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Mannheim curves and their partner curves in Minkowski 3-space E13

Elsharkawy Ayman, Elshenhab Ahmed M.

Demonstratio Mathematica

2022

Vol. 55, nr 1

798--811

The modified orthogonal frame is an important tool to study analytic space curves whose curvatures have discrete zero points. In this article, by using the modified orthogonal frame, Mannheim curves and their partner curves are investigated in Minkowski 3-space. Some characterizations according to the curvatures and torsions of the curves are given. Finally, some relations under the conditions for Mannheim curves and their partner curves to be generalized helices are presented. All the possible cases for the partner curves to be spacelike and timelike are considered in the whole of the article.

On the involute and evolute curves of the timelike curve in Minkowski 3-space

Bukcu B., Karacan K.

Demonstratio Mathematica

2007

Vol. 40, nr 3

721-732

In this study, we have generalized the involute and evolute curves of the timelike curve in Minkowski 3-Space. Firstly, we have shown that, the length between the timelike curve alpha and the spacelike curve beta is constant. Furthermore, the Frenet-Serret frame of the involute curve beta has been found as dependent on curvatures of the curve alpha. We have determined the involute curve beta is planar in which conditions. Secondly, we have found transformation matrix between the evolute curve beta and the curve alpha. Finally, we have computed the curvatures of the evolute curve beta.