In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in $\backslash mathbb^3$ 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.
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** 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
:$t\; \backslash mapsto\; \backslash left(\; t\; -\; \backslash tanh,\; \backslash operatorname\backslash ,\; \backslash right),\; \backslash quad\; \backslash quad\; 0\; \backslash le\; t\; <\; \backslash infty.$
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, 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 pseudosphere and its relation to three other models of hyperbolic geometry
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
:$(x,y)\backslash mapsto\; \backslash big(v(\backslash operatorname\; y)\backslash cos\; x,\; v(\backslash operatorname\; y)\; \backslash sin\; x,\; u(\backslash operatorname\; y)\backslash big)$
where
:$t\backslash mapsto\; \backslash big(u(t)\; =\; t\; -\; \backslash operatorname\; t,v(t)\; =\; \backslash operatorname\; t\backslash big)$
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.

See also

*Hilbert's theorem (differential geometry) *Dini's surface *Gabriel's Horn *Hyperboloid *Hyperboloid structure *Quasi-sphere *Sine–Gordon equation *Sphere *Surface of revolution

References

* * * {{cite book|first1=Edward |last1=Kasner |first2=James |last2=Newman |date=1940 |title=Mathematics and the Imagination |page=140, 145, 155 |publisher=Simon & Schuster

External links

Non Euclid

Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina

Norman Wildberger lecture 16

History of Mathematics, University of New South Wales. YouTube. 2012 May.

at the virtual math museum. Category:Differential geometry Category:Hyperbolic geometry Category:Surfaces Category:Spheres

See also

*Hilbert's theorem (differential geometry) *Dini's surface *Gabriel's Horn *Hyperboloid *Hyperboloid structure *Quasi-sphere *Sine–Gordon equation *Sphere *Surface of revolution

References

* * * {{cite book|first1=Edward |last1=Kasner |first2=James |last2=Newman |date=1940 |title=Mathematics and the Imagination |page=140, 145, 155 |publisher=Simon & Schuster

External links

Non Euclid

Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina

Norman Wildberger lecture 16

History of Mathematics, University of New South Wales. YouTube. 2012 May.

at the virtual math museum. Category:Differential geometry Category:Hyperbolic geometry Category:Surfaces Category:Spheres