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 solely with the three-dimensional case.
There are two approaches to defining the plane at infinity which depend on whether one starts with a projective 3-space or an affine 3-space.
On the other hand, given an affine 3-space, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties.
This point of view emphasizes the internal structure of the plane at infinity, but does make it look "special" in comparison to the other planes of the space.
Since any two projective planes in a projective 3-space are equivalent, we can choose a homogeneous coordinate system so that any point on the plane at infinity is represented as (X:Y:Z:0).
The points on the plane at infinity seem to have three degrees of freedom, but homogeneous coordinates are equivalent up to any rescaling: so that the coordinates (X:Y:Z:0) can be normalized, thus reducing the degrees of freedom to two (thus, a surface, namely a projective plane).
Any pair of parallel lines in 3-space will intersect each other at a point on the plane at infinity.
Also, every line in 3-space intersects the plane at infinity at a unique point.
This second line intersects the plane at infinity at the point (3:0:0:0).
■ Any pair of parallel planes in affine 3-space will intersect each other in a projective line (a line at infinity) in the plane at infinity.
Since the plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e. a sphere in which antipodal points are equivalent: S2/{1,-1} where the quotient is understood as a quotient by a group action (see quotient space).