In
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 ...
and
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, the line at infinity is a
projective line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...
that is added to the
affine plane in order to give
closure to, and remove the exceptional cases from, the
incidence properties of the resulting
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 ...
. The line at infinity is also called the ideal line.
Geometric formulation
In projective geometry, any pair of lines always intersects at some point, but
parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel.
Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the
slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
of the lines, not at all on their
y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects ...
.
In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are
closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s, i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical.
Topological perspective
If we consider the real affine plane, then the line at infinity can be visualized as a circle which surrounds it. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the real affine plane and the line at infinity makes the
real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
,
.
A
hyperbola
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
s of the hyperbola. Likewise, a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the parabola. If the parabola is cut by its vertex into a symmetrical pair of "horns", then these two horns become more parallel to each other further away from the vertex, and are actually parallel to the axis and to each other at infinity, so that they intersect at the line at infinity.
The analogue for the complex projective plane is a 'line' at infinity that is (naturally) a complex
projective line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...
. Topologically this is quite different, in that it is a
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
, which is therefore a 2-
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, being added to a complex affine space of two dimensions over
(so four ''real'' dimensions), resulting in a four-dimensional
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 ...
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 ...
. The result is
orientable, while the real projective plane is not.
History
The complex line at infinity was much used in nineteenth century geometry. In fact one of the most applied tricks was to regard a circle as a
conic
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, thou ...
constrained to pass through two points at infinity, the solutions of
:''X''
2 + ''Y''
2 = 0.
This equation is the form taken by that of any circle when we drop terms of lower order in ''X'' and ''Y''. More formally, we should use
homogeneous 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. ...
:
'X:Y:Z''
and note that the line at infinity is specified by setting
: ''Z'' = 0.
Making equations homogeneous by introducing powers of ''Z'', and then setting ''Z'' = 0, does precisely eliminate terms of lower order.
Solving the equation, therefore, we find that all circles 'pass through' the ''
circular points at infinity''
:''I'' =
:''i'':0and ''J'' =
:−''i'':0
These of course are complex points, for any representing set of homogeneous coordinates. Since the projective plane has a large enough
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
, they are in no way special, though. The conclusion is that the three-parameter family of circles can be treated as a special case of the
linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstractio ...
of conics passing through two given distinct points ''P'' and ''Q''.
See also
*
*
Hyperplane at infinity
*
Parallel postulate
In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
If a line segment intersects two straight lines forming two interior ...
*
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 ...
*
Point at infinity
In geometry, 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 of parallel lines of the plane. Ad ...
References
{{reflist
Projective geometry
Perspective projection
Infinity