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Ingeometry, apoint at infinity orideal point is an idealized limiting point at the "end" of each line.
In the case of anaffine plane (including theEuclidean plane), there is one ideal point for eachpencil of parallel lines of the plane. Adjoining these points produces aprojective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over anyfield, and more generally over anydivision ring.[1]
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 thecomplex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line,CP1, also called theRiemann sphere (when complex numbers are mapped to each point).
In the case of ahyperbolic space, each line has two distinctideal points. Here, the set of ideal points takes the form of aquadric.
In anaffine orEuclidean space of higher dimension, thepoints at infinity are the points which are added to the space to get theprojective completion.[citation needed] The set of the points at infinity is called, depending on the dimension of the space, theline at infinity, theplane at infinity or thehyperplane at infinity, in all cases a projective space of one less dimension.[2]
As a projective space over a field is asmooth 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 amanifold.
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 theirvanishing point.[3]
Inhyperbolic geometry,points at infinity are typically namedideal points.[4] UnlikeEuclidean andelliptic geometries, each line has two points at infinity: given a linel and a pointP not onl, the right- and left-limiting parallelsconvergeasymptotically to different points at infinity.
All points at infinity together form theCayley absolute or boundary of ahyperbolic plane.
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 ofgraphical perspective where aparallel projection arises as acentral projection where the centerC is a point at infinity, orfigurative point.[5] The axiomatic symmetry of points and lines is calledduality.
Though a point at infinity is considered on a par with any other point of aprojective range, in the representation of points withprojective 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.
This construction can be generalized totopological spaces. Different compactifications may exist for a given space, but arbitrary topological space admitsAlexandroff extension, also called theone-pointcompactification when the original space is not itselfcompact. Projective line (over arbitrary field) is the Alexandroff extension of the corresponding field. Thus, the circle is the one-point compactification of thereal line, and the sphere is the one-point compactification of the plane.Projective spacesPn forn > 1 are notone-point compactifications of corresponding affine spaces for the reason mentioned above under§ Affine geometry, and completions of hyperbolic spaces with ideal points are also not one-point compactifications.