A Geometric Aspect of the Two-Parameter Planar Lorentzian Motions

We examined the moving coordinate systems, the polar axes, the density invariance of the polar axis transformation, and the curve plotter points and the support function of the two-parameter planar Lorentzian motion. Furthermore, we were concerned with the determination of the motion using the polar axes and analyzed the motion when the density of the polar axes is zero.


Introduction
As a branch of physics and a subdivision of classical mechanics, kinematics identifies the possible motion of points, objects (bodies), and systems of objects (bodies) geometrically without consideration of the effects (causes) of the motions.It deals with any motion of any object.For mathematics, kinematics is a bridge connecting geometry, physics, and mechanism, we can say that it means geometry of motion.Moreover, kinematics is important to astrophysics, mechanical engineering, physics, biomechanics and robotics.The geometry of such a one-or two-parameter motion of points, bodies, and systems of bodies has a number of applications in physics, geometric design, and design and trajectory of robotics.
Mathematicians, physicists, and mechanists have investigated a rigid body motion in different ways.In general, if a rigid body moves, both its orientation and position vary with time.In the kinematic meaning, these changes are called rotation and translation, respectively.A rigid body, programmed to move in a plane with a one/two independent degree/degrees of freedom is defined as a one-/two-parameter planar motion.
In kinematics, W. Blaschke and H. R. Müller defined the one-parameter planar motions and obtained the relation between absolute, relative, and sliding velocities and accelerations in the Euclidean plane in 1956 [1].Oneparameter motions on Lorentzian plane and Galilean plane are described by A. A. Ergin and M. Akar and S. Yüce, respectively [2,3].They also gave the relations between the velocities and accelerations.Furthermore, A. A. Ergin investigated Lorentzian moving planes and pole points [4].One-parameter planar motions are given in affine Cayley Klein planes (CK-planes) by generalizing of the motions in Euclidean, Galilean, and Lorentzian planes in [5].In [6], the authors expressed the higher-order velocities, accelerations, and poles under the one-parameter planar hyperbolic motions and their inverse motions.The higher-order accelerations and poles are also presented by considering the rotation angle as a parameter of the motion and its inverse motion.
Two-parameter planar motions are investigated with different but equivalent definitions [1,7,8].The two-parameter planar Euclidean motions were introduced by W. Blaschke and H. R. Müller.The polar axes of two-parameter motion, the curve plotter points, the density invariance of the polar axes transformation, the support function of two-parameter motion, the normed coordinate system, and the main oneparameter motion obtained from the two-parameter motion have been studied in the Euclidean plane [1].Local pictures for a general two-parameter planar motions are investigated by C. G. Gibson, W. Hawes, and C. A. Hobbs in [9] and singularities of a general planar motions with two degrees of freedom are studied by C. G. Gibson, D. Marsh, and Y. Xiang in [10].

Mathematical Problems in Engineering
A general planar Euclidean motion is defined by the equations If the functions ,  1 , and  2 are given by continuously differentiable functions of the parameters  1 and  2 , then the motion is called a two-parameter motion.Here, if the parameters  1 and  2 are functions of , then a one-parameter planar motion is obtained [1,7].By accepting a certain asymmetry, we can take  as the parameter.In this case, we obtain a special two-parameter planar Euclidean motion [8].After their contributions, numerous studies conducted hitherto on two-parameter motions in Euclidean and non-Euclidean planes have been examined [11][12][13][14][15][16][17] .In these papers, one-parameter planar motions obtained from twoparameter planar motions, the geometric locus of Hodograph of any points, and acceleration poles of the motions are examined in Euclidean, complex, hyperbolic, Lorentzian, and Galilean planes.It is proved that the pole points at any position lie on a line in the fixed and the moving planes.The instantaneous kinematics of a special two-parameter motion was investigated in [18].Two-parametric motions are given in the Lobatchevski plane in [19].Besides, the two-parameter motion is used for biomechanical modeling of left ventricle (LV) using cardiac tagged magnetic resonance imaging data in biomedical area and for modeling of effective elastic tensor for cortical bone in biomechanics area [20,21].
The purpose of this paper is to combine the field of twoparameter planar motion with Lorentzian geometry.We aim to develop the theory of the two-parameter planar Lorentzian motion by considering geometric aspects.In the second section, we review some of the standard facts on one-and two-parameter planar motions.In Section 3, which is actually the original part of our study, we intend to motivate our investigation of two-parameter planar Lorentzian motions in the way that W. Blaschke and H. R. Müller examine two-parameter motions in the Euclidean plane.Lorentzian plane geometry has similarities and fundamental differences from Euclidean plane geometry in the large.First of all, we define two-parameter planar Lorentzian motion by the help of hyperbolic statements.And then we analyze the moving coordinate systems, the polar axes, the density invariance of the polar axis transformation, zero-density of polar axes, the curve plotter points, and the support function of the motion.All in all, we are interested in the determination of the motion using the polar axes.

