Ergin Bayram

Parametric representation of a surface pencil with a common asymptotic curve

In this paper, we study the problem of finding a surface pencil from a given spatial asymptotic curve. We obtain the parametric representation for a surface pencil whose members have the same curve as a given asymptotic curve. Using the Frenet frame of the given asymptotic curve, we present the surface as a linear combination of this frame and analyse the necessary and sufficient condition for that curve to be asymptotic. We illustrate this method by presenting some examples.

SURFACE PENCIL WITH A COMMON LINE OF CURVATURE IN MINKOWSKI 3-SPACE

In this paper, we analyze the problem of constructing a surface pencil from a given spacelike (timelike) line of curvature. Using the Frenet frame of the given line of curvature 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. We illustrate this method by presenting some examples.

Intrinsic equations for a relaxed elastic line of second kind on an oriented surface

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 H of α is defined by H =∫0lτ2ds. 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 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.

Surface Family with a Common Natural Asymptotic Lift of a Timelike Curve in Minkowski 3-space

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.

Hypersurface Family with a Common Isoasymptotic Curve

We handle the problem of finding a hypersurface family from a given asymptotic curve in . 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.