Pseudosphere

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.

Theoretical pseudosphere

In its general interpretation, a pseudosphere of radius R is any surface of curvature −1/R², 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]

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]

$t \mapsto \left( t - \tanh{t}, \operatorname{sech}\,{t} \right), \quad \quad 0 \le t < \infty.$

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 is a two-dimensional 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 1693 Christiaan 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 4πR² just as it is for the sphere, while the volume is 2/3πR³ and therefore half that of a sphere of that radius.^[4][5]

Universal covering space

The half pseudosphere of curvature −1 is covered by the portion of the hyperbolic upper half-plane with y ≥ 1.^[6] The covering map is periodic in the x direction of period 2π, 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. The precise mapping is

$(x,y)\mapsto (v(\operatorname{arcosh} y)\cos x, v(\operatorname{arcosh} y) \sin x, u(\operatorname{arcosh} y))$

where $t\mapsto (u(t),v(t))$ is the arclength 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.^[7] 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

• Dini's surface
• Gabriel's Horn
• Hyperboloid
• Hyperboloid structure
• Sine–Gordon equation
• Sphere
• Surface of revolution

References

1. Beltrami, Eugenio (1868), "Saggio sulla interpretazione della geometria non euclidea", Gior. Mat. (in Italian) 6: 248–312
   (Also Beltrami, Eugenio, Opere Matematiche (in Italian) 1, pp. 374–405, ISBN 1-4181-8434-9;
   Beltrami, Eugenio (1869), "Essai d'interprétation de la géométrie noneuclidéenne", Annales de l'École Normale Supérieure (in French) 6: 251–288)
2. Bonahon, Francis (2009). Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots. AMS Bookstore. p. 108. ISBN 0-8218-4816-X., Chapter 5, page 108
3. Mangasarian, Olvi L.; Pang, Jong-Shi (1999). Computational optimization: a tribute to Olvi Mangasarian, Volume 1. Springer. p. 324. ISBN 0-7923-8480-6., Chapter 17, page 324
4. Le Lionnais, F. (2004). Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences (2 ed.). Courier Dover Publications. p. 154. ISBN 0-486-49579-5., Chapter 40, page 154
5. Weisstein, Eric W., "Pseudosphere", MathWorld.
6. Thurston, William, Three-dimensional geometry and topology 1, Princeton University Press, p. 62.
7. Hasanov, Elman (2004), "A new theory of complex rays", IMA J Appl Math 69: 521–537, doi:10.1093/imamat/69.6.521, ISSN 1464-3634

• Stillwell J: Sources of Hyperbolic Geometry, 1996, Amer. Math. Soc & London Math. Soc.
• Henderson, D. W. and Taimina, D. (2006). "Experiencing Geometry: Euclidean and Non-Euclidean with History". Aesthetics and Mathematics. Springer-Verlag. External link in |chapter= (help)
• Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 140,145,155, Simon & Schuster.

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)
• Norman Wildberger lecture 16, History of Mathematics, University of New South Wales. YouTube. 2012 May.
• Pseudospherical surfaces at the virtual math museum.