• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1099-1108
• /
• 2013

We define the Mannheim curve in a 3-dimensional Riemannian manifold, which is a generalization of the Mannheim curve in Euclidean space. In particular, we study the Mannheim curves and their partner curves in 3-dimensional space forms.

• 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.

• Honam Mathematical Journal
• /
• v.36 no.1
• /
• pp.113-129
• /
• 2014

In this study, we have investigated the possibility of whether any Frenet plane of a given space curve in a 3-dimensional Euclidean space $\mathbb{E}_3$ also is any Frenet plane of another space curve in the same space. We have obtained some characterizations of a given space curve by considering nine possible case.

• Honam Mathematical Journal
• /
• v.45 no.3
• /
• pp.380-396
• /
• 2023

In this study, we introduce the adjoint curves of framed curves. We examine some characterizations of adjoint curve of a framed curve. In addition, we give the conditions for framed curves and adjoint curves to be Bertrand and Mannheim curves. Then, we introduce adjoint curves of Frenet-type framed curves and give ruled surfaces related to adjoint curves. Finally, we create normal and binormal surfaces of the framed adjoint curves and obtain some characterizations of these surfaces and we support by the results with figures.

• Honam Mathematical Journal
• /
• v.36 no.3
• /
• pp.475-492
• /
• 2014

In this study, we have investigated the possibility of whether any spacelike Frenet plane of a given space curve in Minkowski 3-space $\mathbb{E}_1^3$ also is any spacelike Frenet plane of another space curve in the same space. We have obtained some characterizations of a given space curve by considering nine possible case.