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In [[geometry]], the term '''pseudosphere''' is used to describe various surfaces with constant negative [[Gaussian curvature]]. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid.
__TOC__
 
== Theoretical pseudosphere ==
In its general interpretation, a '''pseudosphere''' of radius ''R'' is any surface of [[Gaussian curvature|curvature]] &minus;1/''R''<sup>2</sup> (precisely, a [[complete metric space|complete]], [[simply connected]] surface of that curvature), by analogy with the sphere of radius ''R'', which is a surface of curvature 1/''R''<sup>2</sup>. The term was introduced by [[Eugenio Beltrami]] in his 1868 paper on models of [[hyperbolic geometry]].<ref>{{Citation
| first=Eugenio
| last=Beltrami
| title=Saggio sulla interpretazione della geometria non euclidea
| journal=Gior. Mat.
| volume=6
| pages=248&ndash;312
| language=Italian
| year=1868
}}<br />
(Also {{Citation
| first=Eugenio
| last=Beltrami
| title=Opere Matematiche
| volume=1
| pages=374&ndash;405
| language=Italian
| isbn=1-4181-8434-9
}};<br />
{{Citation
| first=Eugenio
| last=Beltrami
| title=Essai d'interprétation de la géométrie noneuclidéenne
| journal=Annales de l'École Normale Supérieure
| year=1869
| volume = 6
| pages=251&ndash;288
| language=French
| url=http://smf4.emath.fr/Publications/AnnalesENS/1_6/html/
}})</ref>
 
== Tractricoid ==
[[Image:Pseudosphere.png|right|frame|Tractricoid]]
The term is also used to refer to a certain surface called the '''tractricoid''': the result of [[surface of revolution|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<ref>{{cite book
|title=Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots
|first1=Francis
|last1=Bonahon
|publisher=AMS Bookstore
|year=2009
|isbn=0-8218-4816-X
|page=108
|url=http://books.google.com/books?id=YZ1L8S4osKsC}}, [http://books.google.com/books?id=YZ1L8S4osKsC&pg=PA108 Chapter 5, page 108]
</ref>
:<math>t \mapsto \left( t - \tanh{t}, \operatorname{sech}\,{t} \right), \quad \quad 0 \le t < \infty.</math>
 
It is a [[Mathematical singularity|singular space]] (the equator is a singularity), but away from the singularities, it has constant negative [[Gaussian curvature]] and therefore is locally [[isometry|isometric]] to a [[Hyperbolic space|hyperbolic plane]].
 
The name "pseudosphere" comes about because it is a [[dimension|two-dimensional]] [[surface]] of constant negative curvature just like a sphere with positive Gauss curvature.
Just as the [[sphere]] has at every point a [[positive number|positively]] curved geometry of a [[dome]] the whole pseudosphere has at every point the [[negative number|negatively]] curved geometry of a [[saddle surface|saddle]].
 
As early as 1639 [[Christiaan Huygens]] found that the volume and the surface area of the pseudosphere are finite,<ref>{{cite book
|title=Computational optimization: a tribute to Olvi Mangasarian, Volume 1
|first1=Olvi L.
|last1=Mangasarian
|first2=Jong-Shi
|last2=Pang
|publisher=Springer
|year=1999
|isbn=0-7923-8480-6
|page=324
|url=http://books.google.com/books?id=kJa15IMxAoIC}}, [http://books.google.com/books?id=kJa15IMxAoIC&pg=PA324 Chapter 17, page 324]
</ref> despite the infinite extent of the shape along the axis of rotation. For a given edge [[radius]] ''R'', the [[area]] is 4π''R''<sup>2</sup> just as it is for the sphere, while the [[volume]] is 2/3 π''R''<sup>3</sup> and therefore half that of a sphere of that radius.<ref>{{cite book
|title=Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences
|edition=2
|first1=F.
|last1=Le Lionnais
|publisher=Courier Dover Publications
|year=2004
|isbn=0-486-49579-5
|page=154
|url=http://books.google.com/books?id=pCYDhbhu1O0C}}, [http://books.google.com/books?id=pCYDhbhu1O0C&pg=PA154 Chapter 40, page 154]
</ref><ref>{{MathWorld|title=Pseudosphere|urlname=Pseudosphere}}</ref>
 
