# Time-like Ruled Surface in One-Parameter Hyperbolic Dual Spherical Motions

Authors: Monia F. Naghi; Rashad A. Abdel-Baky; Fatemah Mofarreh
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this work, we introduce a time-like ruled surface in one-parameter hyperbolic dual spherical motions. This provides the ability to derive some formulae of surface theory into line spaces. Then, a time-like Plücker conoid associated with the motion has been obtained, and its kinematic geometry is researched in detail. Consequently, a characterization for a time-like ruled surface to be a constant Disteli-axis is derived and investigated in detail. At last, we have discussed some special cases which lead to some special time-like ruled surfaces such as the time-like helicoids, Lorentzian sphere, and time-like cone.


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## Body

## 1. Introduction

In spatial kinematics, the movement of an oriented line over a curve performs ruled surface. These oriented lines are said to be generators (rulings), and each curve that intersects all the generators is called a directrix (base curve) [1–3]. One of the most suitable methods to study the movement of oriented line space seems to create a relationship between this space, dual numbers, and dual vector calculus. Dual numbers were first introduced by W. Clifford; after him, E. Study used it as an instrument for his research on line geometry and kinematics. He devoted special care to the impersonation of oriented lines by dual unit vectors and defined the mapping that is known by his name. The E. Study map states that “The set of all oriented lines in Euclidean 3-space E3 is directly linked to the set of points on the dual unit sphere in the dual 3-space D3.” Thus, the differential geometry of ruled surfaces based on the E. Study map has rederived the curvature theory of a line trajectory and exposed the fundamental curvature functions that describe the shape of ruled surface (see, for example, [4–11]).In the Minkowski 3-spaceE13, the investigation of ruled surfaces is more motivating than the Euclidean case, since the Euclidean metric can only be positive whereas the Lorentzian metric can be negative, positive, or zero. Therefore, the kinematics and geometric explanations can be widely different. Hence, if we replace the Minkowski 3-space E13 as an alternative of the Euclidean 3-space E3, the E. Study map can be given as follows: “the time-like (resp., space-like) oriented lines are represented with the time-like (resp., space-like) dual points on the hyperbolic (resp., Lorentzian) dual unit sphere in the Lorentzian dual 3-space D13 [10, 11].” It means that a regular curve on ℍ+2 represents a time-like ruled surface at E13. Similarly, the space-like (resp., time-like) curve on S12 represents a time-like (resp., space-like) ruled surface at E13. In view of its relationships with engineering and physical sciences in Minkowski space, many geometers and engineers have studied and gained many ownerships of the ruled surfaces (see [8–13]).However, to the best of the authors’ knowledge, no literature exists regarding the time-like ruled surface with constant Disteli-axis from one-parameter hyperbolic dual spherical motions. Thus, the present study hopes to serve such a need. This work offers an approach for constructing time-like ruled surfaces with constant Disteli-axis by using E. Study map. Hence, the well-known formulae of Hamilton, Mannheim, and Dupin’s indicatrix of surfaces theory are proved for the line space. Moreover, a time-like Plücker conoid associated with the motion has been derived and it is shown that the principal axes of it are located at its center. Then, some characterization equations for special time-like ruled surfaces such as the time-like helicoid, the Lorentzian sphere, and the time-like cone are obtained and investigated.


## 2. Basic Concepts

We start with basic concepts on the Minkowski 3-spaceE13, the theory of dual numbers, dual Lorentzian vectors, and E. Study map, for example, [1–3, 14–17]. A dual number A is a number a+εa∗, where a,a∗ in ℝ and ε is a dual unit with the property that ε2=0. Then, the set
(1)D3=A≔a+εa∗=A1,A2,A3,together with the Lorentzian scalar product
(2)<A,A>=−A12+A22+A32,forms the so-called dual Lorentzian 3-space D13. Thus, a point A=A1,A2,A3t has dual coordinates Ai=ai+εai∗∈D. If A is space-like or time-like dual vector the norm A of A is defined by
(3)A=<A,A>=<a,a>+ε12<a,a><a,a><a,a>·2<a,a∗>=a+ε1a<a,a><a,a><a,a∗>.Ifa is space-like, we have
(4)A=a+ε1a<a,a∗>=a1+ε1a2<a,a∗>.Ifa is time-like, we have
(5)A=a−ε1a<a,a∗>=a1−ε1a2<a,a∗>.Therefore,A is called a space-like (resp., time-like) dual unit vector if <A,A>=1 (resp. <A,A>=−1). The hyperbolic and Lorentzian dual unit spheres, respectively, are
(6)ℍ+2=A∈D13∣−A12+A22+A32=−1,S12=A∈D13∣−A12+A22+A32=1.Theorem 1.
There is a one-to-one correspondence between space-like (resp., time-like) oriented lines in the Minkowski 3-spaceE13 and ordered pairs of vectors a,a∗ such that
(7)A2=±1⇔a2=±1,<a,a∗>=0,where ai,ai∗i=1,2,3 of a and a∗ are called the normed Plücker coordinates of the line.According to Theorem1, four independent parameters locate a line complex, so it is reasonable to intersect any two of line complexes and gain a definite number of lines (line congruence) with common properties. The intersection of two independent linear congruences carries out a differentiable family of straight lines (a ruled surface). Ruled surfaces (such as cylinders and cones) include rulings where the tangent plane relates the surface over the entire line (torsal lines) [1–3]. Also, we have the E. Study’s map: the dual unit spheres are shaped as a pair of conjugate hyperboloids. The ring-shaped hyperboloid correlates with the set of space-like lines, the common asymptotic cone correlates the set of null (light-like) lines, and the oval-shaped hyperboloid forms the set of time-like lines, and opposite points of each hyperboloid perform the pair of obverse vectors on a line (see Figure 1). Therefore, a regular curve on ℍ+2 represents a time-like ruled surface in E13. Similarly, a regular curve on S12 represents a space-like or time-like ruled surface in E13.Figure 1
The hyperbolic and Lorentzian dual unit spheres.


