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Murat Babaarslan

Publications and source records attributed to Murat Babaarslan.

13 recordsLinked to original sources
arXiv2024

Loxodromes and geodesics on rotational surfaces in pseudo-isotropic space

Special curves and surfaces have an important place in mathematics, engineering and other fields of science. Loxodromes are special curves which cut all meridians on the Earth's surface at a constant angle and they are very popular in engineering. Ships sailing and airplanes flying along a fixed magnetic compass course move along this curve. The Mercator projections of the loxodromes on the sphere correspond to the lines and their stereographic projections to the logarithmic spirals. In general, loxodromes are not great circle arcs (geodesics). Geodesics correspond to the shortest distance between two points on the Earth's surface. Since loxodromes do not need to change course, they are important in navigation. Up till now, there have been many important studies of loxodromes on different surfaces (sphere, ellipsoid, rotational, helicoidal, canal, twisted, etc.) and in different ambient spaces (Euclidean, Minkowski, simply isotropic, etc.). However, there is no study on loxodromes in pseudo-isotropic space I_p^3. In I_p^3, different pseudo-isotropic angles can be defined depending on whether the vectors are space-like or time-like as in Minkowski space. In this study, we will first define these angles in I_p^3. Then, the equations of space-like and time-like loxodromes and geodesics on rotational surfaces will be obtained. Some examples will also be provided.

math.DG↗
arXiv2024

$K^\alpha$-translators of offset surfaces

In this paper, we study $K^{\alpha}$--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 $K^{\alpha}$--translators moving with the same speed $w$. Then, we examine $K^{\alpha}$--translators on canal surfaces and we show that if a canal surface is $K^{\alpha}$--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 $K^{\alpha}$--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 $K^{\alpha}$--translator with speed $w$.

math.DG↗
arXiv2022

Isometric Timelike Surfaces in 4--Dimensional Minkowski Space

In this paper, first we study on Bour's theorem for four kinds of timelike helicoidal surfaces in 4-dimensional Minkowski space. Secondly, we analyse the geometric properties of these isometric surfaces having same Gauss map. Also, we present the parametrizations of such isometric pair of surfaces. Finally, we introduce some examples and draw the corresponding graphs by using Wolfram Mathematica 10.4.

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 $\pi/{4}$ becomes an asymptotic curve. Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.

math.DG↗
arXiv2021

Bour's Theorem of Spacelike Surfaces in Minkowski 4--Space

In this paper, we study on three kinds of spacelike helicoidal surfaces in Minkowski $4$--space. First, we give an isometry between such helicoidal surfaces and rotational surfaces which is a kind of generalization of Bour theorem in Minkowski $3$--space to Minkowski $4$--space. Then, we investigate geometric properties for such isometric surfaces having same Gauss map. By using these results, we give the parametrizations of isometric pair of surfaces. As a particular case, we examine the right helicoidal surfaces in view of Bour's theorem. Also, we present some examples by choosing the components of the profile curves and the parameters of the surfaces via Mathematica.

math.DG↗
arXiv2021

Timelike Loxodromes on Lorentzian Helicoidal Surfaces in Minkowski n--Space

In this paper, we examine timelike loxodromes on three kinds of Lorentzian helicoidal surfaces in Minkowski $n$--space. First, we obtain the first order ordinary differential equations which determine timelike loxodromes on the Lorentzian helicoidal surfaces in $\mathbb{E}^n_1$ according to the causal characters of their meridian curves. Then, by finding general solutions, we get the explicit parametrizations of such timelike loxodromes. In particular, we investigate the timelike loxodromes on the three kinds of Lorentzian right helicoidal surfaces in $\mathbb{E}^n_1$. Finally, we give an example to visualize the results.

math.DG↗
arXiv2020

Spacelike Loxodromes on Helicoidal Surfaces in Lorentzian n--Space

In this paper, we study three types of helicoidal surfaces in a Lorentzian n--space $\mathbb{E}^n_1$. First, we find the parametrizations of spacelike loxodromes on such spacelike helicoidal surfaces in $\mathbb{E}^n_1$. Then, we make a similar classification for spacelike loxodromes on such timelike helicoidal surfaces of a Lorentzian n--space $\mathbb{E}^n_1$.

math.DG↗
arXiv2018

Space-like Loxodromes on the Canal Surfaces in Minkowski 3-Space

In this paper, we obtain the differential equations of the space-like loxodromes on the non-degenerate canal surfaces depending on the causal characters of these canal surfaces and their meridians in Minkowski 3-space. Also we give an example by using Mathematica computer programme.

math.DG↗
arXiv2012

Time-like constant slope surfaces and space-like Bertrand curves in Minkowski 3-space

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Sitter evolutes of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ are studied. A relation between space-like Bertrand curves and helices is obtained. De Sitter Darboux images of space-like Bertrand curves are equal to de Sitter evolutes. The relations between time-like constant slope surfaces lying in the space-like cone and space-like Bertrand curves in Minkowski 3-space $\mathbb{R}^{3}_{1}$ are obtained.

math.DG↗
arXiv2012

On Helices and Bertrand Curves in Euclidean 3-Space

In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Similarly, in case of a space curve is a slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix.

math.DG↗
arXiv2012

Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3-Space

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternions and spacelike constant slope surfaces in Minkowski 3-space. We show that spacelike constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit timelike quaternions with the spacelike vector parts and homothetic motions. Subsequently we give some examples to illustrate our main results.

math-ph↗
arXiv2011

On space-like constant slope surfaces and Bertrand curves in Minkowski 3-space

In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$. In addition, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-like Bertrand curves can be constructed from unit speed space-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and hyperbolic space $\mathbb{H}^{2}$, respectively. We obtain the relations between Bertrand curves and helices. Also we show that pseudo-spherical Darboux images of Bertrand curves are equal to pseudo-spherical evolutes in Minkowski 3-space $\mathbb{R}^{3}_{1}$. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in $\mathbb{R}^{3}_{1}$. Finally, we give some examples to illustrate our main results.

math.DG↗