can be obtained as follows:(27)Ks,t=N2h11h22−h122g11g22−g122=μ1r˙2+μ2r˙1r˙1r¨2−r˙2r¨21+μ1r1+μ2r2.Sinceg12=h12=0, the value of one principal curvature is as follows:(28)κ1s0,t=r˙×r¨r˙3=r˙1r¨2−r˙2r¨1.r˙1.The other principal curvature is easy to obtain the following:(29)κ2s,t=Ks,tκ1s0,t=−μ1r˙2+μ2r˙11+μ1r1+μ2r2.On the other hand, the curvature for the s-curves (t-constant) can be obtained as follows:(30)χs,t0=ys×yssys3=−1ρ+r1coshφ+r2sinhφ.In view of Meusnier’s Theorem, the expression ofκ2s,t in (29) is the following:(31)κ2s,t=χs,t0sinhϑ=−sinhϑρ+r1coshφ+r2sinhφ.Hereϑ=sinh−1r˙2coshφ+r˙1sinhφ≥0 is the Lorentzian time-like angle between the vectors υ2 and N. Thus, the Gaussian curvature Ks,t can be obtained in terms of κ1, χ, and ϑ as(32)Ks,t=−κ1s0,tχs,t0sinhϑ.It can be seen that time-like sweeping surface has parabolic curves ifKs,t=0, that is,(33)Ks,t=0⇔κ1s0,tχs,t0sinhϑ=0.The parabolic curves separate elliptic (K>0, locally convex) and hyperbolic (K<0, hence non-convex) parts of M. From (33), three cases emerge when parabolic curves are concerned:Case 1.
Exists whenκ1=0. In this case, the profile curve r=rt is degenerate into a time-like line, from equation (28), it can be seen that(34)κ1t=0⇔r˙×r¨=0⇔r˙‖r¨,which shows that a flat point or an inflection of r=rt traces a parabolic curve s= const. On parts of the time-like sweeping surface.Case 2.
Exists whenχs,t0=0. In this case, an infection or flat point of the spine curve generates an iso-parametric parabolic curve t = const. on the time-like sweeping surface.Case 3.
Exists whensinhϑ=0. These parabolic curves are given by the following condition(35)r2.coshφ+r1.sinhφ=0,is satisfied for all s, and t. In this case, the spine curve γ is not only a spacelike line of curvature but also a geodesic curve on M.
The convexity can be controlled with the integration of (35), that is, we have the following:(36)r2coshφ+r1sinhφ=λs,where λ=λs is an arbitrary function. Then, we have the following equations:(37)r1=−λssinhφ,r2=λscoshφ.
When (37) are substituted to (13), with attention of (5), we directly find that the expression of the parabolic curve is the following:(38)Ps=γs+λsυ3s.
From the above investigation the next corollary can be acquired.Corollary 2.
LetM be a time-like sweeping surface expressed by (13), with spine and profile curves have non-vanishing curvatures anywhere. Then M has precisely one parabolic curve if the spacelike spine curve γs is a geodesic curve.


## 3.1. Singularity

We now study singularity as follows: It can be seen thatyt,s has singularities if(22)yt×ys=1+μ1r1+μ2r2=0.Via (5), the following can be obtained as follows:(23)ρs+r1tcoshφs+r2tsinhφs=0,where ρ=κ−1s is the radius of curvature of γs. By ρs, we have r1=−ρcoshφ, and r2=ρsinhφ. Hence, from (13), the singular locus is as follows:(24)Cs=γs+ρs−coshφξ1+sinhφξ2.In view of the kinematical generation ofM, the singular points occur between the profile curve r=rt and the curvature axis (revolution axis):(25)Lt=r1,r2|ρ0+r1coshφ0+r2sinhφ0=0.Thus, the time-like sweeping surface has a 2nd-order contact with the time-like surface of revolution that is generated by rotating the profile curver=rt around Lt.Hence, we obtain the following corollary:Corollary 1.
The time-like sweeping surfaceM represented by (13), has no singular points if the condition(26)1+μ1r1+μ2r2≠0,is satisfied for all s, and t.


