Course MT3818 Topics in Geometry
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Homogeneous coordinates

To calculate with points and figures in the projective plane we introduce homogeneous coordinates which allow us to see the relationship between points at infinity and ordinary points.

First we will consider the projective line.
Fix the point of projection P to be the origin in R2. Then identify an ordinary point A on the line y = 1 with the line AP. Then every line through P is identified with a point of the line y = 1 except the x-axis itself. We will identify this with the point at infinity.
We can describe a line AP through the origin by taking a vector (α, β) in it. Then the point A has x-coordinate α/β while the point at infinity corresponds to a vector (α, 0).

Definition

The homogeneous coordinates of a point x on the affine line are (α, β) where x = α/β . The point at infinity has homogeneous coordinates (1, 0).
Since any non-zero multiple of the vector (α, β) will also be in AP, homogeneous coordinates are only defined up to multiplication by a non-zero scalar.

Remarks

  1. The pair [0, 0] does not represent any point on the projective line.

  2. We will denote the projective line by RP1.

  3. To do this formally, we use an equivalence relation.
    RP1 = R2 - {(0, 0)}/~ where (α, β) ~ (λα, λβ) for any λ ≠ 0.
    We denote the equivalence class of (α, β) by [α, β] so that [α, β] =[λα, λβ] for any λ ≠ 0.

  4. Ordinary points on the line thus have homogeneous coordinates [x, 1] (since [α, β] = [α/β , 1] if β ≠ 0) and the point at infinity has homogeneous coordinates [1,0].

Similarly, we may introduce homogeneous coordinates in the projective plane RP2. A point has homogeneous coordinates [x, y, z] (with, for example, [1, 2, 3] = [2, 4, 6] etc).
Ordinary affine points have coordinates [x/z , y/z , 1] while points on the line at infinity have coordinates [x, y, 0].

Properties

  1. In the ordinary affine plane a straight line has an equation ax + by + c = 0.
    Take homogeneous coordinates and replace x by x/z and y by y/z and get an equation a x/z + b y/z + c = 0 or ax + by + cz = 0 (a homogeneous equation). Note that this second equation has an extra solution corresponding to z = 0. This is the point at infinity on the line and has homogeneous coordinates [b, -a, 0].

  2. The equation y - x2 = 0 represents a parabola in the affine plane. Using the same trick as above ⇒ y/z - (x/z)2 = 0 ⇒ yz - x2 = 0.
    This homogeneous equation gives a curve in RP2 which meets the line at infinity z = 0 in the single point [0, 1, 0].

  3. The equation xy = 1 representing a rectangular hyperbola in the affine plane ⇒ x/z.y/z = 1 ⇒ xy = z2.
    This homogeneous equation gives a curve in RP2 which meets the line at infinity z = 0 at the two points [1, 0, 0] and [0, 1, 0] (the points at infinity on the asymptotes).

  4. The equation x2 + y2 = 1 representing a circle in the affine plane ⇒ x2 + y2 = z2 and this homogeneous equation gives a curve in RP2 which does not meet the line at infinity.

In fact arguing as above, one can show that ellipses do not meet the line at infinity, hyperbolae meet it at two points and parabolas touch it (meet it at one point).
Of course, if one projects so that the line at infinity is projected into an ordinary line, it is possible to project a parabola into an ellipse, etc.

We will see (soon!) how to get a "topological" picture of a projective space.

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JOC March 2003