Unit dual split quaternions and arcs of dual hyperbolic spherical triangles

Abstract

In this paper we obtain the cosine hyperbolic and sine hyperbolic rules for a dual hyperbolic spherical triangle T(A; ~ B; ~ C~) whose arcs are represented by dual split quaternions.

Keywords

• Hyperbolic spherical trigonometry, split quaternion, Lorentzian space

References

1. S. Altmann, “Rotations, Quaternions, and Double Groups”. Clarendon Press, Oxford 1972.
2. R. Ablamowicz and G. Sobczyky, “Lectures on Cliğord (Geometric) Algebras and Applica- tions” Birkhäuser, Boston 2004.
3. C. Mladenova, “Robot problems over con…gurational manifold of vector-parameters and dual vector-parametes” J. Intelligent and Robotic systems 11 (1994) 117-133.
4. E. Ata, Y. Yayli, “Split Quaternions and Semi-Euclidean Projective Spaces” Caos Solitons Fractals 41 ( 2009), no.4, 1910-1915.
5. E. Ata, Y. Yayli, “Dual Quaternions and Dual Projective Spaces” Chaos Solitons Fractals 40 ( 2009), no.3, 1255-1263.
6. J. P. Ward, “Quaternions and Cayley Numbers” Kluwer Academic Publisher, 1997.
7. G. R. Veldkamp, “On the Use of Dual Numbers, Vectors and Matrices in instantaneous, spatial Kinematics”, Mechanism and Machine Theory, 1976 vol. 11, pp. 141-156.
8. A. F. Beardon, “The Geometry of Discrete Groups”, Springer-Verlag, New York, Berlin 1983.

9. H. H. U¼gurlu and H. Gündo¼gan, “The Cosine Hyperbolic and Sine Hyperbolic Rules for Dual Hyperbolic Spherical Trigonometry”, Mathematical and Computational Applications, Vol. 5, No.3, 185-190, 2000.
10. M. Kazaz, H. H. U¼gurlu, A. Özdemir, “The Cosine Rule II for a Spherical Triangle on the Dual Unit Sphere S2”, Math. Comput. Appl. 10(2005), no.3, 313-320.
11. A. Özdemir, M. Kazaz, “Hyperbolic Sine and Cosine Rules for Geodesic Triangles on The Hyperbolic Unit Sphere H2”, Mathematical and Computational Applications, Vol. 10, No.2, 209, 2005.
12. ”, Mathematical and Computational Applications, Vol. 10, No.2, H. H. U¼gurlu, A. Çalı¸skan, “The Study Mapping for Directed Space-like and Time-like lines in Minkowski 3-Space R1”, Mathematical and Computational Applications, Vol. 1, No. 2 pp.142-148,1996.
13. John G. Ratcliğe, “ Foundation of hyperbolic Manifolds (Graduate Text in Mathematics)” Sipringer-Verlag Newyork, Berlin 1991
14. L. Kula and Y. Yayli, “ Dual Split Quaternions and Screw Motions in Minkowski 3-Space” Iranian Journal of Sciences & Technology, Transaction A. Vol. 30, no. 3, 245-258. Address : Ankara University, Science Faculty, Mathematics Department, Tando¼gan, Ankara