Pseudosphere
In geometry, a pseudosphere is a surface in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} . It is the most famous example of a pseudospherical surface. A pseudospherical surface is a surface piecewise smoothly immersed in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} with constant negative Gaussian curvature. A "pseudospherical surface of radius R" is a surface in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. Examples include the tractroid, Dini's surfaces, breather surfaces, and the Kuen surface.
The term "pseudosphere" was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry.[1]
Tractroid
By "the pseudosphere", people usually mean the tractroid. The tractroid is obtained by 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]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 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 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]
The pseudosphere is an important geometric precursor to mathematical fabric arts and pedagogy.[6]
Line congruence
A line congruence is a 2-parameter families of lines in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^3} . It can be written asFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X(u, v, t)=x(u, v)+t p(u, v)} where each pick of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u, v \in \R} picks a specific line in the family.
A focal surface of the line congruence is a surface that is tangent to the line congruence. At each point on the surface,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det\left(\partial_u X, \partial_v X, p\right)=0} The above equation expands to a quadratic equation in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} :Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det (\partial_u x(u, v)+ t\partial_u p(u, v), \partial_v x(u, v)+ t\partial_v p(u, v), p(u, v)) = 0} Thus, for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u, v) \in \R^2} , there in general exists two choices of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1(u, v), t_2(u, v)} . Thus a generic line congruence has exactly two focal surfaces parameterized by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1(u, v), t_2(u, v)} .
For a bundle of lines normal to a smooth surface, the two focal surfaces correspond to its evolutes: the loci of centers of principal curvature.
In 1879, Bianchi proved that if a line congruence is such that the corresponding points on the two focal surfaces are at a constant distance 1, that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |t_1(u, v) - t_2(u, v)| = 1} , then both of the focal surfaces have constant curvature -1.
In 1880, Lie proved a partial converse. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle X} be a pseudospherical surface. Then there exists a second pseudospherical surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{X}} and a line congruence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{X}} are the focal surfaces of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}} . Furthermore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \hat{X}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathcal{L}} may be constructed from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle X} by integrating a sequence of ODEs.
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 ≥ 1.[7] Then 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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)\mapsto \big(v(\operatorname{arcosh} y)\cos x, v(\operatorname{arcosh} y) \sin x, u(\operatorname{arcosh} y)\big) ,}
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
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.
On a surface, at each point, draw a cross, pointing at the two directions of principal curvature. These crosses can be integrated into two families of curves, making up a coordinate system on the surface. Let the coordinate system be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x, y)} .
At each point on a pseudospherical surface there in general exists two asymptotic directions. Along them, the curvature is zero. Let the angle between the asymptotic directions be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} .
A theorem states thatFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_{xx} \theta - \partial_{yy} \theta = \sin\theta} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} .
This connection between sine-Gordon equations and pseudospherical surfaces mean that one can identify solutions to the equation with surfaces. Then, any way to generate new sine-Gordon solutions from old automatically generates new pseudospherical surfaces from old, and vice versa.
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
See also
- Hilbert's theorem (differential geometry)
- Dini's surface
- Gabriel's Horn
- Hyperboloid
- Hyperboloid structure
- Quasi-sphere
- Sine–Gordon equation
- Sphere
- Surface of revolution
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. 1. Milan: Ulrico Hoepli. XXIV, Template:Pgs. Translated into French as . Translated by J. Hoüel. "Essai d'interprétation de la géométrie noneuclidéenne". Annales Scientifiques de l'École Normale Supérieure. Ser. 1. 6: 251–288. 1869. doi:10.24033/asens.60. Template:EuDML. 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, 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, hdl:11729/142, ISSN 1464-3634, archived from the original on 2013-04-15
- ↑ Wheeler, Nicholas. "From Pseudosphere to sine-Gordon equation" (PDF). Retrieved 24 November 2022.
Further reading
- 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.
- Rogers, C.; Schief, Wolfgang K., eds. (2002). Bäcklund and Darboux transformations: geometry and modern applications in soliton theory. Cambridge texts in applied mathematics. Cambridge New York: Cambridge University Press. ISBN 978-0-521-01288-1.
External links
- Non Euclid
- Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
- Pseudospherical surfaces at the virtual math museum.