Editing Pseudosphere

{{Short description|Geometric surface}}
In [[geometry]], a '''pseudosphere''' is a surface with constant negative [[Gaussian curvature]].

A pseudosphere of radius {{mvar|R}} is a surface in <math>\mathbb{R}^3</math> having [[Gaussian curvature|curvature]] −1/''R''<sup>2</sup> at each point. Its name comes from the analogy with the sphere of radius {{mvar|R}}, which is a surface of curvature 1/''R''<sup>2</sup>. The term was introduced by [[Eugenio Beltrami]] in his 1868 paper on models of [[hyperbolic geometry]].<ref>
{{cite journal
 | first=Eugenio
 | last=Beltrami
 | title=Saggio sulla interpretazione della geometria non euclidea
 | trans-title=Essay on the interpretation of noneuclidean geometry
 | journal=Gior. Mat.
 | volume=6
 | pages=248–312
 | language=it
 | year=1868
}} {{pb}}
(Republished in
{{cite book
 | first=Eugenio
 | last=Beltrami
 | title=Opere Matematiche
 | date=1902
 | volume=1
 | at=[https://archive.org/details/operematematiche01beltuoft/page/374/ XXIV, {{pgs|374–405}}]
 | place=Milan |publisher=Ulrico Hoepli
}} Translated into French as
{{cite journal
 | first=Eugenio
 | last=Beltrami
 | display-authors=0
 | title=Essai d'interprétation de la géométrie noneuclidéenne
 | translator=J. Hoüel 
 | journal=Annales Scientifiques de l'École Normale Supérieure |series=Ser. 1
 | year=1869
 | volume=6
 | pages=251–288
 | doi=10.24033/asens.60 |doi-access=free
 | id={{EuDML|80724}}
}} Translated into English as "Essay on the interpretation of noneuclidean geometry" by [[John Stillwell]], in {{harvnb|Stillwell|1996|pp=7–34}}.)</ref>

__TOC__

== Tractroid ==
[[Image:Pseudosphere.png|right|frame|Tractroid]]
The same surface can be also described as the result of [[surface of revolution|revolving]] a [[tractrix]] about its [[asymptote]].
For this reason the pseudosphere is also called a  '''tractroid'''. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by<ref>{{cite book |title=Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots
|first1=Francis
|last1=Bonahon
|publisher=AMS Bookstore
|year=2009
|isbn=978-0-8218-4816-6
|page=108
|url=https://books.google.com/books?id=YZ1L8S4osKsC}}, [https://books.google.com/books?id=YZ1L8S4osKsC&pg=PA108 Chapter 5, page 108]
</ref>
: <math>t \mapsto \left( t - \tanh t, \operatorname{sech}\,t \right), \quad \quad 0 \le t < \infty.</math>

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

The name "pseudosphere" comes about because it has a [[dimension|two-dimensional]] [[Surface (topology)|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 [[positive number|positively]] curved geometry of a [[dome]] the whole pseudosphere has at every point the [[negative number|negatively]] curved geometry of a [[saddle surface|saddle]].

As early as 1693 [[Christiaan Huygens]] found that the volume and the surface area of the pseudosphere are finite,<ref>{{cite book |title=Mathematics and Its History |edition=revised, 3rd |first1=John |last1=Stillwell |publisher=Springer Science & Business Media |year=2010 |isbn=978-1-4419-6052-8 |page=345 |url=https://books.google.com/books?id=V7mxZqjs5yUC}}, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA345 extract of page 345]</ref> despite the infinite extent of the shape along the axis of rotation. For a given edge [[radius]] {{mvar|R}}, the [[area]] is {{math|4π''R''<sup>2</sup>}} just as it is for the sphere, while the [[volume]] is {{math|{{sfrac|2|3}}π''R''<sup>3</sup>}} and therefore half that of a sphere of that radius.<ref>{{cite book
|title=Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences
|edition=2
|first1=F.
|last1=Le Lionnais
|publisher=Courier Dover Publications
|year=2004
|isbn=0-486-49579-5
|page=154
|url=https://books.google.com/books?id=pCYDhbhu1O0C}}, [https://books.google.com/books?id=pCYDhbhu1O0C&pg=PA154 Chapter 40, page 154]
</ref><ref>{{MathWorld|title=Pseudosphere|urlname=Pseudosphere}}</ref>

The pseudosphere is an important geometric precursor to mathematical [[fabric arts]] and [[pedagogy]].<ref>{{cite news | url=https://www.nytimes.com/2024/01/15/science/mathematics-crochet-coral.html | title=The Crochet Coral Reef Keeps Spawning, Hyperbolically | work=The New York Times | date=15 January 2024 | last1=Roberts | first1=Siobhan }}</ref>

