General helices and a theorem of Lancret

Abstract

Abstract. We present a theorem of Lancret for general helices in a 3-dimensional real-space-form which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3-sphere: the problem of solving natural equations and the closed curve problem.

New Type Direction Curves in 3-Dimensional Compact Lie Group

“…">IntroductionMany authors have made significant contributions to the theory of curves from past to present. Some of these studies indicated that the relationships between the curvatures of the space curves are quite remarkable, and the new special curves are also defined via these relations [1][2][3][4][5]. Helices, one of these special space curves, have been studied by many researchers [6][7][8][9].…”

Letter: Models of Relativistic Particle with Curvature and Torsion Revisited

“…In [4], the second author used the concept of Killing vector field along a curve to define the notion of general helix in a three dimensional real space form, M(C). Then, he obtained the extension of the Lancret program to this framework.…”

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

“…The well-known characterizations obtained for a single curve have allowed to define some special curves such as helix, slant helix, plane curve, spherical curve, etc. [1,11,13,20,23] and these special curves, especially helices, are used in many applications [2,9,10,19]. Similarly, by considering two curves, some special curve pairs such as involute-evolute curves, Bertrand curves, Mannheim curves have been defined and studied so far [4,12,14,15,18,21,22].…”