### 2.1. One-Parameter Hyperbolic Dual Spherical Motions

Consider two hyperbolic dual unit spheresℍ+m2 and ℍ+f2. Suppose that O is the common center, and two orthonormal dual frames O;R1time-like,R2,R3, and O;F1time-like,F2,F3 be rigidly linked to ℍ+m2 and ℍ+f2, respectively. If we suppose that O;F1,F2,F3 is fixed, whereas the elements of the set O;R1,R2,R3 are functions of a real parameter t∈ℝ (say the time), then we say that ℍ+m2 moves with respect to ℍ+f2. This motion is called a one-parameter hyperbolic dual spherical motions and will denoted by ℍ+m2/ℍ+f2. If the dual unit spheres ℍ+m2 and ℍ+f2 correspond to the line spaces Lm and Lf, respectively, then ℍ+m2/ℍ+f2 represents the one-parameter Lorentzian spatial motion Lm/Lf. Therefore, Lm is the moving Lorentzian space with respect to the fixed Lorentzian space Lf in E13. By putting <Fi,Rj>=Lij and introducing the dual matrix L=Lij, we can write the E. Study map in the matrix form as follows:
(8)ℍ+m2/ℍ+f2:F1F2F3=L11L12L13L21L22L23L31L32L33R1R2R3.It then follows that the signature matrixϵ, describing the scalar product in D13, is given by [15]
(9)ϵ=−100010001.The dual matrixL≔Lij=lij+εlij∗ has the property that LT=ϵL−1ϵ,L−1=LTϵL. Hence, we have
(10)LL−1=LϵLTϵ=L−1L=ϵLTϵL=I,where I is the 3×3 unit matrix. This result indicates that when a one-parameter Lorentzian spatial motion is given in E13, we can find a corresponding dual Lorentzian orthogonal 3×3 matrix Lt=Lij, where Lij are dual functions of one variable t∈ℝ. As the set of real Lorentzian orthogonal matrices, the set of Lorentzian dual orthogonal 3×3 matrices, denoted by OD13×3, forms a group with matrix multiplication as the group operation (real Lorentzian orthogonal matrices are a subgroup of Lorentzian dual orthogonal matrices). The identity element of OD13×3 is the 3×3 unit matrix. Since the center of the hyperbolic dual unit sphere in D13 must remain fixed, the transformation group in D13 (the image of Lorentzian motions in the Minkowski 3-space E13) does not contain any translations.The Lie algebraLOD13×3 of the group GL of 3×3 positive orthogonal dual matrices L is the algebra of skew-adjoint 3×3 dual matrices
(11)Ωt≔L′ϵLTϵ=0Ω3−Ω2Ω30Ω1Ω2−Ω10=−Ω1Ω2Ω3,where dash indicates the differential with respect to the real parameter t∈ℝ. Then, the derivative equation of ℍ+m2/ℍ+f2 is
(12)R1′R2′R3′=0Ω3−Ω2Ω30Ω1Ω2−Ω10R1R2R3=Ω×R1R2R3.Ωt=ω+εω∗=Ω1,Ω2,Ω3 is called the instantaneous dual rotation vector of ℍ+m2/ℍ+f2. The real part ω and dual part ω∗, respectively, correspond to the instantaneous rotational differential velocity vector and the instantaneous translational differential velocity vector of the motion Lm/Lf.


## 2.1. One-Parameter Hyperbolic Dual Spherical Motions

Consider two hyperbolic dual unit spheresℍ+m2 and ℍ+f2. Suppose that O is the common center, and two orthonormal dual frames O;R1time-like,R2,R3, and O;F1time-like,F2,F3 be rigidly linked to ℍ+m2 and ℍ+f2, respectively. If we suppose that O;F1,F2,F3 is fixed, whereas the elements of the set O;R1,R2,R3 are functions of a real parameter t∈ℝ (say the time), then we say that ℍ+m2 moves with respect to ℍ+f2. This motion is called a one-parameter hyperbolic dual spherical motions and will denoted by ℍ+m2/ℍ+f2. If the dual unit spheres ℍ+m2 and ℍ+f2 correspond to the line spaces Lm and Lf, respectively, then ℍ+m2/ℍ+f2 represents the one-parameter Lorentzian spatial motion Lm/Lf. Therefore, Lm is the moving Lorentzian space with respect to the fixed Lorentzian space Lf in E13. By putting <Fi,Rj>=Lij and introducing the dual matrix L=Lij, we can write the E. Study map in the matrix form as follows:
(8)ℍ+m2/ℍ+f2:F1F2F3=L11L12L13L21L22L23L31L32L33R1R2R3.It then follows that the signature matrixϵ, describing the scalar product in D13, is given by [15]
(9)ϵ=−100010001.The dual matrixL≔Lij=lij+εlij∗ has the property that LT=ϵL−1ϵ,L−1=LTϵL. Hence, we have
(10)LL−1=LϵLTϵ=L−1L=ϵLTϵL=I,where I is the 3×3 unit matrix. This result indicates that when a one-parameter Lorentzian spatial motion is given in E13, we can find a corresponding dual Lorentzian orthogonal 3×3 matrix Lt=Lij, where Lij are dual functions of one variable t∈ℝ. As the set of real Lorentzian orthogonal matrices, the set of Lorentzian dual orthogonal 3×3 matrices, denoted by OD13×3, forms a group with matrix multiplication as the group operation (real Lorentzian orthogonal matrices are a subgroup of Lorentzian dual orthogonal matrices). The identity element of OD13×3 is the 3×3 unit matrix. Since the center of the hyperbolic dual unit sphere in D13 must remain fixed, the transformation group in D13 (the image of Lorentzian motions in the Minkowski 3-space E13) does not contain any translations.The Lie algebraLOD13×3 of the group GL of 3×3 positive orthogonal dual matrices L is the algebra of skew-adjoint 3×3 dual matrices
(11)Ωt≔L′ϵLTϵ=0Ω3−Ω2Ω30Ω1Ω2−Ω10=−Ω1Ω2Ω3,where dash indicates the differential with respect to the real parameter t∈ℝ. Then, the derivative equation of ℍ+m2/ℍ+f2 is
(12)R1′R2′R3′=0Ω3−Ω2Ω30Ω1Ω2−Ω10R1R2R3=Ω×R1R2R3.Ωt=ω+εω∗=Ω1,Ω2,Ω3 is called the instantaneous dual rotation vector of ℍ+m2/ℍ+f2. The real part ω and dual part ω∗, respectively, correspond to the instantaneous rotational differential velocity vector and the instantaneous translational differential velocity vector of the motion Lm/Lf.


