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Let M3 q(c) denote the 3-dimensional space form of index q = 0, 1, and constant curvature c 6= 0. A curve α immersed in M3 q(c) is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: nonnull Bertrand curves in M3 q(c) correspond with curves for which there exist two constants λ 6= 0 and μ such that λκ + μ = 1, where τ and stand for the curvature and torsion of the curve. As a consequence, non-null helices in M3 q(c) are the only twisted curves in M3 q(c) having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.
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참고문헌 (23)
참고문헌 신청
A. Ferrandez / 2001 / Null helices in Lorentzian space forms / Internat. J. Modern Phys. A 16 (30) : 4845 ~ 4863
A. Gorgulu / 1986 / A generalization of the Bertrand curves as general inclined curves in En / Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist 35 (1) : 53 ~ 60
A. Gray / 1997 / Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapter 3.5, Clothoids / CRC Press : 64 ~ 66
E. Nessovic / 2005 / Curves in Lorentzian spaces / Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat 8 (3) : 685 ~ 696
G. Harary / 2010 / 3D Euler spirals for 3D curve completion / Proceedings of the 2010 Annual Symposium on Computational Geometry SoCG’10 : 393 ~ 402
A. Gorgulu / 1986 / A generalization of the Bertrand curves as general inclined curves in En / Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist 35 (1) : 53 ~ 60
A. Gray / 1997 / Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapter 3.5, Clothoids / CRC Press : 64 ~ 66
E. Nessovic / 2005 / Curves in Lorentzian spaces / Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat 8 (3) : 685 ~ 696
G. Harary / 2010 / 3D Euler spirals for 3D curve completion / Proceedings of the 2010 Annual Symposium on Computational Geometry SoCG’10 : 393 ~ 402
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