On the two-parameter Lorentzian homothetic motions
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On the two-parameter Lorentzian homothetic motions ˘ Muhsin Çelik* , Dogan Ünal and Mehmet Ali Güngör * Correspondence: [email protected] Department of Mathematics, Faculty of Arts and Sciences, Sakarya University, Sakarya, Turkey
Abstract In this study, sliding velocity, pole lines, hodograph, and acceleration poles of two-parameter Lorentzian homothetic motions at ∀(λ, μ) positions are obtained. By defining two-parameter Lorentzian homothetic motion along a curve in Lorentzian space L3 , the theorems related to this motion and characterizations of the trajectory surface are given. MSC: 53A17; 53B30; 14H50 Keywords: two-parameter motion; planar motion; Lorentz plane and space
1 Introduction 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 [–]. 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 L is indefinite. In the theory of relativity, the geometry of indefinite metric is very crucial. Thus, by taking a Lorentzian plane L instead of an Euclidean plane E , Ergin [] has introduced one-parameter planar motion in the Lorentzian plane. In [] all one-parameter motions obtained from twoparameter motion on the Lorentzian plane are investigated. In this paper, firstly we introduce two-parameter homothetic motions in a Lorentzian plane L 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 L 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. ©2014 Çelik et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Çelik et al. Advances in Difference Equations 2014, 2014:42 http://www.advancesindifferenceequations.com/content/2014/1/42
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2 Two-parameter homothetic motions in Lorentzian plane The Lorentzian homothetic motion is examined by
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