In the paper, three types of surfaces of revolution in the Galilean 3space are defined and studied. The construction of the well-known surface of revolution, defined as the trace of a planar curve rotated about an axis in the supporting plane of the curve, is given for the Galilean 3-space. Then we classify the surfaces of revolution with vanishing Gaussian curvature or vanishing mean curvature in the Galilean 3-space.

In this paper, we introduce a new version of tubular surfaces. We first define a new adapted frame along a space curve, and denote this the q-frame. We then reveal the relationship between the Frenet frame and the q-frame. We give a parametric representation of a directional tubular surface using the q-frame. Finally, some comparative examples are shown to confirm the effectiveness of the proposed method.Mathematics Subject Classification: 53A04, 53A05

In this paper, …rst of all, the de…nition of parallel surfaces in Galilean space is given. Then, the relationship between the curvatures of the parallel surfaces in Galilean space is determined. Moreover, the …rst and second fundamental forms of parallel surfaces are found in Galilean space. Consequently, we obtained Gauss curvature and mean curvature of parallel surface in terms of those curvatures of the base surface.

Abstract. In this paper, we investigate the parallel surfaces of the ruled surfaces in Galilean space. There are three types of ruled surfaces in Galilean space. We derive the necessary conditions for each type of the ruled surfaces of the parallel surfaces to be ruled. Consequently, we construct some examples.

In this paper, we introduce the tube surfaces in pseudo-Galilean 3-space. We classify the tube surfaces into two types according to the spine curve which generates the tube surface in pseudo-Galilean space. Moreover, we show that both types of tube surfaces are spacelike surfaces. Then, we begin to study the local geometry of the tube surfaces. We obtain the first and second fundamental forms of the tube surfaces. In addition, we investigate the Gauss and mean curvatures of the tube surfaces in pseudo-Galilean space.

In this work, we define twisted surfaces in Galilean 3-space. In order to construct these surfaces, a planar curve is subjected to two simultaneous rotations, possibly with different rotation speeds. The existence of Euclidean rotations and isotropic rotations leads to three distinct types of twisted surfaces in Galilean 3-space. Then we classify twisted surfaces in Galilean 3-space with zero Gaussian curvature or zero mean curvature.

The trajectory of a robot end-effector is described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way, the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The curvature theory of a ruled surface generated by a line fixed in the end-effector referred to as the tool line is used for more accurate motion of a robot end-effector. In the present paper, we first defined tool trihedron in which tool line is contained for timelike ruled surface with timelike ruling, and transition relations among surface trihedron: tool trihedron, generator trihedron, natural trihedron, and Darboux vectors for each trihedron, were found. Then differential properties of robot end-effector's motion were obtained by using the curvature theory of timelike ruled surfaces with timelike ruling.

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.