Narrow Results

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf ℰ on S and a map of ℰ to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group C*, and we determine its fixed-point set, which leads to explicit formulas for the rational homology of the moduli space.

We study complex projective surfaces admitting a Poisson structure; we prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.

Some situations that change the parameters of the kinematic structure may cause the robot end effector to deviate [Merlet [2005] Parallel Robots, Vol. 128 (Springer Science & Business Media, Germany)] from the desired trajectory. This effect is called the robustness of the robot by Merlet. One of the ways to correct the robustness is by updating the robot trajectory. The jerk vector of the robot end effector is the third-order positional variation of the TCP and defined as thus the time derivative of the acceleration vector. If there is a high curvature on the transition curve trajectory of robot, then there is a tangential jerk along the trajectory. In this study, the geometrically offset trajectory of the robot end effector from the current trajectory was obtained by using the curvature theory. The angular velocity and angular acceleration of the offset trajectory were calculated. An example of the main trajectory of robot end effector and its offset is given. Also, the jerk of the robot end effector of the offset trajectory was calculated according to the curvature of the trajectory surface in case of a jerk problem caused by a high curvature in the transition curve along the offset trajectory curve.

We study the non-lightlike ruled surfaces in Minkowski 3-space with non-lightlike base curve $c(s)=\int (\alpha t+\beta n+\gamma b)ds$, where $t$, $n$, $b$ are the tangent, principal normal and binormal vectors of an arbitrary timelike curve $\mathrm{\Gamma }(s)$. Some important results of flat, minimal, II-minimal and II-flat non-lightlike ruled surfaces are studied. Finally, the following interesting theorem is proved: the only non-zero constant mean curvature (CMC) non-lightlike ruled surface is developable timelike ruled surface generated by binormal vector.

Developable surfaces are defined to be locally isometric to a plane. These surfaces can be formed by bending thin flat sheets of material, which makes them an active research topic in computer graphics, computer aided design, computational origami and manufacturing architecture. We obtain condition for developable and minimal ruled surfaces using rotation frame. Also, the validity of the theorems is illustrated with examples.

In this study, we take the striction line which only exists on a ruled surface as an optical fiber. We examine Berry phase model of an optical fiber with a geodesic Frenet frame. We find the direction of the polarized light concerning the plane in which the electric field $Ê$ is located. Also, the electromagnetic trajectories of optical fiber are obtained. Finally, we give the electromagnetic flux ruled surfaces formed by the Killing vector fields of the EM-trajectories of the optical fiber and numerical examples.

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere $S^2$. In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame $\{T_e,N_e,B_e\}$ of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for $T_e,N_e,B_e$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces.

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given.