Pseudosphere

In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.

A pseudosphere of radius $R$ is a surface in $\mathbb {R} ^{3}$ having curvature −1/R² at each point. Its name comes from the 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]

Tractroid

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 a tractroid. 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 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,^[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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠2/3⁠πR3$ and therefore half that of a sphere of that radius.^[4]^[5]

The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.^[6]

Universal covering space

The half pseudosphere of curvature −1 is covered by the interior of a horocycle. In the Poincaré half-plane model one convenient choice is the portion of the half-plane with $y \geq 1$ .^[7] Then 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.^[8] This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a Minkowski space.

Pseudospherical surfaces

A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in $\mathbb {R} ^{3}$ with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the Dini's surfaces, breather surfaces, and the Kuen surface.

Relation to solutions to the sine-Gordon equation

Pseudospherical surfaces can be constructed from solutions to the sine-Gordon equation.^[9] A sketch proof starts with reparametrizing the tractroid with coordinates in which the Gauss–Codazzi equations can be rewritten as the sine-Gordon equation.

In particular, for the tractroid the Gauss–Codazzi equations are the sine-Gordon equation applied to the static soliton solution, so the Gauss–Codazzi equations are satisfied. In these coordinates the first and second fundamental forms are written in a way that makes clear the Gaussian curvature is −1 for any solution of the sine-Gordon equations.

Then any solution to the sine-Gordon equation can be used to specify a first and second fundamental form which satisfy the Gauss–Codazzi equations. There is then a theorem that any such set of initial data can be used to at least locally specify an immersed surface in $\mathbb {R} ^{3}$ .

A few examples of sine-Gordon solutions and their corresponding surface are given as follows:

Static 1-soliton: pseudosphere
Moving 1-soliton: Dini's surface
Breather solution: Breather surface
2-soliton: Kuen surface

References

^ Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Essay on the interpretation of noneuclidean geometry]. Gior. Mat. (in Italian). 6: 248–312.

(Republished in Beltrami, Eugenio (1902). Opere Matematiche. Vol. 1. Milan: Ulrico Hoepli. XXIV, pp. 374–405. Translated into French as "Essai d'interprétation de la géométrie noneuclidéenne". Annales Scientifiques de l'École Normale Supérieure. Ser. 1. 6. Translated by J. Hoüel: 251–288. 1869. doi:10.24033/asens.60. EuDML 80724. Translated into English as "Essay on the interpretation of noneuclidean geometry" by John Stillwell, in Stillwell 1996, pp. 7–34.)
^ Bonahon, Francis (2009). Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots. AMS Bookstore. p. 108. ISBN 978-0-8218-4816-6., Chapter 5, page 108
^ Stillwell, John (2010). Mathematics and Its History (revised, 3rd ed.). Springer Science & Business Media. p. 345. ISBN 978-1-4419-6052-8., extract of page 345
^ 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.
^ Roberts, Siobhan (15 January 2024). "The Crochet Coral Reef Keeps Spawning, Hyperbolically". The New York Times.
^ 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 (6): 521–537, doi:10.1093/imamat/69.6.521, ISSN 1464-3634, archived from the original on 2013-04-15
^ Wheeler, Nicholas. "From Pseudosphere to sine-Gordon equation" (PDF). Retrieved 24 November 2022.

Stillwell, John (1996). Sources of Hyperbolic Geometry. American Mathematical Society & London Mathematical Society. ISBN 0-8218-0529-0.
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. pp. 140, 145, 155.

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.
Pseudospherical surfaces at the virtual math museum.

[1] Beltrami, Eugenio (1868). "Saggio sulla interpretazione della geometria non euclidea" [Essay on the interpretation of noneuclidean geometry]. Gior. Mat. (in Italian). 6: 248–312.

(Republished in Beltrami, Eugenio (1902). Opere Matematiche. Vol. 1. Milan: Ulrico Hoepli. XXIV, pp. 374–405. Translated into French as "Essai d'interprétation de la géométrie noneuclidéenne". Annales Scientifiques de l'École Normale Supérieure. Ser. 1. 6. Translated by J. Hoüel: 251–288. 1869. doi:10.24033/asens.60. EuDML 80724. Translated into English as "Essay on the interpretation of noneuclidean geometry" by John Stillwell, in Stillwell 1996, pp. 7–34.)

[2] Bonahon, Francis (2009). Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots. AMS Bookstore. p. 108. ISBN 978-0-8218-4816-6., Chapter 5, page 108

[3] Stillwell, John (2010). Mathematics and Its History (revised, 3rd ed.). Springer Science & Business Media. p. 345. ISBN 978-1-4419-6052-8., extract of page 345

[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] Roberts, Siobhan (15 January 2024). "The Crochet Coral Reef Keeps Spawning, Hyperbolically". The New York Times.

[7] Thurston, William, Three-dimensional geometry and topology, vol. 1, Princeton University Press, p. 62.

[8] Hasanov, Elman (2004), "A new theory of complex rays", IMA J. Appl. Math., 69 (6): 521–537, doi:10.1093/imamat/69.6.521, ISSN 1464-3634, archived from the original on 2013-04-15

[wheeler-9] Wheeler, Nicholas. "From Pseudosphere to sine-Gordon equation" (PDF). Retrieved 24 November 2022.

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