Preliminaries
Let us firstly examine the basic concepts related to Lorentzian plane.
The Lorentzian plane  is the vector space R 2 provided with Lorentzian inner product ⟨, ⟩  given by where  = ( Definition 1.A time-like line (or a space-like line) with respect to the coordinate system {; l 1 , l 2 } in Lorentzian plane will be denoted by   (or   ) and it is defined by the equation with the Hesse-coordinates (, Ψ), where  is the distance from the line to the origin and the direction angle Ψ ∈ R [23].
Definition 2. The measure of a set of points  = (, ) is defined by the integral, over the set, of the differential forms which is called the density for points  [23,24].
Definition 3. The measure of a set of non null lines (, Ψ) in the Lorentzian plane is defined by the integral, over the set, of the differential form which is called the density for non null lines  [23].
Let us talk about one-parameter motions in Lorentzian and hyperbolic planes and examine the two-parameter planar Lorentzian motions.It is necessary to describe the oneparameter planar motions in order to obtain the twoparameter planar motions.The classical reference on kinematics is [1].In this book, W. Blaschke and H. R. Müller defined one-parameter motions in the Euclidean and complex planes.A. A. Ergin considered the Lorentzian plane instead of the Euclidean plane and defined the one-parameter planar Lorentzian motions [2].He also gave the relations between absolute, relative, and sliding velocities (and accelerations) in 1991.

One-Parameter Motions in Lorentzian and Hyperbolic
Planes.Let us firstly give one-parameter motions in Lorentzian plane.Definition 4. Let  and   be moving and fixed Lorentzian planes.Let {; l 1 , l 2 } and {  ; l 1  , l 2  } be their orthonormal coordinate systems, respectively.Let us take the points  = (, ) and   = (  ,   ) with respect to the moving and fixed coordinate systems, respectively, and A general planar Lorentzian motion is given by the equations If Θ,  1 , and  2 are given by continuously differentiable functions of a time parameter , then the motion is called oneparameter planar Lorentzian motion.We will use   = /  to denote one-parameter planar motion in the Lorentzian plane [2].
A. A. Ergin also studied three Lorentzian planes and investigated relative, sliding, and absolute velocities [4].Let A and  be moving Lorentzian planes and   be fixed Lorentzian plane and {; b 1 , b 2 }, {; l 1 , l 2 }, and {  ; l 1  , l 2  } be their coordinate systems, respectively.Assume that Φ and Φ  are rotation angles of one-parameter planar Lorentzian motions A/ and A/  .In [4], the pole points of the Lorentzian motions A/ and A/  are examined.Let  = (, ) be a moving point on the plane A. Since the vector equations can be written, we have Assume that  denotes the differential with respect to the Lorentzian plane  and   denotes the differential with respect to the Lorentzian plane   .For the sake of shortness let us use Hence we can give the following definition.
Definition 5. Δ, Δ  , Σ  , and Σ   are called the Pfaff forms of one-parameter planar Lorentzian motion with respect to the time parameter , where 1 ≤  ≤ 2.
The derivative equations of the motion A/ are and the derivative equations of the motion A/  , by taking   a =   a  , are We write differentiation of the point  in the plane  as Definition 6.The velocity of  with respect to  is called the relative velocity and it is denoted by V  = x/.
Definition 7. The velocity of  with respect to   is called the absolute velocity and it is denoted by If V  is equal to zero, then  is a fixed point on  and if V  is equal to zero, then  is a fixed point on   .Therefore, the conditions that the point  is fixed in planes  and   can be obtained as follows: Definition 8.The expression is called the sliding velocity vector of the motion and the sliding velocity is defined by The pole point is characterized by vanishing the sliding velocity at the time .Hence, the relative velocity equals the absolute velocity.During the motion, the pole points do not move in both planes.For the pole points  = ( 1 ,  2 ) ∈  we write or if we take   x as zero, the pole point ( 1 ,  2 ) of the motion is obtained as = (  ,   ) is the image of the point  and are the images of time-like line   (, Ψ) and space-like line   (, Ψ) under the motion, respectively.Here,    (  , Ψ  ) is defined with help of the direction angle Θ + Ψ = Ψ  and the distance   = − 1 cosh Ψ+ 2 sinh Ψ from the origin   and similarly    (  , Ψ  ) is defined with help of the direction angle Θ + Ψ = Ψ  and the distance   =  −  1 sinh Ψ +  2 cosh Ψ from the origin   [25].
Let us give the one-parameter motions in hyperbolic plane.Motion in hyperbolic plane is congruent to the motion in Lorentzian plane.Because of the fact that there is a strict correspondence between Lorentzian plane and hyperbolic plane similar to the complex plane and Euclidean plane, S. Yüce and N. Kuruo g lu defined one-parameter motions in the hyperbolic plane [26].
Hyperbolic numbers can be introduced as an extension of the real numbers.This extension is obtained by including the hyperbolic imaginary , where  2 = 1 but  ̸ = ∓1.In this case, the hyperbolic numbers set can be written as follows: The hyperbolic numbers have been also called split-complex numbers, perplex numbers, or double numbers [27].The collection of all hyperbolic numbers is called the hyperbolic plane H.

