Lecture Notes 18: Homogeneous and non-Homogeneous Coordinates

The projective plane constructed in the last section is denoted by PG(2,D). If D is a finite field of order q, we use the short-hand notation, PG(2,q).

The points of PG(2,D) are the 1-dimensional subspaces of D3. Any non-zero vector in this 1-dimensional subspace is called a homogeneous coordinate of the corresponding point. A point thus has several homogeneous coordinates, but they all differ only by a (left) scalar multiple. Lines are represented by linear homogeneous equations, and a point is on a line if and only if any (and hence all) of its homogeneous coordinates satisfy the equation of the line.

Suppose now that we chose the canonical forms as the homogeneous coordinates of the points of PG(2,D), i.e., (p,q,1), (1,m,0) or (0,1,0). All the points with last coordinate 0 lie on the single line with equation z = 0. If we remove this line (and the points on it) from the projective plane, we obtain the affine plane, which we denote by AG(2,D). Under our assumption about canonical forms, every point of AG(2,D) has homogeneous coordinates which end with a 1. Since this last position doesn't change, we can drop it without any loss of information. Doing this gives non-homogeneous coordinates for the points of the affine plane. That is, a point with homogeneous coordinates (p,q,1) has non-homogeneous coordinates (p,q). Unlike homogeneous coordinates, non-homogeneous coordinates are uniquely associated with their points. In order to provide non-homogeneous coordinates for all points of the projective plane, we would have to assign some symbol to the points of z = 0. Unfortunately, we can not use ordered pairs, since these are all associated to points of the affine plane. The usual assignment is to label the points (1,m,0) with (m) and the point (0,1,0) with ().

We will now consider the equations of lines in terms of non-homogeneous coordinates. Before we do this, notice that in D, x(-m) = -(xm) for all x,m in D. This can be seen from the calculation xm + x(-m) = x(m -m) = x(0) = 0, hence x(-m) is the additive inverse of xm.

In homogeneous coordinates, the point (x,y,1) is on the line [1,0,-k] if and only if, x(1) + y(0) + 1(-k) = 0, i.e., x -k = 0 or x = k. Notice that (0,1,0) is also on this line for any value of k. These are the vertical lines. Next, the point (x,y,1) is on the line [-m,1,-b] if and only if, x(-m) + y(1) + 1(-b) = 0, i.e., -(xm) + y -b = 0, or y = xm + b. Note here that the point (1,m,0) is on the line [-m,1,-b] for all choices of b, since 1(-m) + m(1) + 0(-b) = -m + m + 0 = 0. The equations of the form x = k, and y =xm + b are called non-homogeneous equations. The vertical lines (in the projective plane) pass through the point (), and the lines with fixed m of the form y = xm + b (again in the projective plane) pass through the point (m).