• Honam Mathematical Journal
• /
• v.43 no.3
• /
• pp.453-463
• /
• 2021

In this study, tubular surfaces that play an important role in technological designs in various branches are examined for the case of the base curve is not satisfying the fundamental theorem of the differential geometry. In order to give an alternative perspective to the researches on tubular surfaces, the modified orthogonal frame is used in this study. Firstly, the relationships between the Serret-Frenet frame and the modified orthogonal frame are summarized. Then the definitions of the tubular surfaces, some theorems, and results are given. Moreover, the fundamental forms, the mean curvature, and the Gaussian curvature of the tubular surface are calculated according to the modified orthogonal frame. Finally, the properties of parameter curves of the tubular surface with modified orthogonal frame are expressed and the tubular surface is drawn according to the Frenet frame and the modified orthogonal frame.

• Honam Mathematical Journal
• /
• v.45 no.4
• /
• pp.668-677
• /
• 2023

In this study, we bring forth a new general formula for inextensible flows of Euclidean curves as regards modified orthogonal frame (MOF) in E³. For an inextensible curve flow, we provide the necessary and sufficient conditions, which are denoted by a partial differential equality containing the curvatures and torsion.

• Honam Mathematical Journal
• /
• v.44 no.3
• /
• pp.419-431
• /
• 2022

Mannheim introduced the concept of a pair of curves, called as Mannheim partner curves, in 1878. Until now, Mannheim partner curves have been studied widely in the literature. In this study, we take into account of this concept according to Positional Adapted Frame (PAF) for the particles moving in the 3-dimensional Euclidean space. We introduce a new type special trajectory pairs which are called Mannheim partner P-trajectories in the Euclidean 3-space. The relationships between the PAF elements of this pair are investigated. Also, the relations between the Serret-Frenet basis vectors of Mannheim partner P-trajectories are given. Afterwards, we obtain the necessary conditions for one of these trajectories to be an osculating curve and for other to be a rectifying curve. Moreover, we provide an example including an illustrative figure.

• International Journal of Naval Architecture and Ocean Engineering
• /
• v.12 no.1
• /
• pp.832-842
• /
• 2020

The coordination control of multiple Underactuated Underwater Vehicles (UUVs) moving in three dimensional space is investigated in this paper. The coordinated path following control task is decomposed into two sub tasks, that is, path following control and coordination control. In the spatial curve path following control task, path following error dynamics is build in the Serret-Frenet coordinate frame. The virtual reference object can be chosen freely on the desired spatial path. Considering the speed of the UUV, the line-of-sight navigation is introduced to help the path following errors quickly converge to zero. In the coordination control sub task, the communication topology of multiple UUVs is described by the graph theory. The speed of each UUV is adjusted to achieve the coordination. The path following system and the coordination control system are viewed as the feedback connection system. Input-to-state stable of the coordinated path following system can be proved by small gain theorem. The simulation experiments can further demonstrate the good performance of the control method.