On the two-parameter Lorentzian homothetic motions
© Çelik et al.; licensee Springer. 2014
Received: 11 October 2013
Accepted: 6 January 2014
Published: 27 January 2014
In this study, sliding velocity, pole lines, hodograph, and acceleration poles of two-parameter Lorentzian homothetic motions at positions are obtained. By defining two-parameter Lorentzian homothetic motion along a curve in Lorentzian space , the theorems related to this motion and characterizations of the trajectory surface are given.
MSC:53A17, 53B30, 14H50.
To investigate the geometry of the motion of a line or a point in the motion of space is important in the study of space kinematics or spatial mechanisms or in physics. The geometry of such a motion of a point or line has a number of applications in geometric modeling and model-bored manufacturing of mechanical products or in the design of robotic motion. These are specifically used to generate geometric models of shell-type objects and thick surfaces [1–3].
Muller has introduced one- and two-parameter planar motions and obtained the relations between absolute, relative, sliding velocity, and pole curves of these motions . Moreover, two-parameter motions in three-dimensional space are defined by  and . Lorentzian metric in three-dimensional Minkowski space is indefinite. In the theory of relativity, the geometry of indefinite metric is very crucial. Thus, by taking a Lorentzian plane instead of an Euclidean plane , Ergin  has introduced one-parameter planar motion in the Lorentzian plane. In  all one-parameter motions obtained from two-parameter motion on the Lorentzian plane are investigated.
In this paper, firstly we introduce two-parameter homothetic motions in a Lorentzian plane and we calculate the pole points obtained from Lorentzian homothetic motion. Additionally, we give some theorems and corollaries as regards these pole points. Similarly, we calculate the acceleration poles of Lorentzian homothetic motions. By considering the above mentioned, we define two-parameter homothetic motion along a curve in Lorentzian space and we give the equation of the trajectory surfaces formed by Lorentzian homothetic motions. Finally, we obtain the parametrizations of the trajectory surfaces and give some examples for these trajectory surfaces.
2 Two-parameter homothetic motions in Lorentzian plane
for . Also, there can be given some special results of and .
where and are coordinate functions of the fixed plane and moving planes, respectively. If λ and μ in are given by the differential functions of the time parameter t, then homothetic motions are obtained and called homothetic motions obtained from homothetic motions on the Lorentzian plane.
Here, at the initial time and , the coordinate systems of the moving and fixed planes are congruent.
The pole points of Lorentzian homothetic motions obtained from Lorentzian homothetic motions on a moving plane at the position of give the following results. □
at the position of and when .
and the equation of the pole points (2.10) is obtained. The pole points of Lorentzian homothetic motions obtained from Lorentzian homothetic motions on a fixed plane at the position of give the following results. □
Corollary 2.5 As a special case in Corollary 2.4, if , the pole points of the fixed and moving planes are congruent.
Corollary 2.6 If is constant, the pole points of fixed planes lie on the line equation (2.9) .
Corollary 2.7 As a special case in Corollary 2.2, if , the pole points of moving planes are congurent to pole lines of fixed plane in Corollary 2.6.
Corollary 2.8 If is a constant never vanishing and the pole axis is the y-axis, then the pole ray and the sliding velocity are perpendicular .
at the position of each .
Then, the length of the sliding velocity vector is obtained. □
Corollary 2.9 If , then we obtain .
at the position of each .
From the equality of the last two equations, we obtain equation (2.17). □
Definition 2.2 When the sliding velocity vectors of a fixed point are carried to the initial point, without changing the directions, then the locus of the end points of these vectors is a curve called a hodograph.
Finally, if we find the values of and and substitute these values into the equation , and the following theorem is found.
Theorem 2.6 In all Lorentzian homothetic motions obtained from Lorentzian homothetic motions , the locus of the hodograph is a hyperbola at the position of .
That is, the locus of the hodograph is a hyperbola. □
3 The acceleration pole of the homothetic motions
at position and when .
Proof Setting in equation (3.1) gives us the desired equation. Therefore, we can give the following corollaries at the position of . □
Corollary 3.1 The acceleration pole points on the moving plane lie on the line given by equation (2.7) if is constant.
Corollary 3.2 The acceleration pole points on the moving plane lie on the line given by equation (2.8) if is constant.
Corollary 3.3 The acceleration pole points on the moving plane lie on the line given by equation (2.9) if , .
Corollary 3.4 If is constant, the pole line on the moving planes obtained from Corollary 2.2 and the acceleration pole line on the moving planes obtained from Corollary 3.2 are congruent .
at position and when .
If we take in the last equation, we have equation (3.4).
So, we can give the following corollaries at the position of . □
Corollary 3.5 The acceleration pole points on the fixed plane lie on the line given by equation (2.12) if is constant.
Corollary 3.6 As a special case, if and is constant, the acceleration pole points on the moving plane and the acceleration pole points on the fixed plane are congruent.
Corollary 3.7 The acceleration pole points on the fixed plane lie on the line given by equation (2.9) if is constant.
Corollary 3.8 If the acceleration pole line of a moving plane obtained from Corollary 3.2 and the acceleration pole line of a fixed plane obtained from Corollary 3.5 are congruent.
Corollary 3.9 As is seen from Corollaries 2.4, 3.5 and 2.6, 3.7, the pole line of a fixed plane and the acceleration pole line of a fixed plane are congruent.
4 Two-parameter homothetic motion along a curve in Lorentzian space
In this section, we define two-parameter homothetic motion along a curve in a Lorentzian space and obtain the characterization of the same trajectory surface.
where , , are the basis of the space . Semi-orthogonal matrices provide a rotation by the angle (hyperbolic) t around the vector . The shape of the matrix depends on the type of the vector as seen in .
- i.If is a spacelike vector(4.3)
- ii.If is a timelike vector(4.4)
is the orthogonal matrix defined via the semi-skew symmetric matrices corresponding to the vector .
If we calculate the normals of these surfaces, there are two states depending on whether we have the timelike and spacelike cases.
1. If is a spacelike curve, then the tangent is a spacelike and we have the following cases.
If is a constant that is never vanishing, then the normal of this surface is in a normal plane which is perpendicular to the tangent vector field of the curve .
and the geometric position of the selected point p which is over a principal normal vector of the α spacelike curve is a spacelike surface.
the geometric position of the selected point p which is over a principal binormal vector of the α spacelike curve is a timelike surface.
2. If is a timelike curve, then the tangent is timelike and we have the following case.
the geometric position of the selected point p which is over a principal binormal vector of the α timelike curve is a timelike surface.
5 Parametrizations of trajectory surfaces
In this section, we find some parametrizations of the trajectory surfaces obtained from two-parameter motions in a Lorentzian space.
5.1 Cylinder surface
5.1.1 is spacelike
5.1.2 is timelike
5.2 Hyperboloid surface
5.2.1 is spacelike
5.2.2 is timelike
5.3 Tor surface
5.3.1 is spacelike
5.3.2 is timelike
The results we have presented deal with Lorentzian homothetic motions in which the position of the moving object depends on two parameters. The hodographs of two-parameter Lorentzian homothetic motions were obtained. A hodograph is the locus of the end points of the velocity of a particle and it is the solution of the first order equation which is Newton’s Law. The locus of the hodograph of a Lorentzian homothetic motion was found as a hyperbola in this study.
Also this paper deals with trajectory surfaces (cylinder, hyperboloid and tor surfaces) generated by a point, the moving body, and figures of these surfaces were drawn by using MATLAB software.
We would like to thank to the reviewers whose careful reading, helpful suggestions and valuable comments helped us to improve the manuscript.
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