### 3.1.1. Convexity and Parabolic Curves

Conditions that guarantee the convexity or curves which produce parabolic points of a surface are desired in several implementations (such as manufacturing of sculptured surfaces, or layered manufacturing). Therefore, we will investigate in what conditions the parametric curves are parabolic curves as in the following: The Gaussian curvature ofM at a regular point can be obtained as follows:(27)Ks,t=N2h11h22−h122g11g22−g122=μ1r˙2+μ2r˙1r˙1r¨2−r˙2r¨21+μ1r1+μ2r2.Sinceg12=h12=0, the value of one principal curvature is as follows:(28)κ1s0,t=r˙×r¨r˙3=r˙1r¨2−r˙2r¨1.r˙1.The other principal curvature is easy to obtain the following:(29)κ2s,t=Ks,tκ1s0,t=−μ1r˙2+μ2r˙11+μ1r1+μ2r2.On the other hand, the curvature for the s-curves (t-constant) can be obtained as follows:(30)χs,t0=ys×yssys3=−1ρ+r1coshφ+r2sinhφ.In view of Meusnier’s Theorem, the expression ofκ2s,t in (29) is the following:(31)κ2s,t=χs,t0sinhϑ=−sinhϑρ+r1coshφ+r2sinhφ.Hereϑ=sinh−1r˙2coshφ+r˙1sinhφ≥0 is the Lorentzian time-like angle between the vectors υ2 and N. Thus, the Gaussian curvature Ks,t can be obtained in terms of κ1, χ, and ϑ as(32)Ks,t=−κ1s0,tχs,t0sinhϑ.It can be seen that time-like sweeping surface has parabolic curves ifKs,t=0, that is,(33)Ks,t=0⇔κ1s0,tχs,t0sinhϑ=0.The parabolic curves separate elliptic (K>0, locally convex) and hyperbolic (K<0, hence non-convex) parts of M. From (33), three cases emerge when parabolic curves are concerned:Case 1.
Exists whenκ1=0. In this case, the profile curve r=rt is degenerate into a time-like line, from equation (28), it can be seen that(34)κ1t=0⇔r˙×r¨=0⇔r˙‖r¨,which shows that a flat point or an inflection of r=rt traces a parabolic curve s= const. On parts of the time-like sweeping surface.Case 2.
Exists whenχs,t0=0. In this case, an infection or flat point of the spine curve generates an iso-parametric parabolic curve t = const. on the time-like sweeping surface.Case 3.
Exists whensinhϑ=0. These parabolic curves are given by the following condition(35)r2.coshφ+r1.sinhφ=0,is satisfied for all s, and t. In this case, the spine curve γ is not only a spacelike line of curvature but also a geodesic curve on M.
The convexity can be controlled with the integration of (35), that is, we have the following:(36)r2coshφ+r1sinhφ=λs,where λ=λs is an arbitrary function. Then, we have the following equations:(37)r1=−λssinhφ,r2=λscoshφ.
When (37) are substituted to (13), with attention of (5), we directly find that the expression of the parabolic curve is the following:(38)Ps=γs+λsυ3s.
From the above investigation the next corollary can be acquired.Corollary 2.
LetM be a time-like sweeping surface expressed by (13), with spine and profile curves have non-vanishing curvatures anywhere. Then M has precisely one parabolic curve if the spacelike spine curve γs is a geodesic curve.


## 3.1.1. Convexity and Parabolic Curves

Conditions that guarantee the convexity or curves which produce parabolic points of a surface are desired in several implementations (such as manufacturing of sculptured surfaces, or layered manufacturing). Therefore, we will investigate in what conditions the parametric curves are parabolic curves as in the following: The Gaussian curvature ofM at a regular point can be obtained as follows:(27)Ks,t=N2h11h22−h122g11g22−g122=μ1r˙2+μ2r˙1r˙1r¨2−r˙2r¨21+μ1r1+μ2r2.Sinceg12=h12=0, the value of one principal curvature is as follows:(28)κ1s0,t=r˙×r¨r˙3=r˙1r¨2−r˙2r¨1.r˙1.The other principal curvature is easy to obtain the following:(29)κ2s,t=Ks,tκ1s0,t=−μ1r˙2+μ2r˙11+μ1r1+μ2r2.On the other hand, the curvature for the s-curves (t-constant) can be obtained as follows:(30)χs,t0=ys×yssys3=−1ρ+r1coshφ+r2sinhφ.In view of Meusnier’s Theorem, the expression ofκ2s,t in (29) is the following:(31)κ2s,t=χs,t0sinhϑ=−sinhϑρ+r1coshφ+r2sinhφ.Hereϑ=sinh−1r˙2coshφ+r˙1sinhφ≥0 is the Lorentzian time-like angle between the vectors υ2 and N. Thus, the Gaussian curvature Ks,t can be obtained in terms of κ1, χ, and ϑ as(32)Ks,t=−κ1s0,tχs,t0sinhϑ.It can be seen that time-like sweeping surface has parabolic curves ifKs,t=0, that is,(33)Ks,t=0⇔κ1s0,tχs,t0sinhϑ=0.The parabolic curves separate elliptic (K>0, locally convex) and hyperbolic (K<0, hence non-convex) parts of M. From (33), three cases emerge when parabolic curves are concerned:Case 1.
Exists whenκ1=0. In this case, the profile curve r=rt is degenerate into a time-like line, from equation (28), it can be seen that(34)κ1t=0⇔r˙×r¨=0⇔r˙‖r¨,which shows that a flat point or an inflection of r=rt traces a parabolic curve s= const. On parts of the time-like sweeping surface.Case 2.
Exists whenχs,t0=0. In this case, an infection or flat point of the spine curve generates an iso-parametric parabolic curve t = const. on the time-like sweeping surface.Case 3.
Exists whensinhϑ=0. These parabolic curves are given by the following condition(35)r2.coshφ+r1.sinhφ=0,is satisfied for all s, and t. In this case, the spine curve γ is not only a spacelike line of curvature but also a geodesic curve on M.
The convexity can be controlled with the integration of (35), that is, we have the following:(36)r2coshφ+r1sinhφ=λs,where λ=λs is an arbitrary function. Then, we have the following equations:(37)r1=−λssinhφ,r2=λscoshφ.
When (37) are substituted to (13), with attention of (5), we directly find that the expression of the parabolic curve is the following:(38)Ps=γs+λsυ3s.
From the above investigation the next corollary can be acquired.Corollary 2.
LetM be a time-like sweeping surface expressed by (13), with spine and profile curves have non-vanishing curvatures anywhere. Then M has precisely one parabolic curve if the spacelike spine curve γs is a geodesic curve.