== Universal covering space ==
[[Image:Geodesics on the pseudosphere and three other models of hyperbolic geometry.png|right|thumb|The pseudosphere and its relation to three other models of hyperbolic geometry]]
The half pseudosphere of curvature −1 is [[covering space|covered]] by the interior of a [[horocycle]]. In the [[Poincaré half-plane model]] one convenient choice is the portion of the half-plane with {{math|''y'' ≥ 1}}.<ref>{{citation|first=William|last=Thurston|title=Three-dimensional geometry and topology|volume=1|publisher=Princeton University Press|page=62}}.</ref>  Then the covering map is periodic in the {{mvar|x}} direction of period 2{{pi}}, and takes the horocycles {{math|1=''y'' = ''c''}} to the meridians of the pseudosphere and the vertical geodesics {{math|1=''x'' = ''c''}} to the tractrices that generate the pseudosphere.  This mapping is a local isometry, and thus exhibits the portion {{math|''y'' ≥ 1}} of the upper half-plane as the [[universal covering space]] of the pseudosphere.  The precise mapping is
: <math>(x,y)\mapsto \big(v(\operatorname{arcosh} y)\cos x, v(\operatorname{arcosh} y) \sin x, u(\operatorname{arcosh} y)\big) ,</math>
where
: <math>t\mapsto \big(u(t) = t - \operatorname{tanh} t,v(t) =  \operatorname{sech} t\big)</math>
is the parametrization of the tractrix above.

== Hyperboloid ==
[[File:Deforming a pseudosphere to Dini's surface.gif|thumb|506x506px|Deforming the pseudosphere to a portion of [[Dini's surface]]. In differential geometry, this is a [[Sine-Gordon equation#New solutions from old|Lie transformation]]. In the corresponding solutions to the [[sine-Gordon equation]], this deformation corresponds to a [[Lorentz Boost]] of the static 1-[[Soliton (topological)|soliton]] solution.]]
In some sources that use the [[hyperboloid model]] of the hyperbolic plane, the hyperboloid is referred to as a '''pseudosphere'''.<ref>
{{citation
 | first=Elman
 | last=Hasanov
 | year=2004
 | title=A new theory of complex rays
 | journal=IMA J. Appl. Math.
 | volume=69
 | issue=6
 | pages=521&ndash;537
 | issn=1464-3634
 | url=http://imamat.oxfordjournals.org/cgi/reprint/69/6/521
 | archive-url=https://archive.today/20130415131937/http://imamat.oxfordjournals.org/cgi/reprint/69/6/521
 | url-status=dead
 | archive-date=2013-04-15
 | doi=10.1093/imamat/69.6.521
}}</ref>
This usage of the word is because the hyperboloid can be [[hyperboloid#Relation to the sphere|thought of as a sphere]] of imaginary radius, embedded in a [[Minkowski space]].

== Pseudospherical surfaces ==
A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in <math>\mathbb{R}^3</math> with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the [[Dini's surface]]s, [[breather surface]]s, and the [[Kuen surface]].

== Relation to solutions to the sine-Gordon equation ==
Pseudospherical surfaces can be constructed from solutions to the [[sine-Gordon equation]].<ref name="wheeler">{{cite web |last1=Wheeler |first1=Nicholas |title=From Pseudosphere to sine-Gordon equation |url=https://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous%20Math/Geometric%20Origin%20of%20Sine-Gordon/Pseudosphere%20to%20Sine-Gordon.pdf |access-date=24 November 2022 }}</ref> 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 fundamental form|first]] and [[second fundamental form]]s 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 <math>\mathbb{R}^3</math>.

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 ==
* [[Hilbert's theorem (differential geometry)]]
* [[Dini's surface]]
* [[Gabriel's Horn]]
* [[Hyperboloid]]
* [[Hyperboloid structure]]
* [[Quasi-sphere]]
* [[Sine–Gordon equation]]
* [[Sphere]]
* [[Surface of revolution]]

== References ==
{{reflist}}
* {{cite book|last=Stillwell |first=John |author-link=John Stillwell |title=Sources of Hyperbolic Geometry |date=1996 |publisher=American Mathematical Society & London Mathematical Society |isbn=0-8218-0529-0}}
* {{cite book|last1=Henderson |first1=D. W.|last2=Taimina |first2=D.|author2-link= Daina Taimiņa |title=Aesthetics and Mathematics|publisher=Springer-Verlag|year=2006|url=http://dspace.library.cornell.edu/bitstream/1813/2714/1/2003-4.pdf |chapter=Experiencing Geometry: Euclidean and Non-Euclidean with History}}
* {{cite book|first1=Edward |last1=Kasner |first2=James |last2=Newman |date=1940 |title=[[Mathematics and the Imagination]] |pages=140, 145, 155 |publisher=[[Simon & Schuster]]}}

== External links ==
* [http://www.cs.unm.edu/~joel/NonEuclid/pseudosphere.html Non Euclid]
* [http://www.cabinetmagazine.org/issues/16/crocheting.php Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina]
* [http://virtualmathmuseum.org/Surface/gallery_o.html#PseudosphericalSurfaces Pseudospherical surfaces] at the virtual math museum.

[[Category:Differential geometry of surfaces]]
[[Category:Hyperbolic geometry]]
[[Category:Surfaces]]
[[Category:Spheres]]
[[Category:Surfaces of revolution of constant negative curvature]]