## 3. Main Results

During the motionLm/Lf, any fixed time-like line X∈Lm traces a time-like ruled surface in Lf which will be denoted by X. In spatial kinematics, such time-like ruled surface is called time-like line trajectory. In order study the geometrical properties of X, we choose a moving frame, called the Blaschke frame, associated with point on ℍ+m2 as
(13)X=Xt,Tt≔tt+εt∗t=X′X′,Gt≔gt+εg∗t=X×T.Then, we find(14)−<X,X>=<T,T>=<G,G>=1,<X,T>=<X,G>=<G,T>=0,G×X=T,T×G=−X.X, T, and G are three dual unit vectors corresponding to three concurrent mutually perpendicular lines E13, and they intersect at a point C on X called the central point. G is the limit position of the common perpendicular to Xt and Xt+dt and is called the central tangent of X at the central point. The line T is called the central normal of X at the central point. Thus, the motion ℍ+m2/ℍ+f2 is given by [9–11]
(15)ℍ+m2/ℍ+f2:X′T′G′=0P0P0Q0−Q0XTG=Ω×XTG,where
(16)Pt=pt+εp∗t=X′,Qt≔qt+εq∗t=detX,X′,X′′X′−2,are called Blaschke invariants of X. The tangent of the striction curve may be written as
(17)C′=−q∗x+p∗g.The invariantspt, p∗t, qt, and q∗t are called the structure functions of the ruled surface X. The distribution parameters of the ruled surfaces X, T, and G, respectively, are
(18)μt=p∗p,δt=pp∗+qq∗q2+p2,Γt=q∗q.Definition 2.
A nondevelopable ruled surface is defined as a constant parameter ruled surface if the structure functionsμt, δt, and Γt are all constant.
Now, under the hypothesis thatQ>P, we have the evolute of X∈ℍ+2 as
(19)Bt=bt+εb∗t=ΩΩ=QX+PGQ2−P2.
It is clear thatB is the Disteli-axis (curvature-axis or striction-axis) of X. Let Ψ=ψ+εψ∗≥0 stand for the radius of curvature between X and B. Then, we can write
(20)Bt=coshΨX+sinhΨG,withcothΨ=QP.
One of the invariants of the dual curveX∈ℍ+2 is
(21)Σt≔γt+εΓt−μtγt=QP,called the dual geodesic curvature. Here, γt=q/p is the geodesic curvature of the hyperbolic spherical image curve t∈I↦xt of X. The trigonometric hyperbolic function Ψ can be written as
(22)cothΨ=cothψ−εψ∗1sinh2ψ.
Thus, from the real and dual parts of Equations (21) and (22), respectively, we find
(23)γt=cothψ=qp,(24)ψ∗t=12μ−Γsinh2ψ.
ψ∗ is the normal distance along T measured from B to X.


### 3.1. Kinematic-Geometry and Time-like Plücker Conoid

We now are interested in researching the kinematic geometry of (X). Therefore, we are going to make a detailed study of the Blaschke invariants Pt and Qt. To carry this out, from Equation (15), we can write the following equations:
(25)X′t=ΩB×X,T′t=ΩB×T,G′t=ΩB×G.Hence, at any instant, it is seen thatΩ=Ω=ω+εω∗ is the dual angular speed of the motion ℍ+m2/ℍ+f2 about the Disteli-axis B. Thus,
(26)ωt=q2−p2,ω∗t=qq∗−pp∗q2−p2,withq>p,corresponding to the rotational angular speed and translational angular speed of the motion Lm/Lf along B, respectively.Hence, the next corollary can be given:Corollary 3.
During the motionLm/Lf, at any instant t, the pitch can be given by
(27)ht≔<ω,ω∗>ω2=Γcosh2ψ−μsinh2ψ.It is clear that if the dual vectorΩ=ω+εω∗ is given, then the following can be specified:
(i)
The time-like Disteli-axisB is specified by Equation (20)(ii)
The dual angular speed isΩ=ω1+εh(iii)
Ify is a point on the time-like Disteli-axis B, then(28)yt,v=b×b∗+vb,v∈ℝ,is a nondevelopable time-like ruled surface B(iv)
If the motionLm/Lf is pure rotation, that is, hu=0, then(29)Bt=bt+εb∗t=1ωω+εω∗.It is interesting to note that ifht=0, and ω2=−1, then Ω is a time-like oriented line. However, in the case when the motion is pure translational, i.e., Ω=0+εω∗, we set ω∗=Ω∗,ω∗b=ω∗ and select an arbitrary b∗under ω∗≠0; otherwise, the time-like unit vector b can be chosen arbitrarily, too.The Equations in (24) and (27), respectively, are Minkowski versions of the Mannheim and Hamilton formulae of surface theory in Euclidean 3-space. Now, we give geometric interpretations of these formulae. Next, the parametric representation of ψ∗ is time-like Plücker conoid or cylindroid as follows: consider T is coincident with the space-like y-axis of a fixed Minkowski frame oxyz, while the position of the time-like dual unit vector B is given by angle ψ and distance ψ∗ on the space-like positive y-axis. The time-like dual unit vector X and the space-like dual unit vector G can be selected in sense of x- and z-axes, respectively. This shows that the dual unit vectors X and G together with T form the coordinate system, as shown in Figure 2. If y is a point on this time-like Plücker conoid, then we have
(30)M:yt,v=0,ψ∗,0+vcoshψ,0,sinhψ,v∈ℝ.Figure 2
B=coshΨX+sinhΨG.Using such representation, the time-like dual unit vectorsB are obviously visible crossing through the y-axis. It is easily seen from the last equation and (24) that
(31)ψ∗≔y=12μ−Γsinh2ψ,x=vcoshψ,z=vsinhψ.Here,ψ∗ gives us the intersection point of the oriented lines X and G which locates at a half of the conoid height. By direct calculations,
(32)x2−z2y−μ−Γxz=0,which is the algebraic equation for time-like Plücker conoid. The time-like Plücker conoid given by Equation (32) which has two structure functions of the first order and it depends only on their difference; μ−Γ=1,−0.9≤ψ≤0.9, −1.5≤v≤1.5 (Figure 3). Furthermore, one can get a second-order equation in x/z in which its solutions are given by
(33)xz=12yμ−Γ±μ−Γ2−4y2.Figure 3
Time-like Plücker conoid.By equating the discriminant of Equation (33) to zero, we get the limits of the time-like Plücker conoid. Hence, the two extreme positions are given by
(34)y=±μ−Γ2,which gives the positions of the two torsal planes π1, π2, and each one of them contains one torsal line L. Further, hu in Equation (27) is not a periodic function and has at most two extreme values, the structure functions Γ and μ. Therefore, the oriented lines X and G are principal axes of the time-like Plücker conoid. However, the geometric aspects are discussed as follows:
(1)
Ifht≠0, then there are two real torsal lines L1 and L2 passing through the point 0,y,0 only if y<μ−Γ/2; for the two limit points y=±μ−Γ/2, they coincide with the principal axes X and G(2)
Ifht=0, then the two torsal lines L1 and L2 are represented by(35)xz≔cothψ=±μΓ,y=±Γμ.Equation (35) means that the two torsal lines L1 andL2 are orthogonal each other in the Lorentzian sense. Furthermore, transition from polar coordinates to Cartesian coordinates could be completed by substituting
(36)x=coshψh,z=sinhψh,into Hamilton’s formula; one obtains the equation
(37)D:Γx2−μz2=1,of a conic section. This conic section is a Minkowski version of the Dupin indicatrix of surfaces theory in Euclidean 3-space.


