geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, 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 In 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 dista ...

(including the Euclidean plane), 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 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. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...

, 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, 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 multiplicative inverse, that is, an element ...

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

(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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...

(when complex numbers are mapped to each point). In the case of a

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...

, each line has two distinct ideal points. Here, the set of ideal points takes the form of a

quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...

Affine geometry

In an

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

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

of higher dimension, the points at infinity are the points which are added to the space to get the

projective completion In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

. The set of the points at infinity is called, depending on the dimension of the space, the

line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The ...

, 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 contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned ...

or the

hyperplane at infinity In geometry, any hyperplane ''H'' of a projective space ''P'' may be taken as a hyperplane at infinity. Then the set complement is called an affine space. For instance, if are homogeneous coordinates for ''n''-dimensional projective space, then ...

, 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 de ...

, 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

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 on the image plane of a perspective drawing where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicul ...

Hyperbolic geometry

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

, points at infinity are typically named ideal points. Unlike Euclidean and

elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

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 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 *CONVERGE CFD s ...

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 In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the '' projection plane'' or ''image plane'', where the '' rays'', known as ' ...

arises as a central projection where the center ''C'' is a point at infinity, or figurative point. G. B. Halsted (1906
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 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 generalisations

This construction can be generalized to

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

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. 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, 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