Lecture 3 19

Figure 1.13. Multiple geodesics between antipodal points.

The projective plane with distance inherited from the

sphere7

is

called the elliptic plane—it will be one of the star exhibits of this

course. We can motivate its definition by considering the sphere as a

geometric object, on which the notion of a line in Euclidean space is

to be replaced by the concept of a geodesic; one key property of the

former is that it is the shortest path between two points, and so infor-

mally at least, geodesics are simply curves which have this property.

On the sphere, we will see that the geodesics are great circles, and so

we may attempt to formulate various geometric propositions in this

setting. However, this turns out to have some undesirable features

from the point of view of conventional geometry; for example, every

pair of geodesics intersects in two (diametrically opposite) points, not

just one. Further, any two diametrically opposite points on the sphere

can be joined by infinitely many geodesics (Figure 1.13), in stark con-

trast to the “two points determine a unique line” rule of Euclidean

geometry.

Both of these diﬃculties are related to pairs of diametrically oppo-

site points; the solution turns out to be to identify such points with

each other. Identifying each point on the sphere with its antipode

yields a quotient space, which is the projective plane described at the

end of the first lecture. Alternatively, we can consider the flip map

I : (x, y, z) → (−x, −y,−z), which is an isometry of the sphere with-

out fixed points. Declaring all members of a particular orbit of I to

7This

simply means that the distance between two points in the projective plane

is taken to be the minimum of pairwise distances between points in the sphere repre-

senting those points.