Template:Short description In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius Template:Mvar is a surface in <math>\mathbb{R}^3</math> having curvature −1/R2 at each point. Its name comes from the analogy with the sphere of radius Template:Mvar, which is a surface of curvature 1/R2. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.<ref> Template:Cite journal Template:Pb (Republished in Template:Cite book Translated into French as Template:Cite journal Translated into English as "Essay on the interpretation of noneuclidean geometry" by John Stillwell, in Template:Harvnb.)</ref>
TractroidEdit
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<ref>Template:Cite book, 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>Template:Cite book, extract of page 345</ref> despite the infinite extent of the shape along the axis of rotation. For a given edge radius Template:Mvar, the area is Template:Math just as it is for the sphere, while the volume is Template:Math and therefore half that of a sphere of that radius.<ref>Template:Cite book, Chapter 40, page 154 </ref><ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Pseudosphere%7CPseudosphere.html}} |title = Pseudosphere |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>
The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.<ref>Template:Cite news</ref>
Universal covering spaceEdit
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 Template:Math.<ref>Template:Citation.</ref> Then the covering map is periodic in the Template:Mvar direction of period 2Template:Pi, and takes the horocycles Template:Math to the meridians of the pseudosphere and the vertical geodesics Template:Math to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion Template:Math 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.
HyperboloidEdit
In some sources that use the hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere.<ref> Template:Citation</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.
Pseudospherical surfacesEdit
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 surfaces, breather surfaces, and the Kuen surface.
Relation to solutions to the sine-Gordon equationEdit
Pseudospherical surfaces can be constructed from solutions to the sine-Gordon equation.<ref name="wheeler">{{#invoke:citation/CS1|citation |CitationClass=web }}</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 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>.
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 alsoEdit
- Hilbert's theorem (differential geometry)
- Dini's surface
- Gabriel's Horn
- Hyperboloid
- Hyperboloid structure
- Quasi-sphere
- Sine–Gordon equation
- Sphere
- Surface of revolution
ReferencesEdit
External linksEdit
- Non Euclid
- Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
- Pseudospherical surfaces at the virtual math museum.