Note on Surfaces of Revolution with an Affine-Linear Relation between their Curvature Radii

TL;DR

This paper derives explicit parametrizations for surfaces of revolution with specific curvature relations, including affine-linear relations between curvature radii and constant principal curvature ratios, and explores their algebraic properties.

Contribution

It provides explicit parametrizations for these surfaces, including their offsets and special cases with constant principal curvature ratios, expanding understanding of their geometric properties.

Findings

Explicit parametrizations for surfaces with affine-linear curvature radius relations.

Identification of algebraic surfaces among these parametrizations.

Parametrizations for surfaces with constant principal curvature ratios and fixed angle between parameter curves.

Abstract

This note derives parametrizations for surfaces of revolution that satisfy an affine-linear relation between their respective curvature radii. Alongside, parametrizations for the uniform normal offsets of those surfaces are obtained. Those parametrizations are found explicitly for a countably-infinite many of them, and of those, it is shown which are algebraic. Lastly, for those surfaces which have a constant ratio of principal curvatures, parametrizations with a constant angle between the parameter curves are found.