#### 3.1.1. Serret–Frenet Motion

IfX is a time-like developable ruled surface, that is, μ=0, in this case, Dupin’s indicatrix is a set of parallel lines represented by x2=1/Γ, and the Blaschke frame {x,t,g} coincides with the classical Serret–Frenet frame, and then, the striction curve C becomes the edge of regression of X. Hence, p and q are the curvature κ and the torsion τ of C, respectively. Moreover, q∗=1 and Γτ=1 (Γ is the radius of torsion of C). By similar argument, we can also have the following equations:
(38)ht=1τcosh2ψ=ττ2−κ2,ψ∗t=−12Γsinh2ψ=−κτ2−κ2.The corresponding time-like Plücker conoid is(39)M:yt,v=0,ψ∗,0+vcoshψ,0,sinhψ,v∈ℝ.This yields(40)x2−z2y+Γxz=0.


### 3.2. Time-like Ruled Surfaces with Constant Disteli-Axis

A time-like ruled surfaceX is defined as a constant Disteli-axis ruled surface if the dual angle between the ruling of X and the Disteli-axis is always constant. Thus, when we say X is a time-like constant Disteli-axis, we mean that all the rulings of X have a constant Lorentzian dual angle from its Disteli-axis.The dual arc lengthds^=ds+εds∗ of Xt∈ℍ+2 is
(41)s^t=∫0tPdt=∫0tp1+εμdt.After that, we will use the dual arc length parameters^ instead of t. If the prime means to differentiation as s^, then from Equation (15), we get
(42)X′T′G′=01010Σ0−Σ0XTG=Ω×XTG,where Ω=ΣX+G. Thus, we may write the following relationships:
(43)κ^s^≔κ+εκ∗=Σ2−1=1sinhψ=1ρ~,τ^s^≔τ+ετ∗=±Σ′Σ2−1=±Ψ′,whereκ^ and τ^ are the dual curvature function and the dual torsion function of Xs^∈ℍ+2, respectively. The terms found in Equation (43) are such as to their counterparts in 3-dimensional hyperbolic spherical geometry.Definition 4.
For a one-parameter hyperbolic dual motion, at an instants^∈D, an oriented time-like line Z in fixed space will be said to be time-like Bk-Disteli-axis of X if for all i such that 1≤i≤k, <Z,Xis^>=0, but <Z,Xk+1s^>≠0. Here, Xi denotes the i-th derivatives of X.Via this definition, consider the Lorentzian dual angle(44)ρ~=cosh−1<Z,X>,such that X and Z have the same time orientation and Z and ρ~ stay fixed up to the second order at s^=s^0, i.e.,
(45)ρ~′s^=s^0=0,X′s^=s^0=0,ρ~′′s^=s^0=0,X′′s^=s^0=0.We have for the first order(46)<X′,Z>=0,and for the second-order properties
(47)<X′′,Z>∣=0.Then,ρ~ will be invariant in the second approximation if and only if Z is the time-like Disteli-axis B of X, that is,
(48)ρ~′=ρ~′′=0⇔Z=X′×X′′X′×X′′=±B.By the definition of the time-like Disteli-axis, we have the dual frame(49)U1=Bs^,U2s^=B′B′,U3s^=B×U2,as the Blaschke frame along B. Thus, the calculations give that
(50)U1U2U3=coshΨ0sinhΨsinhΨ0coshΨ0−10XTG.The variations of this frame are analogous to Equation (15) and are given by
(51)U1′U2′U3′=0Ψ′0Ψ′0κ^0−κ^0U1U2U3=Ω~×U1U2U3,where Ω~s^=κ^U1+Ψ′U3. Comparing Equation (42) with Equation (51), we have that the relative dual velocity is
(52)Ω−Ω~=Ψ′T.This shows that the Blaschke frame involves a further rotation around the central tangentT, whose speed equals the dual torsion τ^s. Hence, we obtain that if τ^s^=τ+ετ∗=0(Σ′t=γ′t+εΓt−μtγt′=0), that is, ψ and ψ∗ are constants, then the time-like Disteli-axis is fixed up to the second order and the time-like line X moves on it with constant pitch h. Thus kinematically, the time-like ruled surface X is traced by a hyperbolic one-parameter screw motion of constant pitch h along the constant time-like Disteli-axis B, by the time-like line X at a constant Lorentzian distance ψ∗ and constant Lorentzian angle ψ relative to B. Hence, we have the following.Theorem 5.
(X) is a time-like ruled surface with constant Disteli-axis if and only if (a) γ=constant and (b) Γ−μγ=constant.Now, we construct the time-like ruled surfaces for which the Disteli-axis is constant. Thus, from Equation (42), we have the following ordinary differential equation:
(53)X′′′+κ~2X′=0.Then, without loss of generality, we may assumeX′0=0,1,0. Under such initial condition, a space-like dual unit vector X′ is given by
(54)X′s^=A1sinκ~s^F1+cosκ~s^+A2sinκ~s^F2+A3sinκ~s^F3,where A1, A2, and A3 are some dual constants satisfying A12−A32=1 and A2=0. From this, we can obtain
(55)Xs^=−ρ~A1cosκ~s^+D1F1+ρ~sinκ~s^F2+−ρ~A3cosκ~s^+D3F3,where D1, D3 are some dual constants satisfying A3D3−A1D1=0 and D32−D12=ρ~2+1. If we adopt the dual coordinate transformation such that
(56)X1¯X2¯X3¯=A10−A3010−A30A1X1X2X3,with respect to the new coordinates Xi¯, the dual unit vector Xs^ becomes
(57)Xs^=ρ~cosκ~s^F1+ρ~sinκ~s^F2+DF3,for a constant dual D=A1D3−A3D1, with D=∓coshΨ. It is noted that Xs^ does not depend on the choice of the lower sign or upper sign of ∓. Therefore, through the paper, we choose the upper sign, that is,
(58)XΘ=sinhΨcosκ~s^F1+sinhΨsinκ~s^F2−coshΨF3,where Θ=ϑ+εϑ∗=κ~s^. This means that the time-like lines B and F3 are coincident, and
(59)ψ=f1realconst.,ψ∗=f2realconst..Sinceϑ and ϑ∗ are two-independent parameters, we can say that X is, in general, a time-like line congruence in Lf-space. However, by separating the real and dual parts of Equation (58), respectively, we have
(60)xϑ,ϑ∗=sinhψcosϑ,sinhψsinϑ,−coshψ,(61)x∗ϑ,ϑ∗=x1∗x2∗x3∗=ψ∗cosϑcoshψ−ϑ∗sinϑsinhψψ∗sinϑcoshψ+ϑ∗cosϑsinhψ−ψ∗sinhψ.Letαα1,α2,α3 denote a point on X. Since α×x=x∗, we have the system of linear equations in α1,α2, and α3:
(62)−α3sinϑsinhψ−α2coshψ=x1∗,α3cosϑsinhψ+α1coshψ=x2∗,−α1sinϑsinhψ+α2cosϑsinhψ=x3∗.The matrix of coefficients of unknownsα1,α2, and α3 is
(63)0−coshψ−sinϑsinhψcoshψ0cosϑsinhψ−sinϑsinhψcosϑsinhψ0,and therefore, its rank is 2 with ψ≠0 and ϑ≠0. In addition, the rank of the augmented matrix
(64)0coshψ−sinϑsinhψx1∗−coshψ0cosϑsinhψx2∗−sinϑsinhψcosϑsinhψ0x3∗,is 2. Hence, this system has infinitely many solutions represented with
(65)cα1=ψ∗sinϑ+ϑ∗−α3tanhψcosϑ,(66)α2=−ψ∗cosϑ+ϑ∗−α3tanhψsinϑ,(67)−α1sinϑsinhψ+α2cosϑsinhψ=x3∗.Sinceα3 is taken at random, then we may take ϑ∗−α3=0. In this case, Equation (65) reduces to
(68)α1ϑ=ψ∗sinϑ,α2ϑ=−ψ∗cosϑ,α3ϑ=−ϑ∗.Thus, the director surface of this time-like line congruence is given by(69)αϑ,ϑ∗=ψ∗sinϑ,−ψ∗cosϑ,−ϑ∗.Letmm1,m2,m3 denote a point on this time-like line congruence. Hence, we obtain
(70)mϑ,ϑ∗,v=αϑ,ϑ∗+vxϑ,ϑ∗,v∈ℝ,which consists of a family of time-like ruled surfaces mϑ,ϑ0∗,v, mϑ0,ϑ∗,v, and mϑt,ϑ∗t,v. Here, ϑ0∗, ϑ0, and t are real constants. By means of Equations (60), (68), and (70), we simply find that
(71)cm1=ψ∗sinϑ+vsinhψcosϑ,m2=−ψ∗cosϑ+vsinhψsinϑ,m3=−ϑ∗−vcoshψ,or by eliminating ϑ, we have
(72)X:m12ψ∗2+m22ψ∗2−M32n2=1,where n=ψ∗cothψ and M3=m3+ϑ∗. Then, X is two-parameter Lorentzian spheres. The intersection of each Lorentzian sphere and the corresponding space-like plane M3≔m3+ϑ∗=0 is m12+m22=ψ∗2. Therefore the envelope of X is the time-like cylinder m12+m22=ψ∗2. Notice that if ϑ∗=0, then
(73)X:m12ψ∗2+m22ψ∗2−m32n2=1.


