A note on helices in the Euclidean space and in the hyperbolic space

TL;DR

This paper characterizes helices in Euclidean and hyperbolic spaces, showing they have constant curvature and torsion, and identifies their parametrizations and geodesic properties in hyperbolic space.

Contribution

It provides a unified analysis of helices in Euclidean and hyperbolic spaces, including explicit parametrizations and their relation to geodesics.

Findings

Helices in Euclidean space have constant curvature and torsion.

Explicit parametrizations of hyperbolic space helices are derived.

Helices in hyperbolic space are geodesics on certain surfaces.

Abstract

We prove that general helices in Euclidean space for Killing vector fields associated to rotations are helices, that is, curves with constant curvature and constant torsion. In hyperbolic space $\h^3$, we obtain the parametrization of helices for the Killing vector fields associated to hyperbolic rotations, spherical rotations and parabolic rotations. It is proved that these helices are geodesics in suitable surfaces of $\h^3$.