View source for Point at infinity

{{short description|Concept in geometry}}
{{More citations needed|date=July 2017}}[[Image:Real projective line.svg|right|thumb|150px|The real line with the point at infinity; it is called the [[real projective line]].]]
In [[geometry]], a '''point at infinity''' or '''ideal point''' is an idealized limiting point at the "end" of each line.

In the case of an [[affine plane]] (including the [[Euclidean plane]]), there is one ideal point for each [[Pencil (mathematics)|pencil]] of parallel lines of the plane. Adjoining these points produces a [[projective plane]], in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any [[Field (mathematics)|field]], and more generally over any [[division ring]].{{r|mathworld}}

In the real case, a point at infinity completes a line into a topologically closed curve.  In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong.  A point at infinity can also be added to the [[complex line]] (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, '''C'''P<sup>1</sup>, also called the [[Riemann sphere]] (when complex numbers are mapped to each point).

In the case of a [[hyperbolic space]], each line has two distinct [[ideal point]]s.  Here, the set of ideal points takes the form of a [[Quadric (projective geometry)|quadric]].

== Affine 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]].{{citation needed|date=March 2023}} 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.{{r|atinfinity}}

As a projective space over a field is a [[smooth algebraic variety]], the same is true for the set of points at infinity. Similarly, if the ground field is the real or the complex field, the set of points at infinity is a [[manifold]].

=== 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]].{{r|vanishing}}

== Hyperbolic geometry ==
{{main|Ideal point}}
In [[hyperbolic geometry]], '''points at infinity''' are typically named [[ideal point]]s.{{r|hyperbolic}} Unlike [[Euclidean geometry|Euclidean]] and [[Elliptic geometry|elliptic]] geometries, each line has two points at infinity: given a line ''l'' and a point ''P'' not on ''l'', the right- and left-[[limiting parallel]]s [[Convergence (mathematics)|converge]] [[asymptotically]] to different points at infinity.

All points at infinity together form the [[Cayley absolute]] or boundary of a [[hyperbolic plane]].

==Projective geometry==
A symmetry of points and lines arises in a projective plane: just as a pair of points determine a line, so a pair of lines determine a point. The existence of parallel lines leads to establishing a point at infinity which represents the intersection of these parallels. This axiomatic symmetry grew out  of a study of [[graphical perspective]] where  a [[parallel projection]] arises as a [[central projection]] where the center ''C'' is a point at infinity, or '''figurative point'''.{{r|figurative}} The axiomatic symmetry of points and lines is called [[duality (projective geometry)|duality]].

Though a point at infinity is considered on  a par with any other point of a [[projective range]], in the representation of points with [[projective coordinates]], distinction is noted: finite points are represented with a 1 in the final coordinate while a point at infinity has a 0 there. The need to represent points at infinity requires that one extra coordinate beyond the space of finite points is needed.

== Other generalizations ==
{{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;>&nbsp;1 are not ''one-point'' compactifications of corresponding affine spaces for the reason mentioned above under {{section link||Affine geometry}}, and completions of hyperbolic spaces with ideal points are also not one-point compactifications.

== See also == 
*[[Division by zero]]
*[[Sphere at infinity]]
*{{multi-section link|Midpoint|Generalizations}}
*{{multi-section link|Asymptote|Algebraic curves}}

== References ==
{{reflist|30em|refs=
<ref name="atinfinity">{{cite book
 |last      = Coxeter |first = H. S. M. |author-link = H. S. M. Coxeter
 |title     = Projective Geometry
 |url       = https://books.google.com/books?id=gjAAI4FW0tsC&pg=PA109
 |year      = 1987
 |page      = 109
 |publisher = Springer-Verlag
 |edition   = 2nd|isbn = 978-0-387-40623-7 }}</ref>

<ref name="figurative">{{cite book
 |last      = Halsted |first = G. B. |author-link = G. B. Halsted
 |year      = 1906
 |url       = https://archive.org/details/syntheticproject00halsuoft/page/i/mode/2up
 |title     = Synthetic Projective Geometry
 |page      = 7|publisher = New York Wiley }}</ref>

<ref name="hyperbolic">{{cite book
 |last      = Kay |first = David C.
 |year      = 2011
 |title     = College Geometry: A Unified Development
 |url       = https://books.google.com/books?id=0xrSBQAAQBAJ&pg=PA548
 |page      = 548
 |publisher = CRC Press|isbn = 978-1-4398-9522-1
 }}</ref>

<ref name="mathworld">{{cite web
 |last1     = Weisstein |first1 = Eric W.
 |title     = Point at Infinity
 |url       = http://mathworld.wolfram.com/PointatInfinity.html
 |website   = mathworld.wolfram.com
 |publisher = Wolfram Research
 |access-date = 28 December 2016
 |language  = en}}</ref>

<ref name="vanishing">{{cite book
 |last1 = Faugeras |first1 = Olivier |author-link1 = Olivier Faugeras
 |last2 = Luong |first2 = Quang-Tuan |author-link2 = Quang-Tuan Luong
 |title = The Geometry of Multiple Images: The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications
 |year = 2001
 |url = https://books.google.com/books?id=vauYE0nlFGEC&pg=PA19
 |publisher = MIT Press
 |page = 19
 |isbn = 978-0262062206}}</ref>
}}

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

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