A New Approach to Mannheim Curve in Euclidean 3-Space

In this article, a new approach is given for Mannheim curves in 3-dimensional Euclidean space. Thanks to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Mannheim curve in E³. In addition, related examples and graphs are given by showing that there can be Mannheim curves in Salkowski or anti-Salkowski curves as well as giving Mannheim mate curves, which are not in literature. Finally, the Mannheim partner curves are characterized in E³.

Some special curves belonging to Mannheim curves pair

In this paper, we investigate special Smarandache curves with regard to Sabban frame for Mannheim partner curve spherical indicatrix. We create Sabban frame belonging to this curves. Smarandache curves are explained by taking position vector as Sabban vectors belonging to this curves. Then, we calculate geodesic curvatures of this Smarandache curves. Found results are expressed depending on the Mannheim curve.

A pair of kinematically related space curves

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a “radius” function and a “pitch” function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve.

Generalized Timelike Mannheim Curves in Minkowski Space-Time

We give the definition of generalized timelike Mannheim curve in Minkowski space-time . The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

No Null-Helix Mannheim Curves in the Minkowski SpaceE13

We study a null Mannheim curve with time-like or space-like Mannheim partner curve in the Minkowski 3-spaceE13. We get the characterization of a null Mannheim curve. Then, we investigate there is no null-helix Mannheim curve inE13.