Pseudosphere

Partial Pseudosphere

In geometry, a pseudosphere of radius R is a surface of curvature −1/R² (precisely, a complete, simply connected surface of that curvature), by 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^[1].

The term is also used to refer to what is traditionally called a tractricoid: the result of revolving a tractrix about its asymptote.

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 two-dimensional surface of constant negative curvature just like a sphere with positive Gauss curvature. It has same formulas for area and volume (R = edge radius) 4πR² and 4πR³/3 of the full surface in spite of the opposite Gauss curvature sign. 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.

References

See also

• Dini's surface
• Hyperboloid structure
• Sine-Gordon equation
• Sphere

References

1. ^ E. Beltrami, Saggio sulla interpretazione della geometria non euclidea, Gior. Mat. 6, 248–312 (Also Op. Mat. 1, 374-405; Ann. École Norm. Sup. 6 (1869), 251-288).

External links

• Non Euclid
• Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
• Prof. C.T.J. Dodson's web site at University of Manchester
• Interactive demonstration of the pseudosphere (at the University of Manchester)