Introduction

The projective plane provides one alternative to Euclidean geometry. A second alternative is explored in this chapter.

The three geometries are contrasted in the following example: Take a segment P1P2 as shown in Figure 7.1. Erect equal segments P1Q1 and P2Q2 perpendicular to P1P2.

In E2 the segment Q1Q2 will have length equal to that of P1P2. However, in P2, the length of Q1Q2 will be less than that of P1P2. In H2 we shall see that Q1Q2 will be longer than P1P2.

This construction is also related to the question of parallelism. Let ℓ0 be a line, and let P be a point not on ℓ0. Drop a perpendicular PP0 from P to ℓ0, and let ℓ be the line through P perpendicular to PP0. (See Figure 7.2.)

In E2, ℓ will be parallel to ℓ0. In P2, ℓ will meet ℓ0. In H2 it will turn out that ℓ does not meet ℓ0.

We will now proceed to construct the geometry H2. It will again consist of “points” and “lines” with a “distance” function defined for each pair of points. As in the case of E2 and P2, we find that isometries of H2 are generated by reflections and satisfy the three reflections theorems.

Algebraic preliminaries

Our model of spherical geometry was a certain subset of R3, and the usual inner product of R3 played an important role.