# Characterization of Spherical and Plane Curves Using Rotation Minimizing Frames

## TL;DR

This paper explores the properties of spherical and plane curves in Euclidean and Lorentz-Minkowski spaces using rotation minimizing frames, providing new characterizations and applications to special curve types.

## Contribution

It introduces novel characterizations of spherical and plane curves via RM frames and applies these to classify Bertrand curves and slant helices.

## Key findings

- Derived formulas for curvature and torsion of spherical curves
- Characterized curves lying on moving planes spanned by tangent and RM vectors
- Connected natural mates to spherical and general helices

## Abstract

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves and slant helices as curves whose so-called natural mates are spherical and general helices, respectively.