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[[Image:Cyclide.png|thumb|A Dupin cyclide]]
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In [[mathematics]], a '''Dupin cyclide''' or '''cyclide of Dupin''' is any [[Inversive geometry|geometric inversion]] of a [[standard torus]], [[cylinder]] or [[cone|double cone]]. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by (and named after) [[Charles Dupin]] in his 1803 dissertation under [[Gaspard Monge]].<ref>{{Harvnb|O'Connor|Robertson|2000}}</ref> The key property of a Dupin cyclide is that it is a [[channel surface]] (envelope of a one parameter family of spheres) in two different ways. This property means that Dupin cyclides are natural objects in [[Lie sphere geometry]].
 
Dupin cyclides are often simply known as "cyclides", but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the [[Laplace equation]] in three dimensions.
 
==Definitions and properties==
 
There are several equivalent definitions of Dupin cyclides. In <math>\R^3</math>, they can be defined as the images under any inversion of tori, cylinders and double cones.  This shows that the class of Dupin cyclides is invariant under [[Möbius transformation|Möbius (or conformal) transformation]]s.  
In complex space <math>\C^3</math> these three latter varieties can be mapped to one another by inversion, so Dupin cyclides can be defined as inversions of the torus (or the cylinder, or the double cone).
 
Since a standard torus is the orbit of a point under a two dimensional [[abelian group|abelian]] [[subgroup]] of the Möbius group, it follows that the cyclides also are, and this provides a second way to define them.
 
A third property which characterizes Dupin cyclides is the fact that their [[curvature line]]s are all circles (possibly through the [[point at infinity]]). Equivalently, the [[curvature sphere]]s, which are the spheres [[tangent]] to the surface with radii equal to the [[Multiplicative inverse|reciprocals]] of the [[principal curvature]]s at the point of tangency, are constant along the corresponding curvature lines: they are the tangent spheres containing the corresponding curvature lines as [[great circle]]s. Equivalently again, both sheets of the [[focal surface]] degenerate to conics.<ref>{{Harvnb|Hilbert|Cohn-Vossen|1999}}</ref> It follows that any Dupin cyclide is a [[channel surface]] (i.e., the envelope of a one parameter family of spheres) in two different ways, and this gives another characterization.
 
The definition in terms of spheres shows that the class of Dupin cyclides is invariant under the larger group of all [[Lie sphere transformation]]s. In fact any two Dupin cyclides are [[Lie sphere geometry|Lie equivalent]]. They form (in some sense) the simplest class of Lie invariant surfaces after the spheres, and are therefore particularly significant in [[Lie sphere geometry]].<ref>{{Harvnb|Cecil|1992}}</ref>
 
The definition also means that a Dupin cyclide is the envelope of the one parameter family of spheres tangent to three given mutually tangent spheres. It follows that it is tangent to infinitely many [[Soddy's hexlet]] configurations of spheres.
 
==Cyclides and separation of variables==
 
Dupin cyclides are a special case of a more general notion of a cyclide, which is a natural extension of the notion of a [[quadric surface]]. Whereas a quadric can be described as the zero-set of second order polynomial in Cartesian coordinates (''x''<sub>1</sub>,''x''<sub>2</sub>,''x''<sub>3</sub>), a cyclide is given by the zero-set of a second order polynomial in (''x''<sub>1</sub>,''x''<sub>2</sub>,''x''<sub>3</sub>,''r''<sup>2</sup>), where
''r''<sup>2</sup>=''x''<sub>1</sub><sup>2</sup>+''x''<sub>2</sub><sup>2</sup>+''x''<sub>3</sub><sup>2</sup>. Thus it is a quartic surface in Cartesian coordinates, with an equation of the form:
:<math>
A r^4 + \sum_{i=1}^3 P_i  x_i r^2 + \sum_{i,j=1}^3 Q_{ij}  x_i  x_j + \sum_{i=1}^3 R_i  x_i + B = 0
</math>
where ''Q'' is a 3x3 matrix, ''P'' and ''R'' are a 3-dimensional [[vector (geometric)|vectors]], and ''A'' and ''B'' are constants.<ref>{{Harvnb|Miller|1977}}</ref>
 
Families of cyclides give rise to various cyclidic coordinate geometries.
 
