Delaunay Surfaces

TL;DR

This paper provides a unified parametrization of Delaunay surfaces of revolution derived from conic sections, enabling simple curvature expressions and the construction of new constant mean curvature surfaces.

Contribution

It introduces a uniform approach to parametrizing Delaunay surfaces directly from conic roulette parametrizations, simplifying curvature calculations and surface construction.

Findings

Derived explicit formulas for mean and Gaussian curvatures.

Presented a method to generate new constant mean curvature surfaces.

Unified treatment for different types of conics (parabolic, elliptic, hyperbolic).

Abstract

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads to simple expressions for the mean and Gaussian curvatures of the surfaces as well as the construction of new surfaces.