Surfaces of constant principal-curvatures ratio in isotropic geometry

TL;DR

This paper investigates surfaces with a constant principal-curvature ratio in Euclidean and isotropic geometries, classifying various types of such surfaces using advanced differential geometry methods.

Contribution

It characterizes and classifies surfaces with constant principal-curvature ratio in both Euclidean and isotropic geometries, including rotational, channel, ruled, helical, and translational surfaces.

Findings

Classification of surfaces with constant principal-curvature ratio.

Characterization of rotational, channel, ruled, helical, and translational surfaces.

Application of differential geometry methods to isotropic and Euclidean geometries.

Abstract

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions (the latter two cases only in isotropic geometry). We use the interlacing of various methods of differential geometry, including line geometry and Lie sphere geometry, ordinary differential equations, and elementary algebraic geometry.