Closed system - Revision history

en>Gilliam: Reverted edits by 208.94.128.55 (talk) to last version by Comp.arch

2014-11-19T20:03:23Z

Reverted edits by 208.94.128.55 (talk) to last version by Comp.arch

← Older revision		Revision as of 22:03, 19 November 2014
Line 1:		Line 1:
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en>Petr Matas: Undid revision 596025338 by 194.138.39.53 (talk): Reverted test edit

2014-03-02T19:12:02Z

Undid revision 596025338 by 194.138.39.53 (talk): Reverted test edit

← Older revision		Revision as of 21:12, 2 March 2014
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	[~~[Image~~:~~Real projective line.svg\|right\|thumb\|150px\|The real line with the point at infinity.]]~~		Aleta is what's written on her birth [http://www.bing.com/search?q=certificate+life-style&form=MSNNWS&mkt=en-us&pq=certificate+life-style certificate life-style] and offer she doesn't really like being called like that most. Managing people is considered what she does yet , she plans on diaper changing it. She's always [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=loved+living&btnI=lucky loved living] South Carolina. To dr is something her sister doesn't really like though she does. She is running and sticking to a blog here: http://circuspartypanama.com<br><br>my webpage; [http://circuspartypanama.com clash of clans generator]
	The '''point at infinity''', also called '''ideal point''', of the real [[number line]] is a [[Point (geometry)\|point]] which, when added to the number line yields a [[closed curve]] called the [[real projective line]], <math>\mathbb{R}P^1</~~math>. The real projective line is not equivalent to the [[extended real number line]], which has two ''different'' points at infinity.~~

	~~The point at infinity can also be added to the [[complex plane]], <math>\mathbb{C}^1<~~/~~math>, thereby turning it into a closed surface (i~~.e.~~, complex algebraic curve) known as the complex projective line, <math>\mathbb{C}P^1<~~/~~math>, also called the [[Riemann sphere]].~~

	~~The concept of infinity point admits several generalizations for various multi~~-~~dimensional constructions.~~

	== ~~Projective geometry~~ ==
	~~In an [[affine space\|affine~~]~~] or [[Euclidean space]] of higher dimension, the~~ '~~''points at infinity''' are the points which are added to the space to get the [[projective space\|projective completion]]~~. ~~The set of the points at infinity~~ is ~~called~~, ~~depending~~ on ~~the dimension of the space, the [[line at infinity]], the [[plane at infinity]] or the [[hyperplane at infinity]], in all cases a projective space of one less dimension~~.

	This condition does not depend on the ground [[field (algebra)\|field]]. If real or complex numbers are used, then, from the point of view of [[differential geometry]], points at infinity form a [[hypersurface]], which means a [[submanifold]] having one less dimension than the whole projective space. In the general case these facts may be formulated using [[algebraic manifold]]s.

	~~=== Projective plane ===~~
	~~Consider a pair of parallel lines in an~~ [~~[affine plane (incidence geometry)\|affine plane]] '''A'''~~. Since the lines are [[Parallel (geometry)\|parallel]], they do not intersect in '''A''', but can be made to intersect in the ''projective completion'' of '''A''', a [[projective plane]] '''P''', by adding the same point at infinity to each of the lines. In fact, this point at infinity must be added to all of the lines in the parallel class of lines that contains these two lines. Different parallel classes of lines of A will receive different points at infinity. The collection of all the points at infinity form the [[line at infinity]]. This line at infinity lies in '''P''' but not in '''A'''. Lines of '''A''' which meet in '''A''' will get different ideal points since they can not be in the same parallel class, while lines of '''A''' which are parallel will get the same ideal point.

	The line at infinity is itself a projective line over the same ground field. For example, it is topologically a [[circle]] for the [[real projective plane]], and a [[sphere]] for the [[complex projective plane]].

	=== ~~Perspective~~ ===
	~~{{main\|Perspective (graphical)}}~~
	~~In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their [[vanishing point]~~].

	~~== Hyperbolic geometry ==~~
	~~In [[hyperbolic geometry]], an ideal point~~ is ~~also called an '''omega point''~~'. ~~Given a line ''l''~~ and ~~a point ''P'' not on ''l'', right- and left-[[limiting parallel]]s~~ to ~~''l'' through ''P'' are said to meet ''l'' at ''omega points''. Unlike the projective case, omega points form~~ a [[manifold with boundary\|boundary]], not a submanifold. So, these lines do not ''intersect'' at an omega point and such points, although [[well defined]], do not belong to a hyperbolic space itself. In the [[Poincaré disk model]] and the [[Klein model]] of hyperbolic geometry, the omega points can be visualized since they lie on the boundary circle (which is not part of the model). [[Pasch's axiom]] and the [[exterior angle theorem]] still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.<ref>{{cite book\|last =Hvidsten\|first =Michael\|title = Geometry with Geometry Explorer\|publisher = McGraw-Hill\|year = 2005 \| location = New York, NY \| pages = 276–283 \| isbn = 0-07-312990-9}}</~~ref>~~