Definition 9.
Let H and H  be moving and fixed hyperbolic planes.Let {; h 1 , h 2 } and {  ; h 1  , h 2  } be their orthonormal coordinate systems, respectively.Then, the one-parameter planar hyperbolic motion is defined by the equation and denoted by H/H  , where Θ is the rotation angle of the motion and the hyperbolic numbers x =  + , x  =   +   represent the point  ∈ H with respect to the moving and the fixed rectangular coordinate systems, respectively.The hyperbolic number O  O = u  = −u Θ represents the origin point  of the moving system in the fixed coordinate system.
Moreover, the angle Θ and x  , x, and u  are continuously differentiable functions of a time parameter .The vector equation is called the absolute velocity of the motion.To avoid the pure translation, we assume that Θ/ = Θ ̸ = 0.The vector equation is called the sliding velocity of the motion.Suppose that V  = 0, and then we get the pole points ( 1 ,  2 ) ∈ H and   (  1 ,   2 ) ∈ H  : 2.2.Two-Parameter Planar Lorentzian Motions.Let us examine the two-parameter planar Lorentzian motions.
Here, if  1 and  2 are functions of the time parameter , then one-parameter planar Lorentzian motion is obtained.Two-parameter planar Lorentzian motion given by ( 6) can be written in the form where  ∈ (2,1),  = − 1 cosh Θ −  2 sinh Θ,  = − 1 sinh Θ−  2 cosh Θ, and   and  are the position vectors of the same point  = (, ) and  is the translation vector.By taking derivatives of (23), we get where the velocities V a = d  , V f = +, and V r = d are called the absolute, the sliding, and the relative velocities of the point , respectively.The solution of the equation V f = 0 gives us the pole points  = ( 1 ,  2 ).M. K. Karacan and Y. Yaylı investigated one-parameter planar Lorentzian motions obtained from two-parameter planar Lorentzian motion [12].It is proved that the pole points at any position lie on a line in the fixed and the moving planes and the lengths of the velocity vectors of pole axes are the same.Furthermore, the locus of Hodograph of any point and acceleration poles of the motion are examined.

A New Aspect of the Two-Parameter Planar Lorentzian Motion with regard to the Pole Axes
This section is the original part of our paper.Our purpose is to give firstly the hyperbolic statement of two-parameter planar Lorentzian motion.After that, we study the moving coordinate systems, the polar axes, the density invariance of the polar axis transformation, the motion with zero-density of the polar axes, the curve plotter points, and the support function.(iv) If u/Θ ̸ = 0, then we can take the angle Θ as a parameter of the motion.Hence, we can write  1 = Θ and  2 = .In this case, the motion is called special twoparameter planar Lorentzian motion.We denoted it briefly by   .If  is a function of the angle Θ, special one-parameter planar Lorentzian motions   are obtained.

Two-Parameter
During the motion   , obtained from the motion   , the point  ∈  generally draws a curve segment in the plane   .

Example 12. The two-parameter planar Lorentzian motion
describes a surface in the plane   , Figure 1.
The one-parameter motions obtained from the motion in ( 25) describe the red curve and the blue curve in the plane   , respectively.These curves can be seen in Figure 1.

