In ^{1}, also called the

Synthetic Projective Geometry

page 7 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

^{} 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.

geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine planeIn geometry, an affine plane is a two-dimensional affine space.
Examples
Typical examples of affine planes are
*Euclidean planes, which are affine planes over the real number, reals, equipped with a metric (mathematics), metric, the Euclidean distan ...

(including the Euclidean plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

), there is one ideal point for each pencil
A pencil is a writing
Writing is a medium of human communication that involves the representation of a language with written symbols. Writing systems are not themselves human languages (with the debatable exception of computer language ...

of parallel lines of the plane. Adjoining these points produces a projective plane
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, and more generally over any division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...

.
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 lineIn mathematics, a complex line is a one-dimensional affine space, affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has complex dimension, dimension one over C (hence the term "line") ...

(which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CPRiemann sphere
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(when complex numbers are mapped to each point).
In the case of a hyperbolic space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, each line has two distinct ideal point
200px, Three Ideal triangles in the Poincaré disk model, the vertex (geometry), vertices are ideal points
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space.
Giv ...

s. Here, the set of ideal points takes the form of a quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimension
thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics
Physics is the natural science that s ...

.
Affine geometry

In anaffine
Affine (pronounced /əˈfaɪn/) relates to 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 substitution cipher
*Aff ...

or Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

of higher dimension, the points at infinity are the points which are added to the space to get the projective completion
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...

. The set of the points at infinity is called, depending on the dimension of the space, the line at infinity
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

, the plane at infinity
In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any Plane (geometry), plane contained in the hyperplane at infinity of any projective space of higher dimension. This article wil ...

or the hyperplane at infinityIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

, in all cases a projective space of one less dimension.
As a projective space over a field is a smooth algebraic variety
In the Mathematics, mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly def ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.
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 theirvanishing point
A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections (or drawings) of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lin ...

.
Hyperbolic geometry

Inhyperbolic 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 an ...

, points at infinity are typically named ideal point
200px, Three Ideal triangles in the Poincaré disk model, the vertex (geometry), vertices are ideal points
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well defined point outside the hyperbolic plane or space.
Giv ...

s. Unlike Euclidean
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of:
Geometry
*Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...

and elliptic
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...

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
frame, The two lines through a given point ''P'' and limiting parallel to line ''R''.
In neutral or absolute geometry, and in hyperbolic geometry
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

s converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
See also
...

asymptotically
250px, A curve intersecting an asymptote infinitely many times.
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc ...

to different points at infinity.
All points at infinity together form the Cayley absoluteCayley may refer to:
* Cayley (surname)
*Cayley, Alberta, Canada, a hamlet
*Mount Cayley, a volcano in southwestern British Columbia, Canada
*Cayley (crater), a lunar crater
*Cayley computer algebra system, designed to solve mathematical problems, p ...

or boundary of a hyperbolic plane
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
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 ofgraphical perspective
Graphics () are visual perception, visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustration, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of dat ...

where a parallel projection
A parallel projection is a projection of an object in three-dimensional space onto a fixed Plane (mathematics), plane, known as the ''projection plane'' or ''image plane'', where the ''rays'', known as ''lines of sight'' or ''projection lines'', are ...

arises as a central projection
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

where the center ''C'' is a point at infinity, or figurative point.G. B. Halsted
George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) ...

(1906Synthetic Projective Geometry

page 7 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 rangeIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, in the representation of points with projective coordinates
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, 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 generalisations

This construction can be generalized totopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s. Different compactifications may exist for a given space, but arbitrary topological space admits Alexandroff extension In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population i ...

, 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
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

. 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, and the sphere is the one-point compactification of the plane. Projective space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s PSee also

*Division by zero
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

* Sphere at infinity
*
*
References

{{reflist Projective geometry Hyperbolic geometry Infinity it:Glossario di geometria descrittiva#Punto improprio