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.
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