Pseudosphere

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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 (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]

Tractricoid

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 2/3 πR3 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.

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. ^
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