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Advances in Applied Clifford Algebras

, Volume 18, Issue 2, pp 279–285 | Cite as

One-Parameter Plane Hyperbolic Motions

  • Salim YüceEmail author
  • Nuri Kuruoğlu
Article

Abstract.

Müller [3], in the Euclidean plane \({{\mathbb{E}}}^2\), introduced the one parameter planar motions and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Müller [11] provided the relation between the velocities (in the sense of Complex) under the one parameter motions in the Complex plane \({\mathbb{C}} := \{x + iy | x, y \in {\mathbb{R}}, i^2 = -1\}\).

Ergin [7] considering the Lorentzian plane \({{\mathbb{L}}}^2\), instead of the Euclidean plane \({{\mathbb{E}}}^2\), and introduced the one-parameter planar motion in the Lorentzian plane and also gave the relations between both the velocities and accelerations.

In analogy with the Complex numbers, a system of hyperbolic numbers can be introduced: \({\mathbb{H}} := \{x + jy | x, y \in {\mathbb{R}}, j^2 = 1\}\). Complex numbers are related to the Euclidean geometry, the hyperbolic system of numbers are related to the pseudo-Euclidean plane geometry (space-time geometry), [5,15].

In this paper, in analogy with Complex motions as given by Müller [11], one parameter motions in the hyperbolic plane are defined. Also the relations between absolute, relative, sliding velocities (and accelerations) and pole curves are discussed.

Mathematics Subject Classification (2000).

53A17 11E88 

Keywords.

Kinematics one-parameter motion hyperbolic numbers hyperbolic angle 

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Copyright information

© Birkhauser 2008

Authors and Affiliations

  1. 1.Faculty of Arts and Science, Department of MathematicsYıldız Technical UniversityİstanbulTurkey
  2. 2.Faculty of Arts and Science, Department of Mathematics and Computer SciencesUniversity of BahçeşehirİstanbulTurkey

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