20 1. Various Ways of Representing Surfaces and Examples
Figure 1.14. Determining distances in RP
via central angles.
be the same point, we obtain the quotient space
which is again
the projective plane, or the elliptic plane when we are interested in
In the elliptic plane, there is no such notion as the sign of an angle;
we cannot consistently determine which angles are positive and which
are negative. All the other geometric notions carry over, however; the
distance between two points can still be found as the magnitude of
the (acute) central angle they make (Figure 1.14), and the notions of
angle between geodesics and length of geodesics are still well defined.
Exercise 1.10. Write at least five propositions from Euclidean ge-
ometry which are true in the elliptic plane and at least three propo-
sitions which are true in Euclidean geometry and are not true in the
elliptic plane. Each proposition must include statements about con-
figurations of lines and/or isometries, and no two should be trivial
reformulations of each other.
b. A first glance at geodesics. Informally, as mentioned above,
a geodesic is a curve of shortest length between two points; more
precisely, it is a curve γ with the property that given any two points
γ(a) and γ(b) whose parameter values a and b are suﬃciently close
together, any other curve from one point to the other will have length
at least as great as the portion of γ between the two. Later in the
course (Lecture 25), we will consider the question of whether such a
curve always exists between two points, and whether it is unique.
The two most basic examples are the Euclidean spaces
geodesics are straight lines, and the round sphere