Pseudosphere

Image:Pseudosphere.png

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 a tractroid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by{{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]

:

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,{{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] despite the infinite extent of the shape along the axis of rotation. For a given edge radius {{mvar|R}}, the area is {{math|4πR²}} just as it is for the sphere, while the volume is {{math|{{sfrac|2|3}}πR³}} and therefore half that of a sphere of that radius.{{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]

{{MathWorld|title=Pseudosphere|urlname=Pseudosphere}}

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

Image:Geodesics on the pseudosphere and three other models of hyperbolic geometry.png

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}}.{{citation|first=William|last=Thurston|title=Three-dimensional geometry and topology|volume=1|publisher=Princeton University Press|page=62}}. 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

: $(x,y)\backslash mapsto\; \backslash big(v(\backslash operatorname\{arcosh\}\; y)\backslash cos\; x,\; v(\backslash operatorname\{arcosh\}\; y)\; \backslash sin\; x,\; u(\backslash operatorname\{arcosh\}\; y)\backslash big)\; ,$

where

: $t\backslash mapsto\; \backslash big(u(t)\; =\; t\; -\; \backslash operatorname\{tanh\}\; t,v(t)\; =\; \backslash operatorname\{sech\}\; t\backslash big)$

is the parametrization of the tractrix above.

File:Deforming a pseudosphere to Dini's surface.gif. 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.

{{citation

| first=Elman

| last=Hasanov

| year=2004

| title=A new theory of complex rays

| journal=IMA J. Appl. Math.

| volume=69

| issue=6

| pages=521–537

| 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

}}

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

A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in $\backslash mathbb\{R\}^3$ 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.

Pseudospherical surfaces can be constructed from solutions to the sine-Gordon equation.{{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 }} 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 $\backslash mathbb\{R\}^3$.

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

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

{{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}}

• [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