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.
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.
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