The Moving Coordinate Systems.
It is often necessary to describe the motion by a moving frame and plane.
Let A and A  be moving Lorentzian planes,  and   be fixed Lorentzian planes, and {; b 1 , b 2 } and {  ; b  1 , b  2 } and {; l 1 , l 2 } and {  ; l 1  , l 2  } be their coordinate systems, respectively.We will denote the rotation angles of twoparameter planar Lorentzian motions A/ and A  /  by Φ and Φ  , respectively.
Here are the derivative equations of the motions A/ and A  /  , respectively: Definition 13.Δ, Δ  , Σ  , and Σ   are called the Pfaff forms of the motions A/ and A  /  .They are the differential forms depending on two-parameter  1 and  2 , where 1 ≤  ≤ 2.
The Pfaff forms cannot be assumed arbitrarily for the motion; they must satisfy the integrability conditions which follow from the external derivation of ( 27) and ( 28) by Here, Δ and Δ  are complete differentials.The geometric meanings of the Pfaff forms Δ and Δ  are changing in the angles Φ and Φ  , respectively.These results will be needed in next subsections.
Remark 14.Using two different moving planes A and A  is an extension of the theory of moving coordinate system.We explain two-parameter planar Lorentzian motion /  by moving the coordinate system {  ; b  1 , b  2 } to the coordinate system {; b 1 , b 2 }.Therefore, we can have the relation between the planes A and A  .In Section 3.8, we define the function that moves the point  to the point   .

The Polar Axes of the Two-Parameter Planar Lorentzian
Motion.In this section, we will analyze the polar axes of the motion   in two ways.
First way is by using the hyperbolic statements of the motion.
Theorem 15.The pole points of the motions   obtained from   are on a time-like (space-like) line at each position in both moving and fixed planes.These lines are called the polar axes of the motion.Proof.Now, we will investigate the pole points  of the motions   obtained from the motion   .Since u = u(Θ, ) and  = (Θ), we can write and Substituting last equality into (22), we have If the position (Θ, ) of the motion   is fixed and /Θ of the motion   is changed, then u Θ is not equal to zero.We will denote by  and   the polar axes in the planes  and   .From (33), it can be concluded that the pole axis  passes through the point p 0 = u + u Θ and has the directrix vector k = u  .Similarly, the polar axis   passes through the point p  0 =  Θ u Θ and has the directrix vector is a space-like (or a time-like) vector, then the polar axes  and   are time-like (or space-like) lines.These complete the proof.Definition 16.The pole axes  and   correspond to each other in point-to-point.This is called the polar axis transformation.
Because of ‖k‖  = ‖k  ‖  = |u  |, the polar axes  and   are correspond to each other by equal velocity.
Second way is by using the moving coordinate system.
Theorem 17.The pole points of the motions   obtained from   are on a time-like (space-like) line at each position in both moving and fixed planes.These lines are called the polar axes of the motion.
Proof.We can write for the pole point  = ( 1 ,  2 ) of the motion   obtained from the motion   with the help of the moving coordinate system.Since the Pfaff forms are functions of two-parameter  1 and  2 , we can write where  2 / 1 = .Here, , , , and  are functions of the parameters  1 and  2 .From (34), it can be said that the pole point  is on a straight line passing through point (, ) and has the directrix vector (, ).If (, ) is a space-like (or a time-like) vector, then the polar axes  and   are space-like (or time-like) lines.

The Density Invariance of The Polar Axis Transformation
Theorem 18.The density of the time-like (or space-like) polar axis  in the plane  is equal to the density of the time-like (or space-like) polar axis   in the plane   under the motion   at any position.
Proof.The proof will be divided into two parts.We prove this theorem by using time-like and space-like polar axis separately.
(i) The Time-Like Polar Axis   .If we take that the polar axis coincides with the vector axis {; b 2 }, in Figure 2, then we have  1 = 0 for the pole point  =  1 b 1 +  2 b 2 .There is no loss of generality in assuming it.Hence, Σ  2 − Σ 2 = 0 is obtained from (34).If we take differentiation of the equality Σ  2 = Σ 2 and use the integrability conditions, we find that Since we can write Considering the derivative equations, we can write Then, using Δ = Φ = Ψ and  = Δ − Σ 1 , we obtain the following equation for the density of the polar axis   : The same proof works for the density of the polar axis in the plane   .This finishes the proof.
(ii) The Space-Like Polar Axis   .In analogy to the proof of the above part, it can be shown that   =    .

