Topics in Geometry: Geometries from a transformation point of view

We shall begin this course with the discussion of fundamental transformations of Euclidean plane geometry, that is, with motions or distance preserving transformations. One can, in fact, define geometry as the study of those properties of the figures that are not changed by motions. For example, the diagonal of a square with area 2 is of length 2 regardless of the position of the square in the plane. One may, however consider geometry as study of those properties of the figures that are not changed by similarity transformations. This approach (known as the Klein's Erlangen programme) is generalized, by defining two figures to be equivalent relative to a certain group of transformations if they can be mapped onto each other by a transformation from that group. The geometry associated with the group of transformations is then the study of those properties of figures which are preserved under the transformations of the group.

Topics to be covered are:

Geometry of the Euclidean plane (including the classification of isometries and similarities)

Affine geometry (including Ceva's and Menelaus's theorems)

Real projective plane (including a brief study of conics)

Geometry on the sphere

Hyperbolic plane geometry

There will be no text for this course, but Coxeter's Introduction to Geometry will be used as one of the references.