{{Short description|Geometric surface}} In geometry, a '''pseudosphere''' is a surface in <math>\mathbb{R}^3</math>. It is the most famous example of a '''pseudospherical surface'''. A pseudospherical surface is a surface piecewise smoothly immersed in <math>\mathbb{R}^3</math> with constant negative Gaussian curvature. A "pseudospherical surface of radius {{mvar|R}}" is a surface in <math>\mathbb{R}^3</math> having 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>. Examples include the tractroid, Dini's surfaces, breather surfaces, and the Kuen surface.

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

== Tractroid == right|frame|Tractroid By "the pseudosphere", people usually mean the tractroid. The tractroid is obtained by revolving a tractrix about its asymptote. 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 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,<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>

== Line congruence == A '''line congruence''' is a 2-parameter families of lines in <math>\R^3</math>. It can be written as<math display="block">X(u, v, t)=x(u, v)+t p(u, v)</math>where each pick of <math>u, v \in \R</math> picks a specific line in the family.

A '''focal surface''' of the line congruence is a surface that is tangent to the line congruence. At each point on the surface,<math display="block">\det\left(\partial_u X, \partial_v X, p\right)=0</math>The above equation expands to a quadratic equation in <math>t</math>:<math display="block">\det (\partial_u x(u, v)+ t\partial_u p(u, v), \partial_v x(u, v)+ t\partial_v p(u, v), p(u, v)) = 0</math>Thus, for each <math>(u, v) \in \R^2</math>, there in general exists two choices of <math>t_1(u, v), t_2(u, v)</math>. Thus a generic line congruence has exactly two focal surfaces parameterized by <math>t_1(u, v), t_2(u, v)</math>.

For a bundle of lines normal to a smooth surface, the two focal surfaces correspond to its evolutes: the loci of centers of principal curvature.

In 1879, Bianchi proved that if a line congruence is such that the corresponding points on the two focal surfaces are at a constant distance 1, that is, <math>|t_1(u, v) - t_2(u, v)| = 1</math>, then both of the focal surfaces have constant curvature -1.

In 1880, Lie proved a partial converse. Let <math display="inline">X</math> be a pseudospherical surface. Then there exists a second pseudospherical surface <math display="inline">\hat{X}</math> and a line congruence <math display="inline">\mathcal{L}</math> such that <math display="inline">X</math> and <math display="inline">\hat{X}</math> are the focal surfaces of <math display="inline">\mathcal{L}</math>. Furthermore, <math display="inline">\hat{X}</math> and <math display="inline">\mathcal{L}</math> may be constructed from <math display="inline">X</math> by integrating a sequence of ODEs. == Universal covering space == right|thumb|The pseudosphere and its relation to three other models of hyperbolic geometry 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 {{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 Lie transformation. In the corresponding solutions to the sine-Gordon equation, this deformation corresponds to a Lorentz Boost of the static 1-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 | hdl=11729/142 | hdl-access=free }}</ref> This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a Minkowski space.

== 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.

On a surface, at each point, draw a cross, pointing at the two directions of principal curvature. These crosses can be integrated into two families of curves, making up a coordinate system on the surface. Let the coordinate system be written as <math>(x, y)</math>.

At each point on a pseudospherical surface there in general exists two asymptotic directions. Along them, the curvature is zero. Let the angle between the asymptotic directions be <math>\theta</math>.

A theorem states that<math display="block">\partial_{xx} \theta - \partial_{yy} \theta = \sin\theta</math>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 <math>\mathbb{R}^3</math>.

This connection between sine-Gordon equations and pseudospherical surfaces mean that one can identify solutions to the equation with surfaces. Then, any way to generate new sine-Gordon solutions from old automatically generates new pseudospherical surfaces from old, and vice versa.

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}}

== Further reading == * {{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=https://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}} * {{Cite book |title=Bäcklund and Darboux transformations: geometry and modern applications in soliton theory |date=2002 |publisher=Cambridge University Press |isbn=978-0-521-01288-1 |editor-last=Rogers |editor-first=C. |series=Cambridge texts in applied mathematics |location=Cambridge New York |editor-last2=Schief |editor-first2=Wolfgang K.}}

== 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 constant curvature