On the General Helix with First and Second Curvature in Nil 3-Space

Abstract

Nil geometry is one of the eight geometries of Thurston's conjecture. Helix, null helix and slant helix has been examined in geometry as in the papers [1] , [2] and [3], respectively.  In this paper we study in Nil 3-space and the Nil metric with respect to the standard coordinates (x,y,z) is g_Nil3=(dx)²+(dy)²+(dz-xdy)² in IR³. The explicit parametric equation of a general helix is found in this publication. In Nil 3-Space, we also express the explicit equations Frenet vector fields, the first and second curvatures of the general helix. In Nil 3-space, the parametric equation of the Normal and Binormal governed surface of general helix in terms of curvature and torsion has already been explored in [4].