In Maxime Bôcher's 1891 dissertation, ''Ueber die Reihenentwickelungen der Potentialtheorie'', it was shown that the [[Laplace equation]] in three variables can be solved using separation of variables in 17 conformally distinct quadric and cyclidic coordinate geometries.  Many other cyclidic geometries can be obtained by studying R-separation of variables for the Laplace equation.<ref>{{Harvnb|Moon|Spencer|1961}}</ref>
 
==Notes==
{{reflist}}
 
==References==
 
*{{citation | last = Cecil | first = Thomas E. | title = Lie sphere geometry | publisher = Universitext, Springer-Verlag|place= New York | year = 1992|isbn =978-0-387-97747-8}}.
* {{citation|last=Eisenhart|first= Luther P.|chapter=§133 Cyclides of Dupin|title= A Treatise on the Differential Geometry of Curves and Surfaces|place=  New York|publisher= Dover|pages=312&ndash;314|year= 1960}}.
* {{citation | title = Geometry and the Imagination | author1-link = David Hilbert|first1=David|last1=Hilbert|first2=Stephan |last2=Cohn-Vossen |authorlink2=Stephan Cohn-Vossen| year = 1999 | publisher = American Mathematical Society | isbn= 0-8218-1998-4}}.
* {{citation | title = Field Theory Handbook: including coordinate systems, differential equations, and their solutions | first1=Parry|last1= Moon |first2=Domina Eberle|last2= Spencer| year = 1961 | publisher = Springer | isbn=0-387-02732-7}}.
* {{citation| last1=O'Connor|first1= John J.|first2=Edmund F.|last2= Robertson|chapter-url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html|chapter=Pierre Charles François Dupin|title=[[MacTutor History of Mathematics archive]]|year=2000}}.
* {{citation|last=Pinkall|first=Ulrich|chapter=§3.3 Cyclides of Dupin|title=Mathematical Models from the Collections of Universities and Museums|editor=G. Fischer|place= Braunschweig, Germany|pages= 28&ndash;30|year= 1986|publisher=Vieweg}}.
* {{citation | first = Willard | last = Miller | year = 1977 | title = Symmetry and Separation of Variables}}.
 
==External links==
{{commonscat|Dupin cyclide}}
*{{Mathworld|Cyclide|Cyclide}}
*{{cite web|url=http://www.javaview.de/demo/surface/common/PaSurface_DupinCycloid.html|title=Javaview of Dupin Cycloid}}
 
[[Category:Surfaces]]

Revision as of 11:29, 6 February 2014

Have we ever heard that someone mentioned "My computer is getting slower, Assist me?" Are we looking methods on how do I accelerate my computer? Are we tired of wasting too much time because the loading task is absolutely slow? If you wish To recognize how to boost and speed up computer performance, then this article might assist show we several concepts and tricks "What is the cause?" plus How to prevent a computer getting slower?

Install an anti-virus software. If you already have that on you computer then carry out a full program scan. If it finds any viruses found on the computer, delete those. Viruses invade the computer and create it slower. To safeguard the computer from numerous viruses, it's greater to keep the anti-virus software running when you use the web. You may moreover fix the safety settings of your web browser. It may block unknown plus risky sites and block off any spyware or malware trying to receive into your computer.

It doesn't matter whether you're not really well-defined about what rundll32.exe is. However remember that it plays an important role in maintaining the stability of the computers plus the integrity of the system. When some software or hardware could not reply usually to the system operation, comes the rundll32 exe error, that can be caused by corrupted files or lost information inside registry. Usually, error content may shows up at booting or the beginning of running a program.

Registry cleaners have been crafted for one purpose - to wash out the 'registry'. This is the central database which Windows relies on to function. Without this database, Windows wouldn't even exist. It's so important, which your computer is regularly adding plus updating the files inside it, even when you're browsing the Internet (like now). This really is superb, yet the problems happen whenever a few of those files become corrupt or lost. This happens a lot, plus it takes a superior tool to fix it.

Another thing you should check is whether the tuneup utilities program that you are considering has the ability to detect files and programs which are good. One of the registry cleaner programs we might try is RegCure. It is helpful for speeding up and cleaning up issues on a computer.

2)Fix your Windows registry to speed up PC- The registry is a complex section of your computer that holds different types of information within the things we do on a computer daily. Coincidentally, over time the registry will become cluttered with info and/or can obtain some sort of virus. This really is extremely important and you MUST get this issue fixed right away, otherwise we run the risk of your computer being permanently damage and/or the sensitive information (passwords, etc.) is stolen.

In alternative words, when a PC has any corrupt settings inside the registry database, these settings usually make your computer run slower and with a lot of errors. And unluckily, it's the case which XP is prone to saving many settings from the registry in the wrong way, making them unable to run correctly, slowing it down and causing a lot of mistakes. Each time we employ the PC, it has to read 100's of registry settings... and there are often a lot of files open at once that XP gets confuse plus saves countless in the wrong way. Fixing these damaged settings may boost the speed of the system... plus to do that, we should look to employ a 'registry cleaner'.

Thus, the number one thing to do when the computer runs slow is to buy an authentic and legal registry repair tool which would help you eliminate all issues related to registry plus help we enjoy a smooth running computer.

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