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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a pseudosphere is a surface with constant negative
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
. A pseudosphere of radius is a surface in \mathbb^3 having
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canon ...
in each point. Its name comes from the analogy with the sphere of radius , which is a surface of curvature . The term was introduced by Eugenio Beltrami in his 1868 paper on models of
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
. __TOC__


Tractroid

The same surface can be also described as the result of revolving a
tractrix In geometry, a tractrix (; plural: tractrices) 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 pulling point (the ''tractor'') that moves at a right angl ...
about its
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
. For this reason the pseudosphere is also called tractroid. As an example, the (half) pseudosphere (with radius 1) is the surface of revolution of the tractrix parametrized by :t \mapsto \left( t - \tanh, \operatorname\, \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 In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
and therefore is locally
isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...
to a
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
. The name "pseudosphere" comes about because it has a
two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
of constant negative Gaussian curvature, just as a sphere has a surface with constant positive Gaussian curvature. Just as the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
has at every point a positively curved geometry of a
dome A dome () is an architectural element similar to the hollow upper half of a sphere. There is significant overlap with the term cupola, which may also refer to a dome or a structure on top of a dome. The precise definition of a dome has been a ...
the whole pseudosphere has at every point the negatively curved geometry of a
saddle The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not kn ...
. As early as 1693
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists ...
found that the volume and the surface area of the pseudosphere are finite, despite the infinite extent of the shape along the axis of rotation. For a given edge
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
, the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
is just as it is for the sphere, while the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...
is and therefore half that of a sphere of that radius.


Universal covering space

The half pseudosphere of curvature −1 is covered by the interior of a
horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosph ...
. In the
Poincaré half-plane model 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. Equivalently the Poincaré ha ...
one convenient choice is the portion of the half-plane with . Then the covering map is periodic in the direction of period 2, and takes the horocycles to the meridians of the pseudosphere and the vertical geodesics to the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion of the upper half-plane as the
universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of the pseudosphere. The precise mapping is :(x,y)\mapsto \big(v(\operatorname y)\cos x, v(\operatorname y) \sin x, u(\operatorname y)\big) where :t\mapsto \big(u(t) = t - \operatorname t,v(t) = \operatorname t\big) is the parametrization of the tractrix above.


Hyperboloid

In some sources that use the
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 points on the forward sheet ''S''+ of a two-sheeted hyperboloid ...
of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere. This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
.


Pseudospherical surfaces

A pseudospherical surface is a generalization of the pseudosphere. A surface which is piecewise smoothly immersed in \mathbb^3 with constant negative curvature is a pseudospherical surface. The tractroid is the simplest example. Other examples include the
Dini's surface In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: : \begin x&=a \cos u \sin v \\ y&=a \sin u ...
s, 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 The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfa ...
. A sketch proof starts with reparametrizing the tractroid with coordinates in which the
Gauss–Codazzi equations In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced m ...
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 First or 1st is the ordinal form of the number one (#1). First or 1st may also refer to: *World record, specifically the first instance of a particular achievement Arts and media Music * 1$T, American rapper, singer-songwriter, DJ, and reco ...
and
second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...
s are written in a way that makes clear the
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
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^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


See also

*
Hilbert's theorem (differential geometry) In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface S of constant negative gaussian curvature K immersed in \mathbb^. This theorem answers the question for the negative case of which surfaces in ...
*
Dini's surface In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: : \begin x&=a \cos u \sin v \\ y&=a \sin u ...
*
Gabriel's Horn Gabriel's horn (also called Torricelli's trumpet) is a particular geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Judgment Da ...
*
Hyperboloid 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 defo ...
*
Hyperboloid structure Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the grou ...
*
Quasi-sphere In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied ...
*
Sine–Gordon equation The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfa ...
*
Sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
*
Surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...


References

* * * {{cite book, first1=Edward , last1=Kasner , first2=James , last2=Newman , date=1940 , title=
Mathematics and the Imagination ''Mathematics and the Imagination'' is a book published in New York by Simon & Schuster in 1940. The authors are Edward Kasner and James R. Newman. The illustrator Rufus Isaacs provided 169 figures. It rapidly became a best-seller and received ...
, pages=140, 145, 155 , publisher=
Simon & Schuster Simon & Schuster () is an American publishing company and a subsidiary of Paramount Global. It was founded in New York City on January 2, 1924 by Richard L. Simon and M. Lincoln Schuster. As of 2016, Simon & Schuster was the third largest pu ...


External links


Non EuclidCrocheting 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.

at the virtual math museum. Differential geometry Hyperbolic geometry Surfaces Spheres