#### 3.2.1. Constant Parameter Time-like Ruled Surfaces

A relation such asFϑ,ϑ∗=0, between the parameters, restricts Equation (58) (resp. (70)) to a one-parameter set of time-like lines, that is, a time-like ruled surface in the congruence. Therefore, if we select ϑ∗=hϑ, h indicating the pitch of the motion ℍ+m2/ℍ+f2, and ϑ as the motion parameter, then Equation (58) (resp. (70)) performs a time-like ruled surface in Lf-space. Thus,
(74)XTG=sinhΨcosΘsinhΨsinΘ−coshΨ−sinΘcosΘ0coshΨcosΘcoshΨsinΘ−sinhΨF1F2F3.In this case, we get(75)ddϑXTG=01+εhsinhΨ01+εhsinhΨ01+εhcoshΨ0−1+εhcoshΨ0XTG.Thereby, the Blaschke invariants are(76)P=1+εhsinhΨ,Q=1+εhcoshΨ,cothΨ=QP.From the real and dual parts of the last equations, we obtain(77)μ=h+ψ∗cothψ,Γ=h+ψ∗tanhψ,γ=cothψ,where μ, Γ, and γ are constants. Hence, (X) is a constant parameter time-like ruled surface. We now simply find the base curve as
(78)αϑ=ψ∗sinϑ,−ψ∗cosϑ,−hϑ.It can be shown that<α′,x′>=0, so the base curve of (X) is its striction curve. Also, it can be show that αϑ is a space-like (resp., a time-like) if and only if ψ∗>h (resp., ψ∗<h). For the curvature κ, and the torsion τ, we can find the following calculations simply
(79)κϑ=ψ∗ψ∗2−h2,τϑ=hψ∗2−h2.Hence,αϑ is a space-like (resp., a time-like) helix if and only if ψ∗>h (resp. ψ∗<h). Furthermore, we have
(80)X:mϑ,v=ψ∗sinϑ+vc1cosϑ,−ψ∗cosϑ+vc1sinϑ,−hϑ−c2v,where c1=sinhψ, c2=coshψ, ψ=0.7, ϑ∈0,2π, and v∈−4,4. According to Equation (80), we have the following:
(1)
Time-like general helicoids: for h=0.5 (resp., h=1 and ψ∗=1 (resp., ψ∗=0.5), the graph of the surfaces are shown in Figures 4 and 5, respectively(2)
Lorentzian sphere: for h=0, ψ∗=1, the graph of the surface is shown in Figure 6(3)
Time-like cone: Figure 7 shows the surface X with ψ∗=h=0Figure 4
X with h=1 and ψ∗=0.5.Figure 5
X with h=0.5 and ψ∗=1.Figure 6
X with h=0 and ψ∗=1.Figure 7
X with h=0=ψ∗=0.


