Pseudosphere

In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.

A pseudosphere of radius R is a surface in ${\displaystyle \mathbb {R} ^{3}}$ having curvature −1/R² at each point. Its name comes from the analogy with the sphere of radius R, which is a surface of curvature 1/R². The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.^[1]

Tractroid

Tractroid

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^[2]

${\displaystyle t\mapsto \left(t-\tanh t,\operatorname {sech} \,t\right),\quad \quad 0\leq t<\infty .}$

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,^[3] despite the infinite extent of the shape along the axis of rotation. For a given edge radius R, the area is 4πR² just as it is for the sphere, while the volume is ⁠2/3⁠πR³ and therefore half that of a sphere of that radius.^[4][5]

The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.^[6]

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 y ≥ 1.^[7] Then the covering map is periodic in the x direction of period 2π, and takes the horocycles y = c to the meridians of the pseudosphere and the vertical geodesics x = c to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion y ≥ 1 of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping is

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

where

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

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.^[8] 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 surfaces

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

Relation to solutions to the sine-Gordon equation

Pseudospherical surfaces can be constructed from solutions to the sine-Gordon equation.^[9] 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 ${\displaystyle \mathbb {R} ^{3}}$.

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

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