Geometric group theory: Difference between revisions

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In [[plane geometry]], two lines are '''hyperbolic orthogonal''' when they are [[reflection (mathematics)|reflections]] of each other over the asymptote of a given [[hyperbola]].
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Two particular hyperbolas are frequently used in the plane:
:(A) ''x y'' = 1  with ''y'' = 0 as asymptote.
:When reflected in the x-axis, a line ''y = mx'' becomes ''y'' = &minus;''mx''.
:In this case the lines are hyperbolic orthogonal if their [[slope]]s are [[additive inverse]]s.
 
:(B) ''x''<sup>2</sup> &minus; ''y''<sup>2</sup> = 1  with ''y'' = ''x'' as asymptote.
:For lines ''y = mx''  with &minus;1 < ''m'' < 1, when ''x'' = 1/''m'', then ''y'' = 1.
:The point (1/''m'' , 1) on the line is reflected across ''y'' = ''x'' to (1, 1/''m'').
:Therefore the reflected line has slope 1/m and the slopes of hyperbolic orthogonal lines are [[multiplicative inverse|reciprocal]]s of each other.
 
The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class. Thus, for a given hyperbola and asymptote ''A'', a pair of lines (''a,b'') are hyperbolic orthogonal if there is a pair (''c,d'') such that <math>a \rVert c , b \rVert d </math>, and ''c'' is the reflection of ''d'' across ''A''.
 
The property of the radius being orthogonal to the [[tangent]] at the curve, is extended from the circle to the hyperbola by the hyperbolic orthogonal concept. See Lewis & Wilson or Felsagen for the radius-tangent orthogonality.
 
Since [[Hermann Minkowski]]'s foundation for [[spacetime]] study in 1908, the concept of points in a spacetime plane being hyperbolic-orthogonal to a timeline (tangent to a [[World line]]) has been used to define '''simultaneity''' of events relative to the timeline. In Minkowski's development the hyperbola of type (B) above is in use (see the [[wikisource]] link below). Two vectors <math>x, y, z, t \quad \text {and}\quad  x_1, y_1, z_ 1, t_1</math> are ''normal'' (meaning hyperbolic orthogonal) when
:<math>c^{2} t \ t_1 - x \ x_1 - y \ y_1 - z \ z_1 = 0.</math>
When ''c'' = 1 and the y's and z's are zero, x&ne;0, t<sub>1</sub>&ne;0, then <math>\frac t x = \frac{x_1} {t_1}</math>.
 
Given a hyperbola with asymptote ''A'', its reflection in ''A'' produces the ''conjugate hyperbola''. Any diameter of the original hyperbola is reflected to a [[conjugate diameters|conjugate diameter]]. The directions indicated by conjugate diameters are taken for space and time axes in relativity.
As [[E. T. Whittaker]] wrote in 1910, "[the] hyperbola is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters." On this [[principle of relativity]], he then wrote the Lorentz transformation in the modern form using [[rapidity]].
 
[[Edwin Bidwell Wilson]] and [[Gilbert N. Lewis]] developed the concept within [[synthetic geometry]] in 1912. They note (p.415) "in our plane no pair of perpendicular [hyperbolic-orthogonal] lines is better suited to serve as coordinate axes than any other pair".
 
In the terminology of [[projective geometry]], the operation of taking the hyperbolic orthogonal line is a [[involution (mathematics)#Projective geometry|hyperbolic involution]]. Suppose the slope of a vertical line is denoted ∞ so that all lines have a slope in the [[real projective line]]. Then whichever hyperbola (A) or (B) is used, the operation is an [[real projective line#Hyperbolic involutions|example of a hyperbolic involution]].
 