The Two-Parameter Planar Lorentzian Motion with Zerothe Density of the Polar Axis
Theorem 19.The two-parameter planar Lorentzian motion   with zero-the density of time-like (space-like) pole axis is defined by contacting the evolute space-like (time-like) curve  to its envelope space-like (time-like) curve   .Proof.We assume that the non null polar axes of the motion   have zero-density.In this case, and can be written.So that, , Ψ and   , Ψ  are linearly dependent.Thus,  can be expressed as a function of Ψ,  = (Ψ).Likewise, we can write that   =   (Ψ  ).The functions  and   define the one-parameter families of the non null lines ,   in the Lorentzian planes ,   , respectively.Let  and   be the envelope curves of the lines (Ψ),   (Ψ)  , respectively.Let  be an orthogonal curve to the family of lines .Since the tangent of  is perpendicular to the tangent of , and the curve  is called the evolute of the envelope , Figure 3.As a consequence of these, we can say "the motion   with zero pole axis density is defined by contacting the evolute curve  to its envelope curve   ."If the pole axis  is a time-like (or a space-like) line, then the envelope  will be a time-like (or a space-like) and hence the evolute curve  will be a space-like (or a time-like).
If we take the curves   and    which have distance  from the curves  and   , the same   motion is obtained.The motion   is independent from the choice of the distance .
Moreover, the motion can also be explained as follows.
The curves  ∈  and   ∈   roll without sliding on the pole axis  =   .Because of the decomposability of the motion   into two independent rolling movements, the motion is also called the separable.

The Curve Plotter Points
Theorem 20.The pole axes of the two-parameter planar Lorentzian motion are also the locus of the curve plotter points of the motion   .3.8.The Support Function.We have divided the support function into two parts.We give it by using time-like and space-like polar axis, separately.(54) By differentiating this equation, we see that Similarly, can be obtained in the plane   , so that a special case of the function , defined in Section 3.6, can be written as follows: (ii) The Space-Like Polar Axis   .If we take that the space-like polar axis   coincides with the vector axis { =  =   ; b 1 } and the space-like polar axis    coincides with the vector axis {  ; b 1  }, then we obtain that Let us examine how the support function H changes when the vectors k = V 1 l 1 +V 2 l 2 ∈  and k  = V  1 l 1  +V  2 l 2  ∈   rotate by the angles Φ 0 and Φ  0 , respectively.We investigate this by time-like and space-like polar axis separately.
(i) The Time-Like Polar Axis   .If we take into consideration the time-like polar axis   , Figure 6, the following change is obtained: Moreover, the differentiation of the function ∧ H can be written as follows: (ii) The Space-Like Polar Axis   .If we take into consideration the space-like polar axis   , then the following statement is obtained: where ∧  is the distance of k to the space-like polar axis   and ∧   is the distance of k  to the space-like polar axis    .

Conclusion
After defining the hyperbolic statement of the two-parameter planar Lorentzian motion   , the geometric aspects of the motion   are investigated by considering the polar axes and the moving coordinate system.Thereafter using the polar axes the two-parameter planar Lorentzian motion is ascertained.Taking advantage of those verities, our next study will be concentrated on the main one-parameter motions, the geodesic motions, the osculator motions and the sliding motions obtained from   .Eventually, we have faith in that study whose viewpoint sheds some new lights on the study of motions in Euclidean and non-Euclidean planes.

Figure 1 :
Figure 1: The example of the visualisation of the   and the motions   obtained from   .

Figure 3 :
Figure 3: The envelope and evolute curves of the polar axes.

Figure 5 :
Figure 5: The support function on the polar axis   .
(i) The Time-Like Polar Axis   .We take the moving coordinate systems { =  =   ; b 1 , b 2 }, {  ; b 1  , b 2  } in the Lorentzian planes  and   such that the time-like polar axis   coincides with the vector axis { =  =   ; b 2 } and the time-like polar axis    coincides with the vector axis {  ; b 2  }.There is no loss of generality in assuming that the origin of the {  ; b 1  , b 2  } moves to the point  =  =   , Figure 5.We can write the vector equation a = AO = −b 1 .

Figure 6 :
Figure 6: The change of the support function on the polar axis   .

∧
− V 1 cosh Ψ + V 2 sinh Ψ and ∧   =   − V  1 cosh Ψ  + V  2 sinh Ψ  are taken, is the distance of k to the time-like polar axis   and ∧   is the distance of k  to the time-like polar axis    .
Planar Lorentzian Motion.Here is another way of describing the two-parameter planar Lorentzian motion by the help of the hyperbolic statements.Definition 11.Let us consider  1 ,  2 , and Θ; namely, u = ( 1 ,  2 ) and Θ are given by continuously differentiable function of the parameters  1 and  2 in (19).In this way, we obtain what we call the two-parameter planar Lorentzian motion /  .During this motion, the point  ∈  generally draws a surface part in the plane   .For this reason, the motion can be called the surface drawing motion.We denote the motion briefly by   . 1 and  2 are functions of the time parameter  (parameter is ), (ii) if  1 is a function of the parameter  2 (parameter is  2 ), (iii) if  1 is a constant (parameter is  2 ), then one-parameter planar Lorentzian motions   are obtained.