Why Projective Geometry?

Most people who have taken high-school Geometry are familiar with Euclidean compass and straight-edge constructions. For example: “Given two points A and B, construct an equilateral triangle with base AB.”

The most straightforward way of constructing the triangle is to set your compass to the length of AB, then draw two circles, one with center A and the other with center B. Label one of the two intersection points as C. Use your straight-edge to connect the dots and you have the desired equilateral triangle. (Figure 1)

Figure 1

However, this sort of construction is not rigorous by modern mathematical standards. Nowhere did Euclid actually prove that circles had to intersect. To prove this using just the axioms of Euclidean Geometry takes some real effort, and even then one must make some exceptions.

For example, when dealing with just lines and points, Euclidean Geometry is only able to prove the following:

(1.1 )Two distinct lines meet in at most one point.

Whereas if we want to rigorously construct objects based on incidences (i.e., where lines and points meet), we need to guarantee that lines actually meet. That is, we want the following:

(1.2) Two distinct lines meet in one and only one point.

Projective Geometry, which uses only a straight-edge, and thus constructs Geometric figures by finding where lines and points intersect, must make some changes to traditional Euclidean Geometry in order to prove (1.1).

Next: The Parallel Postulate