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Concept
Pseudosphere
Summary
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.
TOC
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 2pi, 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.
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