Samah Gaber Mohamed

Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E3. We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis.

This research focused on studying the flows of a null Cartan curve specified by the velocity and acceleration fields. We have proven that the tangential and normal velocities are influenced by the binormal velocity along the motion. The velocity fields are used to drive the time evolution equations for the Cartan frame and the torsion of the null curve. The objective of this work is to construct a family of inextensible null Cartan curves from the flows of the initial null Cartan curve. The surface formed by this family of inextensible flows of the null Cartan curve is obtained numerically and visualized. In this paper, we refer to the surface traced out by the family of the null Cartan curve as the generated or constructed surface. We present a novel model for the inextensible null Cartan curve, which moves with a constant binormal velocity to describe the process of constructing a family of null Cartan curves. Through this model, the time evolution equation for the torsion of the inextensible null Cartan curve arises as the Korteweg-de Vries (K-dV) equation, and we obtain and visualize the soliton solutions. The soliton solutions represent the torsion of the family of null Cartan curves at various time values. We construct the family of inextensible null Cartan curves and visualize the generated surface. In addition, we investigate the flows of inextensible null Cartan curves specified by acceleration fields, and we obtain the explicit relationships between the acceleration and velocity functions. Finally, we provide an application for the inextensible flows of the null Cartan curve with constant normal acceleration.