## 4. Space-Like Developable Surfaces

Developable surfaces can be briefly defined as special cases of the ruled surfaces. Such surfaces are utilized in different things, for example, in the manufacture of automobile body parts, and ship hulls. As a result, we discuss the Case1, that the profile curve r=rt is degenerate into a time-like line, that is, rt=0,0,t. In this case, (13) can be simplified as follows:(39)Ds,t=γs+tξ2s,s∈I,t∈ℝ.It is obvious thatDs,t is a developable ruled surface, that is,(40)detγ′s,ξ2s,ξ2′s=0.The normal vector is found by the following:(41)∂D∂s×∂D∂t=1−tμ2ξ1s,which means that Ds,t is a spacelike developable ruled surface. Since Ds,0=γs, 0≤s≤L, then Ds,t interpolate the curve γs. Moreover, the curve γs is a spacelike line of curvature on Ds,t.Theorem 1 (Existence and uniqueness).
Under the above notations there exists a unique spacelike developable surface defined by (39).Proof.
For the existence, we have the developable represented by (39). On the other hand, since Ds,t is a ruled surface, we assume that(42)Ds,t=γs+tβs,t∈ℝ,βs=β1sξ1+β2sξ2+βsξ,βs2=β12−β22+β2=1,β′s≠0..
It can be promptly seen thatDs,t is developable if(43)detγ′,β,β′=0⇔−β1β2′+β2β1′+βκβ2coshφ+β1sinhφ=0.
Further, in view of (41), we have the following:(44)∂D∂s×∂D∂t=θs,tξ1,where θ=θs,t is a differentiable function. Further, the normal vector ∂D/∂s×∂D/∂t at the point s,0 is(45)∂D∂s×∂D∂ts,0=β2ξ1+β1ξ2.
Thus, from (44) and (45), one finds that the following:(46)β1=0,β2=θs,0,which follows from (43) that βκβ2coshφ=0, which leads to ββ2=0, with κ≠0. If s,0 is a regular point (i.e., θs,0≠0), then β2s≠0, and β=0. Therefore, we obtain βs=ξ2. This means that the direction of βs is equal to the direction of ξ2s.
Furthermore, assume thatDs,t has a singular point at s0,0. Then θs0,0=β2s0=β1s0=0, and we have βs0=βs0ξs0. If the singular point γs0 is in the closure of the set of points where Ds,t is developable along γs is regular, then there exists a point γs in any neighborhood of γs0 such that the uniqueness of the Ds,t holds at γs. Passing to the limit s⟶s0, uniqueness of the developable at s0. Assume that there exists an open interval J⊆I such that Ds,t is singular at γs for each s∈J. Then Ds,t=γs+tβsξs for any s∈J. This leads to β1s=β2s=0 for s∈J. Then,(47)∂D∂s×∂D∂ts,t=κtβ2coshφξ2−sinhφξ1.
Thus, the normal vector is directed toξ1, i.e., ∂D/∂s×∂D/∂t‖ζ1s if φ=0 for any s∈J. In this case, βs=±ξ2. This means that uniqueness holds.
As an implementation (such as flank milling or cylindrical milling), during the motion of the RMF, let a spacelike cylindrical cutter be rigidly related to this frame. Then the equation of the of spacelike cylindrical cutters, which is obtained by the movement of spacelike cylindrical cutter alongγs, can be given as follows:(48)D¯s,t=Ds,t+fξ1s,where f is cylindrical cutter radius. This surface is a spacelike developable surface offset of Ds,t. The equation of D¯s,t, can therefore be written as the following:(49)D¯s,t=γs+tξ2s+fξ1s.