## 3.1. Kinematic-Geometry and Time-like Plücker Conoid

We now are interested in researching the kinematic geometry of (X). Therefore, we are going to make a detailed study of the Blaschke invariants Pt and Qt. To carry this out, from Equation (15), we can write the following equations:
(25)X′t=ΩB×X,T′t=ΩB×T,G′t=ΩB×G.Hence, at any instant, it is seen thatΩ=Ω=ω+εω∗ is the dual angular speed of the motion ℍ+m2/ℍ+f2 about the Disteli-axis B. Thus,
(26)ωt=q2−p2,ω∗t=qq∗−pp∗q2−p2,withq>p,corresponding to the rotational angular speed and translational angular speed of the motion Lm/Lf along B, respectively.Hence, the next corollary can be given:Corollary 3.
During the motionLm/Lf, at any instant t, the pitch can be given by
(27)ht≔<ω,ω∗>ω2=Γcosh2ψ−μsinh2ψ.It is clear that if the dual vectorΩ=ω+εω∗ is given, then the following can be specified:
(i)
The time-like Disteli-axisB is specified by Equation (20)(ii)
The dual angular speed isΩ=ω1+εh(iii)
Ify is a point on the time-like Disteli-axis B, then(28)yt,v=b×b∗+vb,v∈ℝ,is a nondevelopable time-like ruled surface B(iv)
If the motionLm/Lf is pure rotation, that is, hu=0, then(29)Bt=bt+εb∗t=1ωω+εω∗.It is interesting to note that ifht=0, and ω2=−1, then Ω is a time-like oriented line. However, in the case when the motion is pure translational, i.e., Ω=0+εω∗, we set ω∗=Ω∗,ω∗b=ω∗ and select an arbitrary b∗under ω∗≠0; otherwise, the time-like unit vector b can be chosen arbitrarily, too.The Equations in (24) and (27), respectively, are Minkowski versions of the Mannheim and Hamilton formulae of surface theory in Euclidean 3-space. Now, we give geometric interpretations of these formulae. Next, the parametric representation of ψ∗ is time-like Plücker conoid or cylindroid as follows: consider T is coincident with the space-like y-axis of a fixed Minkowski frame oxyz, while the position of the time-like dual unit vector B is given by angle ψ and distance ψ∗ on the space-like positive y-axis. The time-like dual unit vector X and the space-like dual unit vector G can be selected in sense of x- and z-axes, respectively. This shows that the dual unit vectors X and G together with T form the coordinate system, as shown in Figure 2. If y is a point on this time-like Plücker conoid, then we have
(30)M:yt,v=0,ψ∗,0+vcoshψ,0,sinhψ,v∈ℝ.Figure 2
B=coshΨX+sinhΨG.Using such representation, the time-like dual unit vectorsB are obviously visible crossing through the y-axis. It is easily seen from the last equation and (24) that
(31)ψ∗≔y=12μ−Γsinh2ψ,x=vcoshψ,z=vsinhψ.Here,ψ∗ gives us the intersection point of the oriented lines X and G which locates at a half of the conoid height. By direct calculations,
(32)x2−z2y−μ−Γxz=0,which is the algebraic equation for time-like Plücker conoid. The time-like Plücker conoid given by Equation (32) which has two structure functions of the first order and it depends only on their difference; μ−Γ=1,−0.9≤ψ≤0.9, −1.5≤v≤1.5 (Figure 3). Furthermore, one can get a second-order equation in x/z in which its solutions are given by
(33)xz=12yμ−Γ±μ−Γ2−4y2.Figure 3
Time-like Plücker conoid.By equating the discriminant of Equation (33) to zero, we get the limits of the time-like Plücker conoid. Hence, the two extreme positions are given by
(34)y=±μ−Γ2,which gives the positions of the two torsal planes π1, π2, and each one of them contains one torsal line L. Further, hu in Equation (27) is not a periodic function and has at most two extreme values, the structure functions Γ and μ. Therefore, the oriented lines X and G are principal axes of the time-like Plücker conoid. However, the geometric aspects are discussed as follows:
(1)
Ifht≠0, then there are two real torsal lines L1 and L2 passing through the point 0,y,0 only if y<μ−Γ/2; for the two limit points y=±μ−Γ/2, they coincide with the principal axes X and G(2)
Ifht=0, then the two torsal lines L1 and L2 are represented by(35)xz≔cothψ=±μΓ,y=±Γμ.Equation (35) means that the two torsal lines L1 andL2 are orthogonal each other in the Lorentzian sense. Furthermore, transition from polar coordinates to Cartesian coordinates could be completed by substituting
(36)x=coshψh,z=sinhψh,into Hamilton’s formula; one obtains the equation
(37)D:Γx2−μz2=1,of a conic section. This conic section is a Minkowski version of the Dupin indicatrix of surfaces theory in Euclidean 3-space.


### 3.1.1. Serret–Frenet Motion

IfX is a time-like developable ruled surface, that is, μ=0, in this case, Dupin’s indicatrix is a set of parallel lines represented by x2=1/Γ, and the Blaschke frame {x,t,g} coincides with the classical Serret–Frenet frame, and then, the striction curve C becomes the edge of regression of X. Hence, p and q are the curvature κ and the torsion τ of C, respectively. Moreover, q∗=1 and Γτ=1 (Γ is the radius of torsion of C). By similar argument, we can also have the following equations:
(38)ht=1τcosh2ψ=ττ2−κ2,ψ∗t=−12Γsinh2ψ=−κτ2−κ2.The corresponding time-like Plücker conoid is(39)M:yt,v=0,ψ∗,0+vcoshψ,0,sinhψ,v∈ℝ.This yields(40)x2−z2y+Γxz=0.


## 3.1.1. Serret–Frenet Motion

IfX is a time-like developable ruled surface, that is, μ=0, in this case, Dupin’s indicatrix is a set of parallel lines represented by x2=1/Γ, and the Blaschke frame {x,t,g} coincides with the classical Serret–Frenet frame, and then, the striction curve C becomes the edge of regression of X. Hence, p and q are the curvature κ and the torsion τ of C, respectively. Moreover, q∗=1 and Γτ=1 (Γ is the radius of torsion of C). By similar argument, we can also have the following equations:
(38)ht=1τcosh2ψ=ττ2−κ2,ψ∗t=−12Γsinh2ψ=−κτ2−κ2.The corresponding time-like Plücker conoid is(39)M:yt,v=0,ψ∗,0+vcoshψ,0,sinhψ,v∈ℝ.This yields(40)x2−z2y+Γxz=0.


