Homogeneous co-ordinates

The introduction for projective space of homogeneous co-ordinates makes calculations possible, as Cartesian co-ordinates do in Euclidean space. The homogeneous co-ordinates of a point of projective space of dimension n are usually written as (x:y:z: ... :w), a row vector of length n+1, other than (0:0:0: ... :0). Two sets of co-ordinates that are proportional denote the same point of projective space: for any non-zero scalar c from the underlying field K, (cx:cy:cz: ... :cw) denotes the same point. Therefore this system of co-ordinates can be explained as follows: if the projective space is constructed from a vector space V of dimension n+1, introduce co-ordinates in V by choosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V.

Taking the example of projective space of dimension three, there will be homogeneous co-ordinates (x:y:z:w). The plane at infinity is usually identified with the set of points with w = 0. Away from this plane we can use (x/w, y/w, z/w) as an ordinary Cartesian system; therefore the affine space complementary to the plane at infinity is co-ordinatised in a familiar way, with a basis corresponding to (1:0:0:1), (0:1:0:1), (0:0:1:1).

If we try to intersect the two planes defined by equations x = w and x = 2w then we clearly will derive first w = 0 and then x = 0. That tells us that the intersection is contained in the plane at infinity, and consists of all points with co-ordinates (0:y:z;0). It is a line, and in fact the line joining (0:1:0:0) and (0:0:1:0). It cannot be given by a single equation in the co-ordinates. In fact a line in three-dimensional projective space corresponds to a two-dimensional subspace of the underlying four-dimensional vector space, therefore given by two linear conditions.