In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid.
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Theoretical Pseudosphere
In its general interpretation, a pseudosphere of radius R is any surface of curvature −1/R^{2} (precisely, a complete, simply connected surface of that curvature), by analogy with the sphere of radius R, which is a surface of curvature 1/R^{2}. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry^{[1]}.
Tractricoid
The term is also used to refer to a certain surface called the tractricoid: the result of 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^{[2]}
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 is a twodimensional surface of constant negative curvature just like a sphere with positive Gauss 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 1639 Christian 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 4πR^{2} just as it is for the sphere, while the volume is 2/3 πR^{3} and therefore half that of a sphere of that radius.^{[4]}^{[5]}
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