The time-like unit normal vector can be represented as follows:(50)ξfs,0=D¯s×D¯tD¯s×D¯t=ξ1s.
Also, from (49), we have the following:(51)Ds,t=D¯s,t−fξ1s.
The derivative of (51) with respect to s can be acquired as follows:(52)D¯ss,t=Dss,u−fω×ξ1.
From (52) we can see that the vector D¯ss,t is orthogonal to the normal vector ξ1. Thus, the vector ξ1 is orthogonal to the tool axis vector ξs. Hence, the envelope surface of the spacelike cylindrical cutter and the spacelike developable surface Ds,t have the mutual time-like normal vector and the distance between them is cylindrical cutter radius f.
Hence, we can draw a conclusion as follows:Proposition 3.
Consider a spacelike developable surfaceDs,t defined by (39). Let D¯s,t be the spacelike developable surface offset of spacelike cylindrical cutter at distance f. Then the two surfaces Ds,t and D¯s,t are offset spacelike developable surfaces.In the following section, we discuss the conditions when the spacelike developable surfaceDs,t is cylinder, cone or tangent surface, respectively. From (40), it follows that(53)detγ′s,ξ2s,ξ2′s=<γ′s,ξ2s×ξ2′s>=0.The first case when,(54)ξ2s×ξ2′s=0⇔κsinhφξ1=0.In this case,Ds,t is a cylinder. Since ξ1 is a non-zero time-like unit vector, then Ds,t is a spacelike cylinder if sinhφ=0⇔φs=0. In this case, we have φ′=0, then τs=0. Therefore, the curve γs is a planar spacelike curve, and the corresponding surface is a binormal spacelike surface.Corollary 3.
The spacelike developable surfaceDs,t is a cylinder surface if φs=0.The second case when,(55)ξ2s×ξ2′s≠0.In this case,Ds,t is a non-cylinder. Thus, the first derivative of the directrix is the following:(56)γ′s=α′s+σsξ2′s+σ′sξ2s,where α′ is the first derivative of the striction curve, σs is a differentiable function [3]. Substituting (56) into (53) gives the following::(57)<ξ2s×ξ2′s,α′s>=0.It can be seen that there are two cases that (57) can be satisfied for any s: The first case is when the first derivative of the striction curve is α′=0. Geometrically, this condition implies that the striction curve becomes a fixed point, and Ds,t is a cone. In this case, we have the following:(58)σ=const.=1κsinhφ⇔κsinhφ=κ0sinhφ0,where φ0=φ0, and κ0=κ0. Therefore, if φ is a constant, that is τs=0, then the curve γs is a planar spacelike curve with a constant curvature. Similarly, if κ is also a constant, we can have τs=0 and φ is a constant. Then the curve γs is the arc of a spacelike circle.Corollary 4.
The spacelike developable surfaceDs,t is a cone if κsinhφ=κ0sinhφ0; φ0=φ0, and κ0=κ0.
The second case is whenα′≠0, i.e., κsinhφ≠κ0sinhφ0. From (53), <α′, ξ2×ξ2′>=0, therefore, α′ is in the plane generated by ξ2 and ξ2′. The condition for α to be striction curve is therefore equivalent to <α′,ξ2′>=0. Hence, we may conclude that ξ2 is parallel to α′, which is also the tangent of the striction curve. This ruled surface is referred to as a tangent ruled surface.Corollary 5.
The spacelike developable surfaceDs,t is a tangent surface if κsinhφ≠κ0sinhφ0; φ0=φ0, and κ0=κ0.


