Pseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. A pseudosphere of radius is a surface in formula_1 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface of curvature . The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry. Tractricoid. 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 tractricoid. 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 , the area is just as it is for the sphere, while the volume is and therefore half that of a sphere of that radius. Universal covering space. The half pseudosphere of curvature −1 is covered by the portion of the hyperbolic upper half-plane with . The covering map is periodic in the direction of period 2, and takes the horocycles to the meridians of the pseudosphere and the vertical geodesics  to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping is where is the parametrization of the tractrix above. Hyperboloid. 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.