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

A pseudosphere of radius R is a surface in having curvature −1/*R*^{2} in each point. Its name comes from the 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] }

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 4π*R*^{2} just as it is for the sphere, while the volume is 2/3π*R*^{3} and therefore half that of a sphere of that radius.^{ [4] }^{ [5] }

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.

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.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics, **hyperbolic functions** are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos *t*, sin *t*) form a circle with a unit radius, the points (cosh *t*, sinh *t*) form the right half of the unit hyperbola. Also, just as the derivatives of sin(*t*) and cos(*t*) are cos(*t*) and –sin(*t*), the derivatives of sinh(*t*) and cosh(*t*) are cosh(*t*) and +sinh(*t*).

In mathematics, **non-Euclidean geometry** consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

In geometry, a **hyperboloid of revolution**, sometimes called a **circular hyperboloid**, is the surface generated by rotating a hyperbola around one of its principal axes. A **hyperboloid** is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In differential geometry, the **Gaussian curvature** or **Gauss curvature**Κ of a surface at a point is the product of the principal curvatures, *κ*_{1} and *κ*_{2}, at the given point:

In geometry, **inversive geometry** is the study of *inversion*, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied.

In mathematics, a **hyperbolic space** is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

In non-Euclidean geometry, the **Poincaré half-plane model** is the upper half-plane, denoted below as **H**, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

**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.

A **tractrix** is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Isaac Newton (1676) and Christiaan Huygens (1692).

In hyperbolic geometry, the **angle of parallelism **, is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length *a* between the right angle and the vertex of the angle of parallelism.

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.

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.

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.

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.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

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.

- ↑ 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.) - ↑ 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 - ↑ 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*. - ↑ 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**: 521–537, doi:10.1093/imamat/69.6.521, ISSN 1464-3634, archived from the original on 2013-04-15

- 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.

- 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.

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