In
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 c ...
, 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
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 (mechanical), abra ...
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 ...
, 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 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, CP
1, also called the
Riemann sphere (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. ...
, 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 ...
.
Affine geometry
In an
affine or
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 sp ...
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 ...
.
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 perpen ...
.
Hyperbolic geometry
In
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 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
* Mount Cayley, a vo ...
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
George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and ...
(1906
Synthetic Projective Geometry
page 7 The axiomatic symmetry of points and lines is called
duality
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
.
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 In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Ale ...
, also called the ''one-point
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
{{disambiguation ...
'' 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
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
, 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.
See also
*
Division by zero
*
Sphere at infinity
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their c ...
*
*
References
{{reflist
Projective geometry
Hyperbolic geometry
Infinity
it:Glossario di geometria descrittiva#Punto improprio