## 3.2. Time-like Ruled Surfaces with Constant Disteli-Axis

A time-like ruled surfaceX is defined as a constant Disteli-axis ruled surface if the dual angle between the ruling of X and the Disteli-axis is always constant. Thus, when we say X is a time-like constant Disteli-axis, we mean that all the rulings of X have a constant Lorentzian dual angle from its Disteli-axis.The dual arc lengthds^=ds+εds∗ of Xt∈ℍ+2 is
(41)s^t=∫0tPdt=∫0tp1+εμdt.After that, we will use the dual arc length parameters^ instead of t. If the prime means to differentiation as s^, then from Equation (15), we get
(42)X′T′G′=01010Σ0−Σ0XTG=Ω×XTG,where Ω=ΣX+G. Thus, we may write the following relationships:
(43)κ^s^≔κ+εκ∗=Σ2−1=1sinhψ=1ρ~,τ^s^≔τ+ετ∗=±Σ′Σ2−1=±Ψ′,whereκ^ and τ^ are the dual curvature function and the dual torsion function of Xs^∈ℍ+2, respectively. The terms found in Equation (43) are such as to their counterparts in 3-dimensional hyperbolic spherical geometry.Definition 4.
For a one-parameter hyperbolic dual motion, at an instants^∈D, an oriented time-like line Z in fixed space will be said to be time-like Bk-Disteli-axis of X if for all i such that 1≤i≤k, <Z,Xis^>=0, but <Z,Xk+1s^>≠0. Here, Xi denotes the i-th derivatives of X.Via this definition, consider the Lorentzian dual angle(44)ρ~=cosh−1<Z,X>,such that X and Z have the same time orientation and Z and ρ~ stay fixed up to the second order at s^=s^0, i.e.,
(45)ρ~′s^=s^0=0,X′s^=s^0=0,ρ~′′s^=s^0=0,X′′s^=s^0=0.We have for the first order(46)<X′,Z>=0,and for the second-order properties
(47)<X′′,Z>∣=0.Then,ρ~ will be invariant in the second approximation if and only if Z is the time-like Disteli-axis B of X, that is,
(48)ρ~′=ρ~′′=0⇔Z=X′×X′′X′×X′′=±B.By the definition of the time-like Disteli-axis, we have the dual frame(49)U1=Bs^,U2s^=B′B′,U3s^=B×U2,as the Blaschke frame along B. Thus, the calculations give that
(50)U1U2U3=coshΨ0sinhΨsinhΨ0coshΨ0−10XTG.The variations of this frame are analogous to Equation (15) and are given by
(51)U1′U2′U3′=0Ψ′0Ψ′0κ^0−κ^0U1U2U3=Ω~×U1U2U3,where Ω~s^=κ^U1+Ψ′U3. Comparing Equation (42) with Equation (51), we have that the relative dual velocity is
(52)Ω−Ω~=Ψ′T.This shows that the Blaschke frame involves a further rotation around the central tangentT, whose speed equals the dual torsion τ^s. Hence, we obtain that if τ^s^=τ+ετ∗=0(Σ′t=γ′t+εΓt−μtγt′=0), that is, ψ and ψ∗ are constants, then the time-like Disteli-axis is fixed up to the second order and the time-like line X moves on it with constant pitch h. Thus kinematically, the time-like ruled surface X is traced by a hyperbolic one-parameter screw motion of constant pitch h along the constant time-like Disteli-axis B, by the time-like line X at a constant Lorentzian distance ψ∗ and constant Lorentzian angle ψ relative to B. Hence, we have the following.Theorem 5.
(X) is a time-like ruled surface with constant Disteli-axis if and only if (a) γ=constant and (b) Γ−μγ=constant.Now, we construct the time-like ruled surfaces for which the Disteli-axis is constant. Thus, from Equation (42), we have the following ordinary differential equation:
(53)X′′′+κ~2X′=0.Then, without loss of generality, we may assumeX′0=0,1,0. Under such initial condition, a space-like dual unit vector X′ is given by
(54)X′s^=A1sinκ~s^F1+cosκ~s^+A2sinκ~s^F2+A3sinκ~s^F3,where A1, A2, and A3 are some dual constants satisfying A12−A32=1 and A2=0. From this, we can obtain
(55)Xs^=−ρ~A1cosκ~s^+D1F1+ρ~sinκ~s^F2+−ρ~A3cosκ~s^+D3F3,where D1, D3 are some dual constants satisfying A3D3−A1D1=0 and D32−D12=ρ~2+1. If we adopt the dual coordinate transformation such that
(56)X1¯X2¯X3¯=A10−A3010−A30A1X1X2X3,with respect to the new coordinates Xi¯, the dual unit vector Xs^ becomes
(57)Xs^=ρ~cosκ~s^F1+ρ~sinκ~s^F2+DF3,for a constant dual D=A1D3−A3D1, with D=∓coshΨ. It is noted that Xs^ does not depend on the choice of the lower sign or upper sign of ∓. Therefore, through the paper, we choose the upper sign, that is,
(58)XΘ=sinhΨcosκ~s^F1+sinhΨsinκ~s^F2−coshΨF3,where Θ=ϑ+εϑ∗=κ~s^. This means that the time-like lines B and F3 are coincident, and
(59)ψ=f1realconst.,ψ∗=f2realconst..Sinceϑ and ϑ∗ are two-independent parameters, we can say that X is, in general, a time-like line congruence in Lf-space. However, by separating the real and dual parts of Equation (58), respectively, we have
(60)xϑ,ϑ∗=sinhψcosϑ,sinhψsinϑ,−coshψ,(61)x∗ϑ,ϑ∗=x1∗x2∗x3∗=ψ∗cosϑcoshψ−ϑ∗sinϑsinhψψ∗sinϑcoshψ+ϑ∗cosϑsinhψ−ψ∗sinhψ.Letαα1,α2,α3 denote a point on X. Since α×x=x∗, we have the system of linear equations in α1,α2, and α3:
(62)−α3sinϑsinhψ−α2coshψ=x1∗,α3cosϑsinhψ+α1coshψ=x2∗,−α1sinϑsinhψ+α2cosϑsinhψ=x3∗.The matrix of coefficients of unknownsα1,α2, and α3 is
(63)0−coshψ−sinϑsinhψcoshψ0cosϑsinhψ−sinϑsinhψcosϑsinhψ0,and therefore, its rank is 2 with ψ≠0 and ϑ≠0. In addition, the rank of the augmented matrix
(64)0coshψ−sinϑsinhψx1∗−coshψ0cosϑsinhψx2∗−sinϑsinhψcosϑsinhψ0x3∗,is 2. Hence, this system has infinitely many solutions represented with
(65)cα1=ψ∗sinϑ+ϑ∗−α3tanhψcosϑ,(66)α2=−ψ∗cosϑ+ϑ∗−α3tanhψsinϑ,(67)−α1sinϑsinhψ+α2cosϑsinhψ=x3∗.Sinceα3 is taken at random, then we may take ϑ∗−α3=0. In this case, Equation (65) reduces to
(68)α1ϑ=ψ∗sinϑ,α2ϑ=−ψ∗cosϑ,α3ϑ=−ϑ∗.Thus, the director surface of this time-like line congruence is given by(69)αϑ,ϑ∗=ψ∗sinϑ,−ψ∗cosϑ,−ϑ∗.Letmm1,m2,m3 denote a point on this time-like line congruence. Hence, we obtain
(70)mϑ,ϑ∗,v=αϑ,ϑ∗+vxϑ,ϑ∗,v∈ℝ,which consists of a family of time-like ruled surfaces mϑ,ϑ0∗,v, mϑ0,ϑ∗,v, and mϑt,ϑ∗t,v. Here, ϑ0∗, ϑ0, and t are real constants. By means of Equations (60), (68), and (70), we simply find that
(71)cm1=ψ∗sinϑ+vsinhψcosϑ,m2=−ψ∗cosϑ+vsinhψsinϑ,m3=−ϑ∗−vcoshψ,or by eliminating ϑ, we have
(72)X:m12ψ∗2+m22ψ∗2−M32n2=1,where n=ψ∗cothψ and M3=m3+ϑ∗. Then, X is two-parameter Lorentzian spheres. The intersection of each Lorentzian sphere and the corresponding space-like plane M3≔m3+ϑ∗=0 is m12+m22=ψ∗2. Therefore the envelope of X is the time-like cylinder m12+m22=ψ∗2. Notice that if ϑ∗=0, then
(73)X:m12ψ∗2+m22ψ∗2−m32n2=1.


