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Ferdağ Kahraman Aksoyak

Publications and source records attributed to Ferdağ Kahraman Aksoyak.

9 recordsLinked to original sources
arXiv2024

$K^α$-translators of offset surfaces

In this paper, we study $K^α$--translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space $\mathbb{E}^3$. First, we investigate the condition under which two parallel surfaces can become $K^α$--translators moving with the same speed $w$. Then, we examine $K^α$--translators on canal surfaces and we show that if a canal surface is $K^α$--translator, then it must be a surface of revolution in $\mathbb{E}^3$. We also provide examples for moving a surface of revolution under $K$--flow (Gauss curvature flow) and $K^{-1/2}$--flow (inverse Gauss curvature flow) along a direction $w=(0,0,1)$ and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no $K^α$--translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed $w$, while the such rotational surfaces itself is a $K^α$--translator with speed $w$.

math.DG↗
arXiv2022

Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3--Space

In this paper, we study on the characterizations of loxodromes on the rotational surfaces satisfying some special geometric properties such as having constant Gaussian curvature, flat and minimality in Euclidean 3-space. First, we give the parametrizations of loxodromes parametrized by arc-length parameter on any rotational surfaces in $\mathbb{E}^{3}$ and then, we calculate the curvature and the torsion of such loxodromes. Then, we give the parametrizations of loxodromes on rotational surfaces with constant Gaussian curvature. In particular, we prove that the loxodrome on the flat rotational surface is a general helix. Also, we investigate the loxodromes on the rotational surfaces with a constant ratio of principal curvatures (CRPC rotational surfaces). Moreover, we give the parametrizations of loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces. Then, we show that the loxodrome intersects the meridians of minimal rotational surface by the angle $π/{4}$ becomes an asymptotic curve. Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.

math.DG↗
arXiv2021

Quaternionic (1,3)-Bertrand Curves According to Type 2-Quaternionic Frame in R^4

If there exists a quaternionic Bertrand curve in E4, then its torsion or bitorsion vanishes. So we can say that there is no quaternionic Bertrand curves whose torsion and bitorsion are non-zero. Hence by using the method which is given by Matsuda and Yorozu [13], we give the definition of quaternionic (1,3)-Bertrand curve according to Type 2-Quaternionic Frame and obtain some results about these curves.

math.DG↗
arXiv2016

On rectifying slant helices in Euclidean 3-space

In this paper, we study the position vectors of rectifying slant helices in $E^3$. First, we have found the general equations of the curvature and the torsion of rectifying slant helices. After that, we have constructed a second order linear differential equation and by solving the equation, we have obtained a family of rectifying slant helices which lie on cones.

math.GM↗
arXiv2015

Characterizations of the Quaternionic Bertrand Curve in Euclidean Space E4

In [18], L. R. Pears proved that Bertrand curves in E-n(n > 3) are degenerate curves. This result restate in [16] by Matsuda and Yorozu. They proved that there is no special Bertrand curves in E-n(n > 3) and they define new kind of Bertrand curves called (1, 3)-type Bertrand curves in 4-dimensional Euclidean space. In this study, we define a quaternionic Bertrand curve ?(4) in Euclidean space E4 and investigate its properties for two cases. In the first case; we consider quaternionic Bertrand curve in the Euclidean space E4 for r-K = 0 where r is the torsion of the spatial quaternionic curve ?; K is the principal curvature of the quaternionic curve ?(4): And then, in the other case, we prove that there is no quaternionic Bertrand curve in the Euclidean space E4 for r - K = 0: So, we give an idea of quaternionic Bertrand curve which we call quaternionic (N - B2) Bertrand curve in the Euclidean space E4 by using the similar method in [16] and we give some characterizations of such curves.

math.DG↗
arXiv2014

Generalized Bicomplex Numbers and Lie Groups

In this paper, we denone the generalized bicomplex numbers and give some algebraic properties of them. Also, we show that some hyperquadrics in R4 and R42 are Lie groups by using generalized bicomplex number product and obtain Lie algebras of these Lie groups. Morever, by using tensor product surfaces, we determine some special Lie subgroups of these hyperquadrics.

math.DG↗
arXiv2013

Flat Rotational Surface with Pointwise 1-typeGauss map in E4

In this paper we study general rotational surfaces in the 4- dimensional Euclidean space E4 and give a characterization of flat general rotation surface with pointwise 1-type Gauss map. Also, we show that a non-planar flat general rotation surface with pointwise 1-type Gauss map is a Lie group if and only if it is a Clifford torus.

math.DG↗