Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

TL;DR

This paper develops rotation minimizing frames in isotropic and pseudo-isotropic 3-spaces, enabling the characterization of spherical curves and exploring geometric structures using Lorentz numbers and dual numbers.

Contribution

It introduces RM frames in isotropic geometries, characterizes spherical curves via linear equations, and discusses the use of Lorentz and dual numbers in these contexts.

Findings

RM frames can be constructed in isotropic geometries.

Spherical curves are characterized by linear curvature equations.

Lorentz and dual numbers facilitate geometric analysis in these spaces.

Abstract

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).