Abstract
The Bertrand curves were first studied using a computer by Wu (1987). The same problem was studied using an improved version of Ritt-Wu’s decomposition algorithm by Chou and Gao (1993). This paper investigates the same problem for pseudo null Bertrand curves in Minkowski 3-space \( \mathbb{E}_1^3 \).
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References
B. Saint Venant, Mémoire sur les lignes courbes non planes, Journal de l’Ecole Polytechnique, 1845, 18: 1–76.
J. M. Bertrand, Mémoire sur la théorie des courbes á double courbure, Comptes Rendus, 1850, 36.
John F. Burke, Bertrand curves associated with a pair of curves, Mathematics Magazine, 1960, 34(1): 60–62
L. R. Pears, Bertrand curves in Riemannian space, J. London Math. Soc., 1935, s1–10(2): 180–183.
C. Bioche, Sur les courbes de M. Bertrand, Bull. Soc. Math., 1889, 17: 109–112.
H. Matsuda and S. Yorozu, Notes on Bertrand curves, Yokohama Math. J., 2003, 50(1–2): 41–58.
H. Balgetir, M. Bektaş, and J. Inoguchi, Null Bertrand curves in Minkowski 3-space and their characterizations, Note Mat., 2004/05, 23(1): 7–13.
H. Balgetir, M. Bektaş, and M. Ergüt, Bertrand curves for nonnull curves in 3-dimensional Lorentzian space, Hadronic J., 2004, 27(2): 229–236.
N. Ekmekci and K. İlarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst., 2001, 3(2): 17–24.
Wen-Tsun Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Scientia Sinica, 1978, 21: 159–172.
J. F. Ritt, Differential Algebra, Amer. Math. Soc. Colloquium, New York, 1950.
Wen-Tsun Wu, On the foundation of algebraic differential geometry, Systems Sci. Math. Sci., 1989, 2: 289–312.
Wen-Tsun Wu, A mechanization method of geometry and its applications II: Curve pairs of Bertrand type, Kexue Tongbao, 1987, 32: 585–588.
Wen-Tsun Wu, Mechanical derivation of Newton’s gravitational laws from Kepler’s laws, MMPreprints, MMRC, 1987, 1: 53–61.
S. C. Chou and X. S. Gao, Automated reasoning in differential geometry and mechanics: Part 1: An improved version of Ritt-Wu’s decomposition algorithm, J. of Automated Reasoning, 1993, 10: 161–172.
S. C. Chou and X. S. Gao, Part 2: Mechanical theorem proving, J. of Automated Reasoning, 1993, 10: 173–189.
S. C. Chou and X. S. Gao, Part 3: Mechanical formuladerivation, IFIP Transaction on automated reasoning, 1993: 1–12.
S. C. Chou and X. S. Gao, Part 4: Bertrand curves, System Sciences and Mathematical Sciences, 1993, 6(2): 186–192.
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
J. Walrave, Curves and Surfaces in Minkowski Space, Doctoral thesis, K. U. Leuven, Fac. of Science, Leuven, 1995.
W. B. Bonnor, Curves with Null Normals in Minkowski Space-Time: A Random Walk in Relativity and Cosmology, Wiley Eastern Limited, 1985: 33–47.
H. Liu and F. Wang, Mannheim partner curves in 3-space, Journal of Geometry, 2008, 88: 120–126.
S. C. Chou and X. S. Gao, Ritt-Wu’s decomposition algorithm and geometry theorem proving, CADE-10, M. E. Stickel (Ed.), Lecture Notes in Comput. Sci., Springer-Verlag, Berlin, 1990, 449: 207–220.
Wen-Tsun Wu, Mechanical theorem-proving of differential geometries and its applications in mechanics, J. Aut. Reasoning, 1991, 7: 171–191.
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This paper was recommended for publication by Editor Xiaoshan GAO.
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İlarslan, K., Yildirim, M. An application of Ritt-Wu’s zero decomposition algorithm to the pseudo null Bertrand type curves in Minkowski 3-space. J Syst Sci Complex 24, 358–366 (2011). https://doi.org/10.1007/s11424-011-8267-1
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DOI: https://doi.org/10.1007/s11424-011-8267-1