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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.
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| 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]].
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| The concept of infinity point admits several generalizations for various multi-dimensional constructions.
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| == 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.
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| 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.
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| === Projective plane ===
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| 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.
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| 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]].
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| === Perspective === | |
| {{main|Perspective (graphical)}}
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| 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]].
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| == Hyperbolic geometry ==
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| 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>
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| == Other generalisations ==
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| {{main|Compactification (mathematics)}}
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| 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.
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| == See also ==
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| *[[Division by zero]]
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| *[[Midpoint#Generalizations]]
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| *[[Asymptote#Algebraic curves]]
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| == References ==
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| <references/>
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| [[Category:Projective geometry]]
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| [[Category:Infinity]]
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| [[it:Glossario di geometria descrittiva#Punto improprio]]
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Aleta is what's written on her birth 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 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
my webpage; clash of clans generator