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 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. If a projective 3-space is given, the plane at infinity is any distinguished projective plane of the space. This point of view emphasizes the fact that this plane is not geometrically different than any other plane. 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. Meaning that the points of the plane at infinity are the points where parallel lines of the affine 3-space will meet, and the lines are the lines where parallel planes of the affine 3-space will meet. The result of the addition is the projective 3-space, . 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. If the affine 3-space is real, , then the addition of a real projective plane at infinity produces the real projective 3-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). Any point in the affine 3-space will then be represented as (X:Y:Z:1). 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). Proposition: Any line which passes through the origin (0:0:0:1) and through a point (X:Y:Z:1) will intersect the plane at infinity at the point (X:Y:Z:0).