Homogenous and inhomogenous coordinates

Points on the Euclidean line may be described by a single number, after we choose a point to call 0 and a point to call 1. This is how we ordinarily find our way around the Euclidean line (the real line). When we add a point at infinity, to get a projective line, we can indicate it by the symbol ∞. This number, or ∞, is the inhomogeneous coordinate of the point on the projective line.

On the other hand, we can use homogeneous coordinates to find our way around the projective line. Here we treat the point at infinity on equal footing with the other points. A point on the line is described by two coordinates (x,y), both of them numbers, and not both zero. The pairs of coordinates (x,y) and (kx,ky) denote the same point. So if y ≠ 0, we can divide both coordinates by y to write the coordinates of the point in the form (x,1), while if y = 0, then it may be described as (1,0). So the point whose inhomogeneous coordinate is a number x, rather than ∞, has homogeneous coordinates (x,1), while the point whose inhomogeneous coordinate is ∞, has homogeneous coordinates (1,0).

Similarly, we find our way around the Euclidean plane by choosing an origin, two perpendicular coordinate axes, and a unit length. Then we can label every point uniquely by a pair of real number (x,y) in the usual way. These are inhomogeneous coordinates for points on the plane.

How do we label the points at infinity? We need to label the point that the line y = mx goes through. A natural labeling is (∞, m∞), but you didn't hear it here. The other point at infinity, the one that the y-axis goes through, is naturally labeled (0, ∞).

For the plane, we need three homogeneous coordinates (x,y,z), not all zero. If z ≠ 0, then we can divide all three by z and get coordinates in the form (x,y,1). The inhomogeneous coordinates-the ordinary ones-of this point are (x,y). A point with homogeneous coordinates (x,y,0) is on the line at infinity. Using the inhomogeneous coordinates of the preceding paragraph, we would indicate this point by (x∞, y∞). For example, the line y = 2x goes through all the finite points of the form (t,2t), and the point at infinity (∞, 2∞). This point has homogeneous coordinates (1,2,0)-you can imagine dividing all three numbers (1,2,0) by 0 to get (∞, 2∞,1). I didn't say that.

The equation of a line in homogeneous coordinates is ax+by+cz = 0. We indicated that line by [a,b,c]. A point (x,y,z) is on the line [a,b,c] exactly when ax+by+cz = 0. What is the inhomogeneous form of the same line?

If z ≠ 0, then (x,y,z) is the finite point (x/z,y/z,1) whose inhomogeneous coordinates are (x/z,y/z). If ax+by+cz = 0, then a(x/z)+b(y/z)+c = 0, so the inhomogeneous equation for this line is ax+by+c = 0. If b ≠ 0, we can write this equation as y = (-a/b)x-c/b. If b = 0 and a ≠ 0, then the equation can be written x = -c/a. If both a and b are zero, then the line [a,b,c] is the line at infinity, with equation z = 0, and the inhomogeneous equation ax+by+c = 0 has no solutions (there are no finite points at infinity).

What's the inhomogeneous equation of the line [1,2,3]? This is the line whose homogeneous equation is x+2y+3z = 0. For finite points, z ≠ 0, and the equation becomes x/z+2y/z+3 = 0, so the equation in inhomogeneous coordinates is y = -¹/₂x - ³/₂. The point at infinity on this line is gotten by setting z = 0 in the equation x+2y+3z = 0. So we get x+2y = 0 so that point at infinity is (2,-1,0).

What is the homogeneous form of the line whose inhomogeneous equation (the standard equation) is y = 5x-7? That is, when does a point whose homogeneous coordinates are (x,y,z) lie on this line? If z ≠ 0, then the inhomogeneous coordinates of that point are (x/z,y/z), so the condition is that y/z = 5x/z-7, that is, y = 5x-7z or 5x-y-7z = 0. So the line is [5,-1,-7].

The cross ratio of four real numbers a,b,c,d (in that order) is the quantity

To compute the cross ratio of four collinear points in a projective plane, first project them onto a standard axis. The easiest way to do this is write them in inhomogeneous coordinates, and then use their first coordinates (the projection on the x-axis) or their second coordinates (the projection on the y-axis). So given the four points (1,1,2), (1,0,1), (2,1,3) and (0,1,1) in the real projective plane, we look at their inhomogeneous coordinates: (1/2,1/2), (1,0), (2/3,1/3) and (0,1). The projections on the x-axis are 1/2, 1, 2/3 and 0, whose cross ratio is

Projective transformations of the line. Euclidean transformations have to preserve distance. So the only Euclidean transformations of the x-axis are translations and reflections: transformations given by an equation of the form x′ = ±x+b. The more general transformations of the form x′ = mx+b, with m ≠ 0, are called affine transformations. They don't quite preserve distances, but they change them uniformly. The transformation x′ = 2x-3 doubles all distances. In Euclidean geometry these kinds of transformations take a triangle to a similar triangles rather than to a congruent triangle.

It's easy to see that the transformation x′ = x+e (a translation) and the transformation x′ = mx preserves cross ratio (replace each letter x in the cross-ratio formula by x+e or mx). So cross ratio is preserved by any affine transformation.

In projective geometry the transformation x′ = 1/x is also allowed. This would be impossible in Euclidean geometry because there is no x′ corresponding to x = 0. But on the projective line we have the point ∞ = 1/0. You can easily check that cross ratio is preserved by this transformation.

By combining this transformation with affine transformations we get the equation of the general projective transformation of the line

Exercises

1. Consider the points (1,2), (2,5), (-1,-4) and (10,29) in the Euclidean plane.
   1. Show that they are collinear.

   2. Compute their cross ratio in two ways.

   3. Project them onto the x-axis from the point (0,1). Compute the cross ratio of the resulting four points on the x-axis.

2. Consider the points (4,0,1), (0,4,1), (3,4,3) and (4,3,3) on the projective plane over Z₅.
   1. Show that they are collinear.

   2. Convert to inhomogeneous coordinates, and compute their cross ratio in two ways.

3. Find a projective transformation of the real line that takes the points 0, 1 and 2 to 0, ∞ and 2, respectively.