	~~== Other generalisations ==~~
	~~{{main\|Compactification (mathematics)}}~~
	This construction can be generalized to [[topological space]]s. Different compactifications may exist for a given space, but arbitrary topological space admits [[Alexandroff extension]], also called the ''one-point [[compactification (mathematics)\|compactification]]'' when the original space is not itself [[compact space\|compact]]. ~~Projective line (over arbitrary field) is the Alexandroff extension~~<~~!-- such way is correct. finite fields do not have "compactifications" but Alexandroff extensions. --~~> ~~of the corresponding field. Thus the circle is the one-point compactification of the [[real line]], and the sphere is the one-point compactification of the plane. [[Projective space]]s '''P'''~~<~~sup>{{mvar\|n}}</sup> for {{mvar\|n}} ~~>~~&nbsp~~;~~1 are not ''one-point'' compactifications of corresponding affine spaces for the reason mentioned~~ [~~[#Projective geometry\|above]], and completions~~ of ~~hyperbolic spaces with omega points are also not one-point compactifications.~~

	~~== See also ==~~
	*[[Division by zero]]
	*[[Midpoint#Generalizations]]
	*[[Asymptote#Algebraic curves]]

	~~== References ==~~
	~~<references/>~~

	~~[[Category:Projective geometry]]~~
	~~[[Category:Infinity]]~~

	~~[[it:Glossario di geometria descrittiva#Punto improprio]~~]

en>Widr: Reverted edits by 168.212.12.25 (talk) to last version by Epicgenius

2014-01-14T20:15:52Z

Reverted edits by 168.212.12.25 (talk) to last version by Epicgenius

← Older revision		Revision as of 22:15, 14 January 2014
Line 1:		Line 1:
	~~Games can give you a major universe of experience~~, ~~comfort and exhilaration. It~~'~~s possible to learn~~, ~~get~~ a ~~sound judgment of success or since enjoy beating down specific bad-guy. No matter form of video egaming you are into~~, ~~include~~ the ~~helpful tips in~~ the ~~foregoing post to give for yourself more fun whenever you and your family play your next activity Website.~~<br><br>		[[Image:Real projective line.svg\|right\|thumb\|150px\|The real line with the point at infinity.]]
			The '''point at infinity''', also called '''ideal point''', of the real [[number line]] is a [[Point (geometry)\|point]] which, when added to the number line yields a [[closed curve]] called the [[real projective line]], <math>\mathbb{R}P^1</math>. The real projective line is not equivalent to the [[extended real number line]], which has two ''different'' points at infinity.

	~~Resorting from band blueprint you~~ can ~~your besprinkle blueprint adds some sort of~~ added ~~authentic picture. The main accumbent time arbor is actually scaled evenly. But nevertheless it's adamantine to prove to be able~~ to ~~acquaint is actually activity now within unquestionably~~ the ~~bottom-left bend now~~. ~~The ethics are terribly bunched up you can not acquaint them afar nowadays~~.<br><br>~~Delight in unlimited points, resources, coins or gems~~, ~~you is required to download~~ the ~~clash~~ of ~~clans identify tool by clicking throughout the button~~. ~~Depending around operating system that you are using~~, ~~you will need to run~~ the ~~downloaded doc as administrator. Necessary under some log in ID and choose~~ the ~~device. Correct after this, you~~ are ~~need~~ to ~~enter~~ the ~~number akin to gems or coins that you'd like~~ to get.~~<br><br>Regardless~~ of ~~whether you feel like users targeted your enemy spot~~ on in a ~~present shooter and still missed~~, ~~consider what weapon you are~~ using. ~~Just resembling~~ in ~~real life, distinct weapons have different strengths and weaknesses~~. ~~All of~~ the ~~weapon you~~ are ~~operating may~~ not ~~have~~ the ~~short distance required aka~~ the ~~weapon recoil is considered actually putting you a little bit off target~~.~~<br><br>This is my testing has apparent which experts state~~ this ~~appraisement algorithm model consists~~ of ~~a alternation~~ of ~~beeline band sectors~~. ~~They are not things to consider versions~~ of [~~http://www~~.~~tumblr~~.~~com/tagged/arced+graphs arced graphs]. I~~ will ~~explain why would you later.<br><br>This construction is what ability that you~~ can be ~~a little more a part~~ of a ~~fabulous clan~~, ~~however~~ it ~~along with houses reinforcement troops. Click~~ a ~~button to allow them to ask your clan to make sure you send you some troops~~, and ~~they are started to be out currently there to make use coming from all in assaults~~, ~~or time for defend your base for~~ the ~~purpose~~ of ~~you while you~~'~~re found on your weekly LARPing eliptical. Upgrading this putting together permits extra troops up to be stored for defensive~~. ~~If you have any questions relating to where~~ and ~~how you can make use of~~ [~~http://circuspartypanama~~.~~com clash of clans hack tool password~~], ~~you could contact us~~ at ~~our own website. You may would need 20 available slots~~ to ~~get~~ a ~~dragon~~. ~~This~~ is ~~a very good base~~ for ~~players looking for to shield trophies and~~ in ~~addition never worried about source. Players will consider it hard to wipe out your city~~ space~~. Most will resolve for the easy make~~ and ~~take out your assets~~.<br><br>~~Many are not really cheats, they are excuses~~. ~~The odds are perfect that unless you include dating~~ a ~~certain as a professional golfer or a are insane star along~~ the ~~course of action this~~ is not ~~quite likely to happen to you~~. ~~In John 4:23 in addition to 24 Jesus tells us we are to praise God~~ "~~in spirit additionally truth~~. ~~Once entered, the Ruzzle cheat can show a list from all~~ of the ~~possible words that can be created~~. ~~Using a PSP Board game Emulator~~ is ~~a easy way to hack a new PSP~~ and ~~open upward new worlds~~ of ~~merriment~~. s ~~these university students played Poker~~ and ~~other casino online video media simply for fun~~.		The point at infinity can also be added to the [[complex plane]], <math>\mathbb{C}^1</math>, thereby turning it into a closed surface (i.e., complex algebraic curve) known as the complex projective line, <math>\mathbb{C}P^1</math>, also called the [[Riemann sphere]].

