Week 7
Earth-like metric geometry:
Spherical geometry
During the next three weeks, we will study non-Euclidean geometries,
where all but one of Euclid's postulates are valid: We discard
the parallel axiom. This week's topic, spherical geometry, can easily be
motivated by the fact that our home planet is (almost) spherical.
Similarly to the discussion of
plane geometry, I start by fixing the
notions of spherical distance and lines, as well as angles.
Next, we will study spherical trigonometry - investigate
the properties of spherical triangles and find that in spherical trigonometry,
too, collinearity in fact is a metric property.
Finally, we will study spherical motions. Similarly to the Euclidean
case, our discussion will gradually lead from an (almost)
Euclidean point of view
to a Kleinian one.
Helpful literature:
-
M. Reid, Geometry and Topology, chapters of forthcoming
textbook by M. Reid and B. Szendröi, available from General Office:
Sections 4.1-4.6.
- M. Berger,
Geometry I,
Springer:
Section 1.8.
Note:
Only topics covered in the lectures and exercises
will be asked in the exam.
The list of helpful literature is meant as an aid, also to raise further
interest, but not as an obligation.