Weeks 9 and 10
Laws of perspective:
Projective geometry
The introduction
of projective space can be motivated from the laws of perspective
in photograph and art. Our definition of projective space will give a
mathematical description of those laws. I will explain the notions of
dimension, projective subspace, and in particular lines in projective space,
and derive a formula for the dimension of intersections of projecitve
linear subspaces.
We will study projective transformations and their relation to specific
coordinate choices on projective space, that is to projective frames
of references. Projectivities are understood as special projective
transformations.
The discussion of the cross ratio will show how projective
space is different from the metric geometries we have investigated so far.
I will explain and prove two important theorems of projective geometry:
Desargues' and Pappus' theorem. To end the course, I will finally
discuss some notions of
axiomatic projective geometry.
Helpful literature:
-
M. Reid, Geometry and Topology, chapters of forthcoming
textbook by M. Reid and B. Szendröi, available from General Office:
Section 5.