The Projective Plane

Section 9.1 The Projective Plane

The incidence postulate, slightly reworded, states that:

Every pair of distinct points lies on exactly one line.

What happens if we swap the role of points and lines? Let's see:

Every pair of distinct lines intersect in exactly one point.

But that's the elliptic parallel postulate!

To model a geometry satisfying both postulates, all lines must intersect. In Euclidean geometry, of course, parallel lines exist. We can classify families of parallel lines by their slope \(m\text{.}\) In analogy with the construction of hyperbolic lunes in Section 8.7, we can expand the Euclidean plane by adding an ideal point \(I_m\) for each possible slope \(m\text{.}\) What values can \(m\) take? From the slope-intercept equation of a line, \(y=mx+b\text{,}\) we know that the slope can be any real number. But what about vertical lines? We will say that they have infinite slope, and write \(m=\infty\text{.}\) ¹ So our ideal points are \(I_m\text{,}\) with \(m\in\RR\cup\{\infty\}\text{.}\)

There is no difference between lines with slope \(\pm\infty\text{,}\) and therefore no need to include \(m=-\infty\) as a separate case.

We now construct a model, called the real projective plane, whose points are all Euclidean points together with all ideal points, and whose lines are either Euclidean lines with the appropriate ideal point \(I_m\) added, or the ideal line \(L\text{,}\) containing all ideal points.

We have seen this model before! These points and lines can be identified with the points and lines of the Klein disk, with ideal points corresponding to pairs of antipodal points on the boundary, and the ideal line corresponding to the boundary. However, there is no measurement in projective geometry, the geometry underlying our model. There is therefore no good way to specify angles or line segments, both of which rely on choosing the smaller of the geometric objects formed.

An important feature of projective geometry, which we built in from the start, is a duality between points and lines. Every property of projective geometry continues to hold if the notions of points and lines are interchanged – because the postulates are invariant under that change.