The Liouville parametrization of a triaxial ellipsoid

TL;DR

This paper develops a novel Liouville parametrization for the triaxial ellipsoid, filling a gap in the literature by providing an explicit parametrization using a generalized Jacobi amplitude.

Contribution

It introduces the generalized Jacobi amplitude as the inverse of the elliptic integral of the third kind to construct the Liouville parametrization of the triaxial ellipsoid.

Findings

Provides the first explicit Liouville parametrization for the triaxial ellipsoid.

Utilizes the generalized Jacobi amplitude to achieve this parametrization.

Fills a gap in the mathematical literature on quadrics and Liouville surfaces.

Abstract

In this article we will construct the Liouville parametrization of the triaxial ellipsoid. In the literature quadrics are given as examples of Liouville surfaces, yet no one gives such a parametrization. For this we introduce the generalized Jacobi amplitude as inverse of the elliptic integral of the third kind.