A new approach to rotational Weingarten surfaces

TL;DR

This paper introduces a novel method using geometric linear momentum to analyze rotational Weingarten surfaces, leading to new classification results and simplified proofs of classical curvature theorems.

Contribution

It reduces Weingarten conditions to first order differential equations on the generatrix curve's momentum, enabling new classifications and applications.

Findings

Classified non-degenerated quadric surfaces of revolution.

Characterized elasticoids as rotational elastic curves.

Provided new proofs for classical curvature surface theorems.

Abstract

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, like Euler's theorem about minimal ones, Delaunay's theorem on constant mean curvature ones, and Darboux's theorem about constant Gauss curvature ones.