Pseudosphere

In geometry, a pseudosphere is a surface in ${\displaystyle \mathbb {R} ^{3}}$. It is the most famous example of a pseudospherical surface. A pseudospherical surface is a surface piecewise smoothly immersed in ${\displaystyle \mathbb {R} ^{3}}$ with constant negative Gaussian curvature. A "pseudospherical surface 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. 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.¹

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 by²

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

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.⁴⁵

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

A line congruence is a 2-parameter families of lines in ${\displaystyle \mathbb {R} ^{3}}$. It can be written as$${\displaystyle X(u,v,t)=x(u,v)+tp(u,v)}$$where each pick of ${\displaystyle u,v\in \mathbb {R} }$ 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,$${\displaystyle \det \left(\partial _{u}X,\partial _{v}X,p\right)=0}$$The above equation expands to a quadratic equation in ${\displaystyle t}$:$${\displaystyle \det(\partial _{u}x(u,v)+t\partial _{u}p(u,v),\partial _{v}x(u,v)+t\partial _{v}p(u,v),p(u,v))=0}$$Thus, for each ${\displaystyle (u,v)\in \mathbb {R} ^{2}}$, there in general exists two choices of ${\displaystyle t_{1}(u,v),t_{2}(u,v)}$. Thus a generic line congruence has exactly two focal surfaces parameterized by ${\displaystyle t_{1}(u,v),t_{2}(u,v)}$.

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, ${\displaystyle |t_{1}(u,v)-t_{2}(u,v)|=1}$, then both of the focal surfaces have constant curvature -1.

In 1880, Lie proved a partial converse. Let ${\textstyle X}$ be a pseudospherical surface. Then there exists a second pseudospherical surface ${\textstyle {\hat {X}}}$ and a line congruence ${\textstyle {\mathcal {L}}}$ such that ${\textstyle X}$ and ${\textstyle {\hat {X}}}$ are the focal surfaces of ${\textstyle {\mathcal {L}}}$. Furthermore, ${\textstyle {\hat {X}}}$ and ${\textstyle {\mathcal {L}}}$ may be constructed from ${\textstyle X}$ by integrating a sequence of ODEs.

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.

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 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.

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 ${\displaystyle (x,y)}$.

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 ${\displaystyle \theta }$.

A theorem states that$${\displaystyle \partial _{xx}\theta -\partial _{yy}\theta =\sin \theta }$$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 ${\displaystyle \mathbb {R} ^{3}}$.

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