In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius R is a surface in having curvature −1/R2 in 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. 
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 tractroid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by 
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,  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.  
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.  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
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.  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 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.  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 .
A few examples of Sine-Gordon solutions and their corresponding surface are given as follows:
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The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions.
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Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model. He also developed singular value decomposition for matrices, which has been subsequently rediscovered several times. Beltrami's use of differential calculus for problems of mathematical physics indirectly influenced development of tensor calculus by Gregorio Ricci-Curbastro and Tullio Levi-Civita.
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