==References==
* [[G. D. Birkhoff]] (1923) ''Relativity and Modern Physics'', pages 62,3, [[Harvard University Press]].
* Francesco Catoni, Dino Boccaletti, & Roberto Cannata (2008) ''Mathematics of Minkowski Space'', [[Birkhäuser Verlag]], Basel. See page 38, Pseudo-orthogonality.
*{{Citation|author=Minkowski, Hermann|year=1909|title=[[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]]|journal=Physikalische Zeitschrift|volume=10|pages=75–88}}
:*Various English translations on Wikisource: [[s:Space and Time|Space and Time]]
* {{Citation  | author=Whittaker, E.T. | year=1910 | title= 1. Edition: [http://www.archive.org/details/historyoftheorie00whitrich A History of the theories of aether and electricity] | place =Dublin |publisher=Longman, Green and Co.}}(see page 441)
*Bjørn Felsager (2004), [http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf Through the Looking Glass - A glimpse of Euclid’s twin geometry, the Minkowski geometry], ICME-10 Copenhagen; pages 6 & 7.  
*Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the [[American Academy of Arts and Sciences]] 48:387-507.
 
 
[[Category:Minkowski spacetime]]
[[Category:Angle]]

Latest revision as of 18:51, 15 August 2014

Have you ever heard which someone said "My computer is getting slower, Help me?" Are you seeking methods on how do I accelerate my computer? Are we tired of wasting too much time because the loading task is truly slow? If you want to recognize how to improve and speed up computer performance, then this particular article may assist show we several concepts plus tricks "What is the cause?" plus How to prevent the computer getting slower?

You should recognize certain simple and cheap methods which could solve the problem of your computer and speed it up. The earlier you fix it, the less damage the computer gets. I may tell regarding some practical techniques which usually help we to speed up we computer.

The PC could have a fragmented difficult drive or the windows registry would have been corrupted. It could equally be as a result of the dust plus dirt which should be cleaned. Whatever the issue, you are able to usually find a answer. Here are certain strategies on how to make your PC run quicker.

First, usually clean the PC plus keep it free of dust and dirt. Dirt clogs up all of the fans plus may cause the PC to overheat. You furthermore have to clean up disk area inside purchase to create your computer run faster. Delete temporary plus unnecessary files plus unused programs. Empty the recycle bin plus remove programs you're not utilizing.

The final step is to make certain which we clean the registry of your computer. The "registry" is a large database that shops significant files, settings & choices, and info. Windows reads the files it requirements in order for it to run programs through this database. If the registry gets damaged, infected, or clogged up, then Windows may not be able to correctly access the files it requires for it to load up programs. As this arises, issues plus errors like the d3d9.dll error happen. To fix this plus avoid future setbacks, we have to download and run a registry cleaning tool. The very suggested software is the "Frontline tuneup utilities 2014".

The software finds these problems and takes care of them inside the shortest time possible. Your hard drive could additionally cause issues sometimes, specifically in the event you have 1 which is almost maxed. When your commence the machine, the are a lot of booting processes involved plus having an almost full storage does not enable a bit. You will usually have a slow PC because there are thus many items in the hard disk being processed simultaneously. The best method to resolve this issue is to upgrade. This usually allow the PC certain time to breath and working faster swiftly.

The 'registry' is merely the central database that shops all your settings plus choices. It's a actually important part of the XP system, meaning that Windows is frequently adding and updating the files inside it. The issues happen when Windows actually corrupts & loses a few of these files. This makes your computer run slow, because it tries difficult to obtain them again.

Most folks make the mistake of struggling to fix Windows registry by hand. I strongly suggest you don't do it. Unless we are a computer expert, I bet we will spend hours plus hours understanding the registry itself, let alone fixing it. And why if you waste a precious time in understanding and fixing anything you recognize nothing about? Why not let a smart and pro registry cleaner do it for you? These software programs would be able to work in a far better method! Registry cleaners are very affordable as well; we pay a one time fee plus employ it forever. Also, many expert registry products are pretty reliable plus simple to use. If you require more info on how to fix Windows registry, merely visit my url by clicking the link below!

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