Line at infinity
Encyclopedia
In geometry
and topology
, the line at infinity is a 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 line at infinity is also called the ideal line.
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. The point at which the parallel lines intersect depends only on the slope
of the lines, not at all on their y-intercept
. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines is parallel.
Every line intersects the line at infinity at some point. The point at which a line intersects the line at infinity determines the slope of the line, but not at all its y-intercept.
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, 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.
, which is homeomorphic to a Möbius strip
: diametrically opposite points of the circle are equivalent -- they are the same point. The combination of affine plane and line at infinity makes the real projective plane
,
.
A hyperbola
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
s of the hyperbola. Likewise, a parabola
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
. Topologically this is quite different, in that it is a Riemann sphere
, which is therefore a 2-sphere
, being added to a complex affine space of two dimensions over C (so four real dimensions), resulting in a four-dimensional compact
manifold
. The result is orientable, while the real projective plane is not.
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
and note that the line at infinity is specified by setting
Making equations homogeneous by introducing powers of Z, and then setting Z = 0, does precisely kill off terms of lower order.
Solving the equation, therefore, we find that all circles 'pass through' the circular points at infinity
These of course are complex points, for any representing set of homogeneous coordinates. Since the projective plane has a large enough symmetry group
, 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
of conics passing through two given distinct points P and Q.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the line at infinity is a line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence
Incidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
properties of the resulting projective plane
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 that do not intersect...
. The line at infinity is also called the ideal line.
Geometric formulation
In projective geometry, any pair of lines always intersect at some point. But parallelParallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
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. The point at which the parallel lines intersect depends only on the slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
of the lines, not at all on their y-intercept
Y-intercept
In coordinate geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept is a point where the graph of a function or relation intersects with the y-axis of the coordinate system...
. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines is parallel.
Every line intersects the line at infinity at some point. The point at which a line intersects the line at infinity determines the slope of the line, but not at all its y-intercept.
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, 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
The line at infinity can be visualized as a circle which surrounds the affine plane. However, this circle is actually like cross-capCross-cap
In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection. This self intersection precludes the cross-cap from being topologically equivalent to a Möbius strip...
, which is homeomorphic to a Möbius strip
Möbius strip
The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
: diametrically opposite points of the circle are equivalent -- they are the same point. The combination of affine plane and line at infinity makes the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
,
.A hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
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
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 they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
s of the hyperbola. Likewise, a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
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
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
. Topologically this is quite different, in that it is a Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
, which is therefore a 2-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, being added to a complex affine space of two dimensions over C (so four real dimensions), resulting in a four-dimensional compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. 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 constrained to pass through two points at infinity, the solutions of- X2 + Y2 = 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
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
- [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 kill off terms of lower order.
Solving the equation, therefore, we find that all circles 'pass through' the circular points at infinity
Circular points at infinity
In projective geometry, the circular points at infinity in the complex projective plane are and ....
- I = [1:i:0] and J = [1:−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
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
, 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
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
of conics passing through two given distinct points P and Q.