# Curves of Constant Curvature and Torsion in the 3-Sphere

## TL;DR

This paper characterizes curves with constant curvature and torsion in the 3-sphere, revealing their unique behaviors such as periodicity or density in Clifford tori, contrasting with Euclidean helices.

## Contribution

It provides a detailed description of such curves in the 3-sphere, highlighting their global behaviors and differences from Euclidean counterparts.

## Key findings

- Curves are trajectories of superposed orthogonal circular motions.
- They can be periodic or dense in Clifford tori.
- Behavior differs significantly from Euclidean helices.

## Abstract

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the three-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.