### 4.1. Application

In this subsection, as an application of our main results, some representative examples are given according to three types of spacelike developable surfaces.Example 1.
Suppose we are given a parametric spacelike curve(59)γs=sinhs,0,coshs,−1≤s≤1.
After simple computation, we have the following:(60)υ1s=coshs,0,sinhs,υ2s=sinhs,0,coshs,υ3s=0,−1,0,κs=1,τs=0..
Thenφs=φ0 is a constant. Thus, the spacelike developable family with a common spacelike line of curvature is given by the following:(61)Ds,t=sinhs,0,coshs+tsinhφ0υ2s+coshφ0υ3s.
Ifφ0=0, for example, and −1≤t≤0 the developable surface is a spacelike cylinder (Figure 1). Take φ0=1, and −.7≤t≤0 the corresponding surface is a spacelike cone as shown in Figure 2.Figure 1
Spacelike cylinder.Figure 2
Spacelike cone.Example 2.
Let us consider a parametric spacelike helix(62)γs=asinhsc,bsc,acoshsc,−3≤s≤3,where a,b,c∈ℝ, and a2+b2=c2. It is easy to obtain,(63)υ1s=accoshsc,bc,acsinhsc,υ2s=sinhsc,0,coshsc,υ3s=bccoshsc,−ac,bcsinhsc,κs=ac2,τs=bc2..
Then,φs=−b/c2s+φ0, and κsinhφ≠κ0sinhφ0. According to Corollary 5, the spacelike developable surface Ds,t is a tangent surface. The corresponding spacelike tangent surface is the following:(64)Ds,t=asinhsc,bsc,acoshsc+tsinhφυ2s+coshφυ3s.
Here, we chose−3≤s≤3, and a=2,b=1. For φ0=3, and 0≤t≤1, the surface is shown in Figure 3. Figure 4 shows the surface with φ0=0, and 0≤t≤3.
At the end of this work, we want to shed light on the case of a spacelike sweeping surfaceM. We found that if the profile curve rt is a unit speed spacelike curve, i.e., r12.−r22.=1, a sweeping surface of the type spacelike surface will be produced. And we got convergent and almost similar results. Hence, this is accompanied by a difference in some signs of the equations and the previous results.Figure 3
Spacelike tangent.Figure 4
Spacelike tangent.


## 4.1. Application

In this subsection, as an application of our main results, some representative examples are given according to three types of spacelike developable surfaces.Example 1.
Suppose we are given a parametric spacelike curve(59)γs=sinhs,0,coshs,−1≤s≤1.
After simple computation, we have the following:(60)υ1s=coshs,0,sinhs,υ2s=sinhs,0,coshs,υ3s=0,−1,0,κs=1,τs=0..
Thenφs=φ0 is a constant. Thus, the spacelike developable family with a common spacelike line of curvature is given by the following:(61)Ds,t=sinhs,0,coshs+tsinhφ0υ2s+coshφ0υ3s.
Ifφ0=0, for example, and −1≤t≤0 the developable surface is a spacelike cylinder (Figure 1). Take φ0=1, and −.7≤t≤0 the corresponding surface is a spacelike cone as shown in Figure 2.Figure 1
Spacelike cylinder.Figure 2
Spacelike cone.Example 2.
Let us consider a parametric spacelike helix(62)γs=asinhsc,bsc,acoshsc,−3≤s≤3,where a,b,c∈ℝ, and a2+b2=c2. It is easy to obtain,(63)υ1s=accoshsc,bc,acsinhsc,υ2s=sinhsc,0,coshsc,υ3s=bccoshsc,−ac,bcsinhsc,κs=ac2,τs=bc2..
Then,φs=−b/c2s+φ0, and κsinhφ≠κ0sinhφ0. According to Corollary 5, the spacelike developable surface Ds,t is a tangent surface. The corresponding spacelike tangent surface is the following:(64)Ds,t=asinhsc,bsc,acoshsc+tsinhφυ2s+coshφυ3s.
Here, we chose−3≤s≤3, and a=2,b=1. For φ0=3, and 0≤t≤1, the surface is shown in Figure 3. Figure 4 shows the surface with φ0=0, and 0≤t≤3.
At the end of this work, we want to shed light on the case of a spacelike sweeping surfaceM. We found that if the profile curve rt is a unit speed spacelike curve, i.e., r12.−r22.=1, a sweeping surface of the type spacelike surface will be produced. And we got convergent and almost similar results. Hence, this is accompanied by a difference in some signs of the equations and the previous results.Figure 3
Spacelike tangent.Figure 4
Spacelike tangent.


## 5. Conclusion

In this paper, we have a tendency to investigate a kind of time-like sweeping surface according to Bishop frame satisfying some fascinating and necessary equations in terms of the Gaussian curvature. Then, the necessary and sufficient condition for the time-like sweeping surface to become a time-like developable ruled surface is derived. In particular, we analyze the necessary and sufficient conditions when the resulting time-like developable surface is a cylinder, cone or tangent surface. We hope that these results will be helpful to physicists and those who are studying general relativity theory.



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