geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

), there is one ideal point for each

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of ...

of parallel lines of the plane. Adjoining these points produces a

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any

division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...

. 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, CP¹, 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 points. Here, the set of ideal points takes the form of a quadric.

Affine geometry

In an

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

of higher dimension, the points at infinity are the points which are added to the space to get the 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. 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

Perspective

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 A vanishing point is a point (geometry), point on the projection plane, image plane of a graphical perspective, perspective rendering where the two-dimensional perspective projections of parallel (geometry), parallel lines in three-dimensional ...

Hyperbolic geometry

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, points at infinity are typically named ideal points. Unlike Euclidean and 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 parallels converge asymptotically to different points at infinity. All points at infinity together form the

Cayley absolute Cayley may refer to: __NOTOC__ People * Cayley (surname) * Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow * Cayley Mercer (born 1994), Canadian women's ice hockey player Places * Cayley, Alberta, Canada, a hamlet ** Cayley/A. J. ...

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. The axiomatic symmetry of points and lines is called 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 In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...

, 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

This construction can be generalized to

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s. Different compactifications may exist for a given space, but arbitrary topological space admits Alexandroff extension, also called the ''one-point compactification'' when the original space is not itself

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

. Projective line (over arbitrary field) is the Alexandroff extension of the corresponding field. Thus, the circle is the one-point compactification of the

real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

, and the sphere is the one-point compactification of the plane. Projective spaces P for > 1 are not ''one-point'' compactifications of corresponding affine spaces for the reason mentioned above under , and completions of hyperbolic spaces with ideal points are also not one-point compactifications.

References

{{reflist, 30em, refs= {{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 {{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 {{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 {{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 {{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 Projective geometry Hyperbolic geometry Infinity it:Glossario di geometria descrittiva#Punto improprio

Affine geometry

Perspective

Hyperbolic geometry

Projective geometry

Other generalizations

See also

References