== Universal covering space ==
The half pseudosphere of curvature &minus;1 is [[covering space|covered]] by the portion of the hyperbolic upper half-plane with ''y''&nbsp;≥&nbsp;1.<ref>{{citation|first=William|last=Thurston|title=Three-dimensional geometry and topology|volume=1|publisher=Princeton University Press|year=|page=62}}.</ref>  The covering map is periodic in the ''x'' direction of period 2π, and takes the horocycles ''y''&nbsp;=&nbsp;''c'' to the meridians of the pseudosphere and the vertical geodesics ''x''&nbsp;=&nbsp;''c'' to the tractrices that generate the pseudosphere.  This mapping is a local isometry, and thus exhibits the portion ''y''&nbsp;≥&nbsp;1 of the upper half-plane as the [[universal covering space]] of the pseudosphere. The precise mapping is
:<math>(x,y)\mapsto (v(\operatorname{arcosh} y)\cos x, v(\operatorname{arcosh} y) \sin x, u(\operatorname{arcosh} y))</math>
where <math>t\mapsto (u(t),v(t))</math> is the arclength 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'''.<ref>{{Citation
| first=Elman
| last=Hasanov
| year=2004
| title=A new theory of complex rays
| journal=IMA J Appl Math
| volume=69
| pages=521&ndash;537
| issn=1464-3634
| url=http://imamat.oxfordjournals.org/cgi/reprint/69/6/521
}}</ref>
This usage of the word is because the hyperboloid can be [[hyperboloid#Relation to the sphere|thought of as a sphere]] of imaginary radius, embedded in a [[Minkowski space]].
 
==See also==
*[[Dini's surface]]
*[[Gabriel's Horn]]
*[[Hyperboloid]]
*[[Hyperboloid structure]]
*[[Sine–Gordon equation]]
*[[Sphere]]
*[[Surface of revolution]]
 
==References==
<div class="references-small">
{{Reflist}}
* Stillwell J: ''Sources of Hyperbolic Geometry'', 1996, Amer. Math. Soc & London Math. Soc.
* {{cite book|author=Henderson, D. W. and Taimina, D.|title=Aesthetics and Mathematics|publisher=Springer-Verlag|year=2006|chapter=[http://dspace.library.cornell.edu/bitstream/1813/2714/1/2003-4.pdf Experiencing Geometry: Euclidean and Non-Euclidean with History]}}
* Edward Kasner & James Newman (1940) [[Mathematics and the Imagination]], pp 140,145,155, [[Simon & Schuster]].
</div>
 
==External links==
*[http://www.cs.unm.edu/~joel/NonEuclid/pseudosphere.html Non Euclid]
*[http://www.cabinetmagazine.org/issues/16/crocheting.php Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina ]
*[http://www.maths.manchester.ac.uk/~kd/ Prof. C.T.J. Dodson's web site at University of Manchester]
*[http://www.maths.manchester.ac.uk/~kd/geomview/dini.html Interactive demonstration of the pseudosphere] (at the [[University of Manchester]])
*[http://www.youtube.com/watch?v=6xgtMQ7WSzQ Norman Wildberger lecture 16], History of Mathematics, University of New South Wales. YouTube. 2012 May.
 
[[Category:Differential geometry]]
[[Category:Surfaces]]

Latest revision as of 19:20, 15 November 2013

In geometry, the term pseudosphere is used to describe various surfaces with constant negative Gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid.

Theoretical pseudosphere

In its general interpretation, a pseudosphere of radius R is any surface of curvature −1/R2 (precisely, a complete, simply connected surface of that curvature), by 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

The term is also used to refer to a certain surface called the tractricoid: the result of 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]

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 is a two-dimensional surface of constant negative curvature just like a sphere with positive Gauss 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 1639 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 2/3 πR3 and therefore half that of a sphere of that radius.[4][5]

Universal covering space

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 arclength 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

References

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  7. Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010