Introduction

We now come to a geometric structure that is more abstract than the previous two we have dealt with. The geometry of the projective plane will resemble that of the sphere in many respects. However, we regain the Euclidean phenomenon that two lines can intersect only once. The projective plane will also be a foundation for our study of hyperbolic geometry in Chapter 7.

Although many of the properties of the projective plane are familiar, one that will appear strange is that of nonorientability. In P2 every reflection may be regarded as a rotation. This has the intuitive consequence that an outline of a left hand can be moved continuously to coincide with its mirror image, the outline of a right hand.

The abstraction is involved in the fact that every point of P2 is a pair of points of S2. Two antipodal points of S2 are considered to be the same point of P2.

Definition.The projective planeP2is the set of all pairs {x, −x} of antipodal points ofS2.

Remark: Two alternative definitions of P2, equivalent to the preceding one are

1. i. The set of all lines through the origin in E3.

2. ii. The set of all equivalence classes of ordered triples (x1, x2, x3) of numbers (i.e., vectors in E3) not all zero, where two vectors are equivalent if they are proportional.