On Helical Surfaces with a Constant Ratio of Principal Curvatures

TL;DR

This paper classifies all helical surfaces in three-dimensional space with a constant ratio of principal curvatures, providing explicit solutions and analyzing their shapes and singularities.

Contribution

It is the first to explicitly determine non-rotational CRPC helical surfaces using a novel approach with generating profiles and differential equations.

Findings

Explicit parametric solutions for CRPC helical surfaces.

Classification of shapes and singularities for different ratios.

Extension beyond known rotational CRPC surfaces.

Abstract

We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio $a:=\kappa_1/\kappa_2$ of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. A key ingredient in the successful determination of these surfaces is the proper choice of generating profiles. We employ the contours for parallel projection orthogonal to the helical axis. This has the advantage that the CRPC property can be nicely expressed with the help of the involution of conjugate surface tangents. The arising ordinary differential equation has an explicit parametric solution, which forms the basis for a further study and classification of the possible shapes and the singularities arising for $a>0$.