Pseudosphere

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

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A pseudosphere of radius R is a surface in having curvature 1/R2 in 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]

Tractricoid

Tractricoid Pseudosphere.png
Tractricoid

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 tractricoid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by [2]

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 R2 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 pseudosphere and its relation to three other models of hyperbolic geometry Geodesics on the pseudosphere and three other models of hyperbolic geometry.png
The pseudosphere and its relation to three other models of hyperbolic geometry

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

where

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.

See also

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Eugenio Beltrami

Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model. He also developed singular value decomposition for matrices, which has been subsequently rediscovered several times. Beltrami's use of differential calculus for problems of mathematical physics indirectly influenced development of tensor calculus by Gregorio Ricci-Curbastro and Tullio Levi-Civita.

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Hyperboloid model

In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space and m-planes are represented by the intersections of the (m+1)-planes in Minkowski space with S+. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the n-dimensional hyperbolic space is closely related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group.

Beltrami–Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

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In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.

Horocycle

In hyperbolic geometry, a horocycle is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional example of a horosphere.

Differential geometry of surfaces

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Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all circular arcs contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.

References

  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). 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. Archived from the original on 2016-02-02. Retrieved 2010-07-24.)
  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. 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. 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, archived from the original on 2013-04-15