Yazar "Ucum, Ali" seçeneğine göre listele

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    A new approach to Bertrand curves in Euclidean 3-space
    (Springer Basel Ag, 2020) Camci, Cetin; Ucum, Ali; Ilarslan, Kazim
    In this article, a new approach is given for Bertrand curves in 3-dimensional Euclidean space. According to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Bertrand curve in E-3. In addition, the related examples and graphs are given by showing that general helices and anti-Salkowski curves can be Bertrand curves or their mates, which is their new characterization.
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    A New Class of Bertrand Curves in Euclidean 4-Space
    (Mdpi, 2022) Li, Yanlin; Ucum, Ali; Ilarslan, Kazim; Camci, Cetin
    Bertrand curves are a pair of curves that have a common principal normal vector at any point and are related to symmetry properties. In the present paper, we define the notion of (1, 3)-V Bertrand curves in Euclidean 4-space. Then we find the necessary and sufficient conditions for curves in Euclidean 4-space to be (1, 3)-V Bertrand curves. Some related examples are given.
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    General Helices with Lightlike Slope Axis
    (Univ Nis, Fac Sci Math, 2018) Camci, Cetin; Ilarslan, Kazim; Ucum, Ali
    In this paper, we investigate general helices with lightlike slope axis. We give necessary and sufficient conditions for a general helix to have a lightlike slope axis. We obtain parametric equation of all general helices with lightlike slope axis. Also we give a nice relation between helix with lightlike slope axis and biharmonic curves in Minkowski 3-space E-1(3).
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    General Helices with Timelike Slope Axis in Minkowski 3-Space
    (Springer Basel Ag, 2016) Ucum, Ali; Camci, Cetin; Ilarslan, Kazim
    In the present paper, we consider the general helices in Minkowski 3-space to have a constant timelike slope axis. As a result, we show that there exists no pseudo null helix with constant timelike slope axis or spacelike helix with timelike principal normal and constant timelike slope axis. Moreover, we obtain the parametric equations of spacelike helix with spacelike principal normal, timelike helix and null Cartan helix with timelike slope axis. We also give some related examples and their figures.
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    New results concerning Cartan null and pseudo null curves in Minkowski 3-space
    (Birkhauser, 2023) Camcı, Çetin; Ucum, Ali; İlarslan, Kazım
    In this study, we focus on Cartan null and pseudo null curves in Minkowski 3-space E13 . Firstly we define Cartan null and pseudo null equivalent curves and give the related examples. Then we give a construction method for Cartan null curves and it is shown that every Cartan null curve can be obtained from a timelike curve lying in the Lorentz plane in E13 . Also a construction method is given for pseudo null curves and we show that all pseudo null curves lie in a lightlike plane in E13 . Finally we define a new surface in E13 called “surface of pseudo null curves” and we obtain the geometric properties of such surfaces.
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    Sequential natural mates of Frenet curves in Euclidean 3-space
    (Springer, 2021) Camcı, Çetin; Chen, Bang-Yen; Ilarslan, Kazım; Ucum, Ali
    Associated with a Frenet curve α in Euclidean 3-space E3, there exists the notion of natural mate β of α. In this article, we extend the natural mate β to sequential natural mates {α1,α2,…,αnα} with α1= β. We call each curve αi, i∈ { 1 , 2 , … , nα} , the i-th natural mate. The main purpose of this article is to study the relationships between the given Frenet curve α with its sequential natural mates {α1,α2,…,αnα}.
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    Space curves related by a transformation of combescure
    (Taylor and Francis Ltd., 2021) Camcı, Çetin; Ucum, Ali; Ilarslan, Kazım
    In this paper, the curves associated with the Combescure transform are discussed. With the help of the fact that these curves have a common Frenet frame, an equivalence relation is defined. The equivalence classes obtained by this equivalence relation have been examined for some special curves and it has been obtained that all curves in the equivalence class of a helix curve are also helix curves. This is also true for k-slant helix curves. The important part of this paper consists of the useful construction method to obtain a curve from the given curve a with the help of Combescure transformation. With this method, some special curves such as Bertrand, Mannheim, Salkowski, anti-Salkowski or spherical curve can be obtained from any curve related by a Combescure transform. For example, we obtain an example of Mannheim curves explicitly obtained from an anti-Salkowski curve. It is not easy to find an example of Mannheim curves except circular helix in the literature. In general, the conditions for being Bertrand, Mannheim, Salkowski, anti-Salkowski or spherical curve of the curve beta obtained from the given curve alpha with the help of Combescure transformation were obtained.
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    Twisted Surfaces in Semi-Euclidean 4-Space with Index 2
    (Int Electronic Journal Geometry, 2024) Ucum, Ali; İlarslan, Kazım; Camcı, Çetin
    In this paper, we consider the twisted surfaces in semi -Euclidean 4 -space with index 2 . We classify the twisted surface with respect to their spine curve which are non -null or null curves. So, we study the geometric properties of these surfaces. Also we obtain the family of some special surfaces such as flat surfaces, marginally trapped surfaces.