# Pseudosphere

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

A pseudosphere of radius R is a surface in having curvature −1/*R*^{2} in each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/*R*^{2}. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.^{[1]}

## Tractroid[edit]

The same surface can be also described as the result of revolving a tractrix about its asymptote.
For this reason the pseudosphere is also called **tractroid**. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by^{[2]}

It is a singular space (the equator is a singularity), but away from the singularities, it has constant negative Gaussian curvature and therefore is locally isometric to a hyperbolic plane.

The name "pseudosphere" comes about because it has a two-dimensional surface of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the sphere has at every point a positively curved geometry of a dome the whole pseudosphere has at every point the negatively curved geometry of a saddle.

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. For a given edge radius R, the area is 4π*R*^{2} just as it is for the sphere, while the volume is 2/3π*R*^{3} and therefore half that of a sphere of that radius.^{[4]}^{[5]}

The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.^{[6]}

## Universal covering space[edit]

The half pseudosphere of curvature −1 is covered by the interior of a horocycle. In the Poincaré half-plane model one convenient choice is the portion of the half-plane with *y* ≥ 1.^{[7]} Then the covering map is periodic in the x direction of period 2π, and takes the horocycles *y* = *c* to the meridians of the pseudosphere and the vertical geodesics *x* = *c* to the tractrices that generate the pseudosphere. 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. The precise mapping is

where

is the parametrization of the tractrix above.

## Hyperboloid[edit]

In some sources that use the hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a **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.

## Pseudospherical surfaces[edit]

A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the Dini's surfaces, breather surfaces, and the Kuen surface.

## Relation to solutions to the sine-Gordon equation[edit]

Pseudospherical surfaces can be constructed from solutions to the sine-Gordon equation.^{[9]} A sketch proof starts with reparametrizing the tractroid with coordinates in which the Gauss–Codazzi equations can be rewritten as the sine-Gordon equation.

In particular, for the tractroid the Gauss–Codazzi equations are the sine-Gordon equation applied to the static soliton solution, so the Gauss–Codazzi equations are satisfied. 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.

Then any solution to the sine-Gordon equation can be used to specify a first and second fundamental form which satisfy the Gauss–Codazzi 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:

- Static 1-soliton: pseudosphere
- Moving 1-soliton: Dini's surface
- Breather solution: Breather surface
- 2-soliton: Kuen surface

## See also[edit]

- Hilbert's theorem (differential geometry)
- Dini's surface
- Gabriel's Horn
- Hyperboloid
- Hyperboloid structure
- Quasi-sphere
- Sine–Gordon equation
- Sphere
- Surface of revolution
- Mathematics in the fabric arts

## References[edit]

**^**Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Treatise on the interpretation of non-Euclidean geometry].*Gior. Mat.*(in Italian).**6**: 248–312.

(Also Beltrami, Eugenio (July 2010).*Opere Matematiche*[*Mathematical Works*] (in Italian). Vol. 1. Scholarly Publishing Office, University of Michigan Library. pp. 374–405. ISBN 978-1-4181-8434-6.;

Beltrami, Eugenio (1869). "Essai d'interprétation de la géométrie noneuclidéenne" [Treatise on the interpretation of non-Euclidean geometry].*Annales de l'École Normale Supérieure*(in French).**6**: 251–288. doi:10.24033/asens.60. Archived from the original on 2016-02-02. Retrieved 2010-07-24.)**^**Bonahon, Francis (2009).*Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots*. AMS Bookstore. p. 108. ISBN 978-0-8218-4816-6., Chapter 5, page 108**^**Stillwell, John (2010).*Mathematics and Its History*(revised, 3rd ed.). Springer Science & Business Media. p. 345. ISBN 978-1-4419-6052-8., extract of page 345**^**Le Lionnais, F. (2004).*Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences*(2 ed.). Courier Dover Publications. p. 154. ISBN 0-486-49579-5., Chapter 40, page 154**^**Weisstein, Eric W. "Pseudosphere".*MathWorld*.**^**Roberts, Siobhan (15 January 2024). "The Crochet Coral Reef Keeps Spawning, Hyperbolically".*The New York Times*.**^**Thurston, William,*Three-dimensional geometry and topology*, vol. 1, Princeton University Press, p. 62.**^**Hasanov, Elman (2004), "A new theory of complex rays",*IMA J. Appl. Math.*,**69**(6): 521–537, doi:10.1093/imamat/69.6.521, ISSN 1464-3634, archived from the original on 2013-04-15**^**Wheeler, Nicholas. "From Pseudosphere to sine-Gordon equation" (PDF). Retrieved 24 November 2022.

- Stillwell, J. (1996).
*Sources of Hyperbolic Geometry*. Amer. Math. Soc & London Math. Soc. - Henderson, D. W.; Taimina, D. (2006). "Experiencing Geometry: Euclidean and Non-Euclidean with History".
*Aesthetics and Mathematics*(PDF). Springer-Verlag. - Kasner, Edward; Newman, James (1940).
*Mathematics and the Imagination*. Simon & Schuster. pp. 140, 145, 155.

## External links[edit]

- Non Euclid
- Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
- Norman Wildberger lecture 16, History of Mathematics, University of New South Wales. YouTube. 2012 May.
- Pseudospherical surfaces at the virtual math museum.