### 3.2.1. Constant Parameter Time-like Ruled Surfaces

A relation such asFϑ,ϑ∗=0, between the parameters, restricts Equation (58) (resp. (70)) to a one-parameter set of time-like lines, that is, a time-like ruled surface in the congruence. Therefore, if we select ϑ∗=hϑ, h indicating the pitch of the motion ℍ+m2/ℍ+f2, and ϑ as the motion parameter, then Equation (58) (resp. (70)) performs a time-like ruled surface in Lf-space. Thus,
(74)XTG=sinhΨcosΘsinhΨsinΘ−coshΨ−sinΘcosΘ0coshΨcosΘcoshΨsinΘ−sinhΨF1F2F3.In this case, we get(75)ddϑXTG=01+εhsinhΨ01+εhsinhΨ01+εhcoshΨ0−1+εhcoshΨ0XTG.Thereby, the Blaschke invariants are(76)P=1+εhsinhΨ,Q=1+εhcoshΨ,cothΨ=QP.From the real and dual parts of the last equations, we obtain(77)μ=h+ψ∗cothψ,Γ=h+ψ∗tanhψ,γ=cothψ,where μ, Γ, and γ are constants. Hence, (X) is a constant parameter time-like ruled surface. We now simply find the base curve as
(78)αϑ=ψ∗sinϑ,−ψ∗cosϑ,−hϑ.It can be shown that<α′,x′>=0, so the base curve of (X) is its striction curve. Also, it can be show that αϑ is a space-like (resp., a time-like) if and only if ψ∗>h (resp., ψ∗<h). For the curvature κ, and the torsion τ, we can find the following calculations simply
(79)κϑ=ψ∗ψ∗2−h2,τϑ=hψ∗2−h2.Hence,αϑ is a space-like (resp., a time-like) helix if and only if ψ∗>h (resp. ψ∗<h). Furthermore, we have
(80)X:mϑ,v=ψ∗sinϑ+vc1cosϑ,−ψ∗cosϑ+vc1sinϑ,−hϑ−c2v,where c1=sinhψ, c2=coshψ, ψ=0.7, ϑ∈0,2π, and v∈−4,4. According to Equation (80), we have the following:
(1)
Time-like general helicoids: for h=0.5 (resp., h=1 and ψ∗=1 (resp., ψ∗=0.5), the graph of the surfaces are shown in Figures 4 and 5, respectively(2)
Lorentzian sphere: for h=0, ψ∗=1, the graph of the surface is shown in Figure 6(3)
Time-like cone: Figure 7 shows the surface X with ψ∗=h=0Figure 4
X with h=1 and ψ∗=0.5.Figure 5
X with h=0.5 and ψ∗=1.Figure 6
X with h=0 and ψ∗=1.Figure 7
X with h=0=ψ∗=0.


## 3.2.1. Constant Parameter Time-like Ruled Surfaces

A relation such asFϑ,ϑ∗=0, between the parameters, restricts Equation (58) (resp. (70)) to a one-parameter set of time-like lines, that is, a time-like ruled surface in the congruence. Therefore, if we select ϑ∗=hϑ, h indicating the pitch of the motion ℍ+m2/ℍ+f2, and ϑ as the motion parameter, then Equation (58) (resp. (70)) performs a time-like ruled surface in Lf-space. Thus,
(74)XTG=sinhΨcosΘsinhΨsinΘ−coshΨ−sinΘcosΘ0coshΨcosΘcoshΨsinΘ−sinhΨF1F2F3.In this case, we get(75)ddϑXTG=01+εhsinhΨ01+εhsinhΨ01+εhcoshΨ0−1+εhcoshΨ0XTG.Thereby, the Blaschke invariants are(76)P=1+εhsinhΨ,Q=1+εhcoshΨ,cothΨ=QP.From the real and dual parts of the last equations, we obtain(77)μ=h+ψ∗cothψ,Γ=h+ψ∗tanhψ,γ=cothψ,where μ, Γ, and γ are constants. Hence, (X) is a constant parameter time-like ruled surface. We now simply find the base curve as
(78)αϑ=ψ∗sinϑ,−ψ∗cosϑ,−hϑ.It can be shown that<α′,x′>=0, so the base curve of (X) is its striction curve. Also, it can be show that αϑ is a space-like (resp., a time-like) if and only if ψ∗>h (resp., ψ∗<h). For the curvature κ, and the torsion τ, we can find the following calculations simply
(79)κϑ=ψ∗ψ∗2−h2,τϑ=hψ∗2−h2.Hence,αϑ is a space-like (resp., a time-like) helix if and only if ψ∗>h (resp. ψ∗<h). Furthermore, we have
(80)X:mϑ,v=ψ∗sinϑ+vc1cosϑ,−ψ∗cosϑ+vc1sinϑ,−hϑ−c2v,where c1=sinhψ, c2=coshψ, ψ=0.7, ϑ∈0,2π, and v∈−4,4. According to Equation (80), we have the following:
(1)
Time-like general helicoids: for h=0.5 (resp., h=1 and ψ∗=1 (resp., ψ∗=0.5), the graph of the surfaces are shown in Figures 4 and 5, respectively(2)
Lorentzian sphere: for h=0, ψ∗=1, the graph of the surface is shown in Figure 6(3)
Time-like cone: Figure 7 shows the surface X with ψ∗=h=0Figure 4
X with h=1 and ψ∗=0.5.Figure 5
X with h=0.5 and ψ∗=1.Figure 6
X with h=0 and ψ∗=1.Figure 7
X with h=0=ψ∗=0.


## 4. Conclusion

In this work, we analyze a certain class of time-like ruled with constant Disteli-axis in Minkowski 3-spaceE13. As a result, the time-like ruled surface generated by a time-like line undergoing a Lorentzian screw motion is examined in detail. We hope that the study of spatial kinematics in Minkowski 3-space E3 via line geometry may shed some light on present research problems and probably propose new ones.



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*Source: 9323490-2022-01-17.xml*