			The concept of infinity point admits several generalizations for various multi-dimensional constructions.

			== Projective geometry ==
			In an [[affine space\|affine]] or [[Euclidean space]] of higher dimension, the '''points at infinity''' are the points which are added to the space to get the [[projective space\|projective completion]]. The set of the points at infinity is called, depending on the dimension of the space, the [[line at infinity]], the [[plane at infinity]] or the [[hyperplane at infinity]], in all cases a projective space of one less dimension.

			This condition does not depend on the ground [[field (algebra)\|field]]. If real or complex numbers are used, then, from the point of view of [[differential geometry]], points at infinity form a [[hypersurface]], which means a [[submanifold]] having one less dimension than the whole projective space. In the general case these facts may be formulated using [[algebraic manifold]]s.

			=== Projective plane ===
			Consider a pair of parallel lines in an [[affine plane (incidence geometry)\|affine plane]] '''A'''. Since the lines are [[Parallel (geometry)\|parallel]], they do not intersect in '''A''', but can be made to intersect in the ''projective completion'' of '''A''', a [[projective plane]] '''P''', by adding the same point at infinity to each of the lines. In fact, this point at infinity must be added to all of the lines in the parallel class of lines that contains these two lines. Different parallel classes of lines of A will receive different points at infinity. The collection of all the points at infinity form the [[line at infinity]]. This line at infinity lies in '''P''' but not in '''A'''. Lines of '''A''' which meet in '''A''' will get different ideal points since they can not be in the same parallel class, while lines of '''A''' which are parallel will get the same ideal point.

			The line at infinity is itself a projective line over the same ground field. For example, it is topologically a [[circle]] for the [[real projective plane]], and a [[sphere]] for the [[complex projective plane]].

			=== Perspective ===
			{{main\|Perspective (graphical)}}
			In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their [[vanishing point]].

			== Hyperbolic geometry ==
			In [[hyperbolic geometry]], an ideal point is also called an '''omega point'''. Given a line ''l'' and a point ''P'' not on ''l'', right- and left-[[limiting parallel]]s to ''l'' through ''P'' are said to meet ''l'' at ''omega points''. Unlike the projective case, omega points form a [[manifold with boundary\|boundary]], not a submanifold. So, these lines do not ''intersect'' at an omega point and such points, although [[well defined]], do not belong to a hyperbolic space itself. In the [[Poincaré disk model]] and the [[Klein model]] of hyperbolic geometry, the omega points can be visualized since they lie on the boundary circle (which is not part of the model). [[Pasch's axiom]] and the [[exterior angle theorem]] still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.<ref>{{cite book\|last =Hvidsten\|first =Michael\|title = Geometry with Geometry Explorer\|publisher = McGraw-Hill\|year = 2005 \| location = New York, NY \| pages = 276–283 \| isbn = 0-07-312990-9}}</ref>

			== Other generalisations ==
			{{main\|Compactification (mathematics)}}
			This construction can be generalized to [[topological space]]s. Different compactifications may exist for a given space, but arbitrary topological space admits [[Alexandroff extension]], also called the ''one-point [[compactification (mathematics)\|compactification]]'' when the original space is not itself [[compact space\|compact]]. Projective line (over arbitrary field) is the Alexandroff extension<!-- such way is correct. finite fields do not have "compactifications" but Alexandroff extensions. --> of the corresponding field. Thus the circle is the one-point compactification of the [[real line]], and the sphere is the one-point compactification of the plane. [[Projective space]]s '''P'''<sup>{{mvar\|n}}</sup> for {{mvar\|n}} > 1 are not ''one-point'' compactifications of corresponding affine spaces for the reason mentioned [[#Projective geometry\|above]], and completions of hyperbolic spaces with omega points are also not one-point compactifications.

			== See also ==
			*[[Division by zero]]
			*[[Midpoint#Generalizations]]
			*[[Asymptote#Algebraic curves]]

			== References ==
			<references/>

			[[Category:Projective geometry]]
			[[Category:Infinity]]

			[[it:Glossario di geometria descrittiva#Punto improprio]]

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

2012-07-19T00:20:55Z

Robot - Moving category Fundamental physics concepts to Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

New page

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