Pseudosphere

In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.

The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.

As early as 1693 Christiaan Huygens found that the volume and the surface area of the pseudosphere are finite,[3] despite the infinite extent of the shape along the axis of rotation.

[4][5] The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.

This mapping is a local isometry, and thus exhibits the portion y ≥ 1 of the upper half-plane as the universal covering space of the pseudosphere.

[8] This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a Minkowski space.

In these coordinates the first and second fundamental forms are written in a way that makes clear the Gaussian curvature is −1 for any solution of the sine-Gordon equations.

There is then a theorem that any such set of initial data can be used to at least locally specify an immersed surface in

A few examples of sine-Gordon solutions and their corresponding surface are given as follows: (Republished in Beltrami, Eugenio (1902).

Translated into French as "Essai d'interprétation de la géométrie noneuclidéenne".