arXiv subjects
Publications and source records attributed to Ergin Bayram.
In the present paper, we find a surface family possessing the natural lift of a given timelike curve as a asymptotic in Minkowski 3-space. We express necessary and sufficient conditions for the given curve such that its natural lift is a asymptotic on any member of the surface family. Finally, we illustrate the method with some examples.
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the geodesic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. In addition, developablity along the common geodesic of the members of surface family are discussed. Finally, we illustrate this method by presenting some examples.
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. Furthermore, we prove that there exists no developable surface possessing a given curve as an asymptotic curve except plane. Finally, we illustrate this method by presenting some examples.
In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R^4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.
Let $α$ be an arc on a connected oriented surface $S$ in Minkowski 3-space, parameterized by arc length $s$, with torsion $τ$ and length $l$. The total square torsion $H$ of $α$ is defined by $% H=\int_{0}^{l}τ^{2}ds$. The arc is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the value of $H$ within the family of all arcs of length $l$ on $S$ having the same initial point and initial direction as $α$. In this study, we obtain differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface in Minkowski 3-space.
In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface as a linear combination of this frame and analyse the necessary and sufficient condition for that curve to be geodesic. We illustrate this method by presenting some examples.
Let α(s) be an arc on a connected oriented surface S in E3, parameterized by arc length s, with torsion τ and length l. The total square torsion F of α is defined by T=\int_{0}^{l}τ^{2}ds\ $. . The arc α is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the value of F within the family of all arcs of length l on S having the same initial point and initial direction as α. In this study, we obtain differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface.
In this paper, we express surfaces parametrically through a given spacelike (timelike) asymptotic curve using the Frenet frame of the curve in Minkowski 3-space. Necessary and sufficient conditions for the coefficients of the Frenet frame to satisfy both parametric and asymptotic requirements are derived. We also present some interesting examples to show the validity of this study.
In this paper, we analyze the problem of constructing a surface pencil from a given spacelike (timelike) line of curvature. By using the Frenet frame of the given curve in Minkowski 3-space, we express the surface pencil as a linear combination of this frame and derive the necessary and sufficient conditions for the coefficients to satisfy the line of curvature requirement. To illustrate the method some examples showing members of the surface pencil with their line of curvature are given.