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THE MANHEIM AND LIOVILLE FORMULAE BY THE BLASCHKE VECTORS IN R31
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  • Journal title : Honam Mathematical Journal
  • Volume 36, Issue 4,  2014, pp.739-753
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2014.36.4.739
 Title & Authors
THE MANHEIM AND LIOVILLE FORMULAE BY THE BLASCHKE VECTORS IN R31
Ozyilmaz, Emin;
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 Abstract
In this study, it is aimed to analyze how relationship among Blaschke vectors that the obtained formulae in [2, 3] change if parameter ruled surfaces of the spacelike line congruence are not choosed as principle ruled surfaces. Moreover, using the relation among Blaschke vectors, we obtain Manheim`s and Liouville`s formulae. This new method can be applied to congruences. Thus, we can obtain new formulae in lines space.
 Keywords
dual space;Blaschke trihedron;dual curvature;
 Language
English
 Cited by
 References
1.
Ravani, B., Ku Ts., Bertrand offsets of ruled and developable surface, Computer Aided Geometric Design, Elsevier 23(2) (1991), 145-152. crossref(new window)

2.
Caliskan, A., On the Studying of a line Congruence by Choosing Parameter Ruled Surfaces as Principal Ruled Surface, Journal of Faculty Science of Ege University Series A (1) (1987).

3.
Caliskan, A., The Relation Among Blaschke Vectors of Ruled Surfaces on a Line Congruence And Its Consequance, Commun. Fac. Sci. Univ. Ank. Series An (1-2) (1989), 77-86.

4.
W. Blaschke, Vorlesungen uber Differentialgeometrie und geometrische Grund-lagen von Einsteins Relativit atstheorie, Dover Publications, New York, 1945.

5.
Clifford, W. K., Preliminary sketch of bi-quaternions, Proc. London Math. Soc. 4(64, 65) (1873), 361-395.

6.
Gugenheimer. H. W., Differential Geometry, Graw-Hill, New York, 1956.

7.
Karger, A., Novak, J., Space Kinematics and Line Groups, Gordon and Breach Science Publishers, New York, 1985.

8.
Veldkamp, G. R., On the use of dual numbers, vectors and matrices in instanta-neous spatial kinematics, Mech. Mach. Theory 11 (1976), 141-156. crossref(new window)

9.
Kummer, E., ber die algebraischen Strahlensysteme, insbesondere uber die der ersten undzweiten Ordnung, Abh. K. Preuss. Akad. Wiss. Berlin (1866), 1-120, also in E. E. Kummer. Collected Papers, Springer Verlag, 1975.

10.
Muller, H. R., Kinematik Dersleri, Ankara University Press, 1963.

11.
Ugurlu, H. H., Caliskan, A., On The Geometry of Space-Like Congruence, Commun. Fac. Sci. Univ. Ank. series A1 50 (2001), 9-24.

12.
Yayl, Y., Saracoglu S., Ruled Surfaces with Different Blaschke Approach, Applied Mathematical Sciences 6(59) (2012), 2945-2955.

13.
O'Neil, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.

14.
Birman, G. S., Nomizo, K., Trigonometry in Lorentzian Geometry, Ann. Math. Mont. 91(9) (1984), 543-549. crossref(new window)

15.
Ugurlu, H. H., Caliskan, A., The Study Mapping for Directed Spacelike and Time-like Lines in Minkowski 3-space $R_{13}$, Mathematical and Computational Appli-cations 1(2) (1996), 142-148.

16.
Onder, M., Dual timelike normal and dual timelike spherical curves in dual Minkowski space $D^3_1$, SDU Fen-Edebiyat Fakultesi Fen Dergisi 1(1-2) (2006), 77-86.