• Title, Summary, Keyword: Bertrand curve

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ON TIMELIKE BERTRAND CURVES IN MINKOWSKI 3-SPACE

  • Ucum, Ali;Ilarslan, Kazim
    • Honam Mathematical Journal
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    • v.38 no.3
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    • pp.467-477
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    • 2016
  • In this paper, we study the timelike Bertrand curves in Minkowski 3-space. Since the principal normal vector of a timelike curve is spacelike, the Bertrand mate curve of this curve can be a timelike curve, a spacelike curve with spacelike principal normal or a Cartan null curve, respectively. Thus, by considering these three cases, we get the necessary and sufficient conditions for a timelike curve to be a Bertrand curve. Also we give the related examples.

BERTRAND CURVES IN NON-FLAT 3-DIMENSIONAL (RIEMANNIAN OR LORENTZIAN) SPACE FORMS

  • Lucas, Pascual;Ortega-Yagues, Jose Antonio
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.1109-1126
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    • 2013
  • Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ 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: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ 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.

On Pseudo Null Bertrand Curves in Minkowski Space-time

  • Gok, Ismail;Nurkan, Semra Kaya;Ilarslan, Kazim
    • Kyungpook Mathematical Journal
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    • v.54 no.4
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    • pp.685-697
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    • 2014
  • In this paper, we prove that there are no pseudo null Bertrand curve with curvature functions $k_1(s)=1$, $k_2(s){\neq}0$ and $k_3(s)$ other than itself in Minkowski spacetime ${\mathbb{E}}_1^4$ and by using the similar idea of Matsuda and Yorozu [13], we define a new kind of Bertrand curve and called it pseudo null (1,3)-Bertrand curve. Also we give some characterizations and an example of pseudo null (1,3)-Bertrand curves in Minkowski spacetime.

NOTE ON BERTRAND B-PAIRS OF CURVES IN MINKOWSKI 3-SPACE

  • Ilarslan, Kazim;Ucum, Ali;Aslan, Nihal Kilic;Nesovic, Emilija
    • Honam Mathematical Journal
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    • v.40 no.3
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    • pp.561-576
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    • 2018
  • In this paper, we define null Cartan and pseudo null Bertrand curves in Minkowski space ${\mathbb{E}}^3_1$ according to their Bishop frames. We obtain the necessary and sufficient conditions for pseudo null curves to be Bertand B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Bertrand B-curves, by considering the cases when their Bertrand B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Bertrand B-curve pairs.

CURVES ON THE UNIT 3-SPHERE S3(1) IN EUCLIDEAN 4-SPACE ℝ4

  • Kim, Chan Yong;Park, Jeonghyeong;Yorozu, Sinsuke
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1599-1622
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    • 2013
  • We show many examples of curves on the unit 2-sphere $S^2(1)$ in $\mathbb{R}^3$ and the unit 3-sphere $S^3(1)$ in $\mathbb{R}^4$. We study whether its curves are Bertrand curves or spherical Bertrand curves and provide some examples illustrating the resultant curves.

BERTRAND CURVES AND RAZZABONI SURFACES IN MINKOWSKI 3-SPACE

  • Xu, Chuanyou;Cao, Xifang;Zhu, Peng
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.377-394
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    • 2015
  • In this paper, we generalize some results about Bertrand curves and Razzaboni surfaces in Euclidean 3-space to the case that the ambient space is Minkowski 3-space. Our discussion is divided into three different cases, i.e., the parent Bertrand curve being timelike, spacelike with timelike principal normal, and spacelike with spacelike principal normal. For each case, first we show that Razzaboni surfaces and their mates are related by a reciprocal transformation; then we give B$\ddot{a}$cklund transformations for Bertrand curves and for Razzaboni surfaces; finally we prove that the reciprocal and B$\ddot{a}$cklund transformations on Razzaboni surfaces commute.

SOME ISOTROPIC CURVES AND REPRESENTATION IN COMPLEX SPACE ℂ3

  • Qian, Jinhua;Kim, Young Ho
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.963-975
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    • 2015
  • In this paper, we give a representation formula for an isotropic curve with pseudo arc length parameter and define the structure function of such curves. Using the representation formula and the Frenet formula, the isotropic Bertrand curve and k-type isotropic helices are characterized in the 3-dimensional complex space $\mathbb{C}^3$.

Analysis on Market Power in Power Transaction with Transmission Constraints (송전선 제약조건에 따른 전력거래에서의 시장지배력 연구)

  • Lee, Gwang-Ho
    • The Transactions of the Korean Institute of Electrical Engineers A
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    • v.51 no.8
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    • pp.403-408
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    • 2002
  • As the electricity industry undergoes a process of fundamental restructuring, horizontal market power appears as a potential obstacle to a fully competitive wholesale electricity market. Market power is the ability profitably to maintain prices above competitive levels by restricting output below competitive levels. In models for imperfect competition under the consideration of the transmission constraints, the Nash equilibrium has the form of a mixed strategy. In this paper, the models for analyzing imperfect competition are compared using the solution of pure and mixed equilibria. The relation between market power and the capacity of a transmission line is investigated by imperfect competition analysis methods: Cournot, Bertrand, and Supply Curve model.

Lane Detection Using Road Geometry Estimation

  • Lee, Choon-Young;Park, Min-Seok;Lee, Ju-Jang
    • 제어로봇시스템학회:학술대회논문집
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    • pp.226-231
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    • 1998
  • This paper describes how a priori road geometry and its estimation may be used to detect road boundaries and lane markings in road scene images. We assume flat road and road boundaries and lane markings are all Bertrand curves which have common principal normal vectors. An active contour is used for the detection of road boundary, and we reconstruct its geometric property and make use of it to detect lane markings. Our approach to detect road boundary is based on minimizing energy function including edge related term and geometric constraint term. Lane position is estimated by pixel intensity statistics along the parallel curve shifted properly from boundary of the road. We will show the validity of our algorithm by processing real road images.

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