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 $−.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/R2⁠$ , by analogy with the sphere of radius $R$ , which is a surface of curvature $⁠ 1 / R 2 ⁠$ . 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\leq 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 $4π R 2$ just as it is for the sphere, while the volume is $⁠ 2 / 3 ⁠ π R 3$ 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 $y \geq 1$ .^[6] The covering map is periodic in the $x$ direction of period 2 $π$ , 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 $y \geq 1$ of the upper half-plane as the universal covering space of the pseudosphere. The precise mapping is

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

where

t\mapsto {\big (}u(t)=t-\operatorname {tanh} t,v(t)=\operatorname {sech} t{\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.^[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

^ Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Treatise on the interpretation of non-Euclidean geometry]. Gior. Mat. (in Italian). 6: 248–312.
(Also Beltrami, Eugenio. Opere Matematiche [Mathematical Works] (in Italian). Vol. 1. pp. 374–405. ISBN 1-4181-8434-9.;
Beltrami, Eugenio (1869). "Essai d'interprétation de la géométrie noneuclidéenne" [Treatise on the interpretation of non-Euclidean geometry]. Annales de l'École Normale Supérieure (in French). 6: 251–288.)
^ 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
^ Mangasarian, Olvi L.; Pang, Jong-Shi (1999). Computational optimization: a tribute to Olvi Mangasarian. Vol. 1. Springer. p. 324. ISBN 0-7923-8480-6., Chapter 17, page 324
^ 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
^ Weisstein, Eric W. "Pseudosphere". MathWorld.
^ Thurston, William, Three-dimensional geometry and topology, vol. 1, Princeton University Press, p. 62.
^ 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. (1996). Sources of Hyperbolic Geometry. Amer. Math. Soc & London Math. Soc.
Henderson, D. W.; Taimina, D. (2006). "Experiencing Geometry: Euclidean and Non-Euclidean with History". Aesthetics and Mathematics (PDF). Springer-Verlag.
Kasner, Edward; Newman, James (1940). Mathematics and the Imagination. Simon & Schuster. p. 140, 145, 155.

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.

[1] Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Treatise on the interpretation of non-Euclidean geometry]. Gior. Mat. (in Italian). 6: 248–312.
(Also Beltrami, Eugenio. Opere Matematiche [Mathematical Works] (in Italian). Vol. 1. pp. 374–405. ISBN 1-4181-8434-9.;
Beltrami, Eugenio (1869). "Essai d'interprétation de la géométrie noneuclidéenne" [Treatise on the interpretation of non-Euclidean geometry]. 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. Vol. 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, vol. 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

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