Pseudosphere

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Template:Short description 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 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. 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> 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>

Tractroid

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>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>Template:MathWorld</ref>

The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.<ref>Template:Cite news</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

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 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.

Hyperboloid

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> 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.

Relation to solutions to the sine-Gordon equation

Pseudospherical surfaces can be constructed from solutions to the sine-Gordon equation.<ref name="wheeler">Template:Cite 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.

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:

See also

References

Template:Reflist

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