Messages - Page 4

We finished section 2.4, which contains a verification that the upper half-plane with H-lines and congruence defined by M?bius transformations is a geometry satisfying Hilbert's axioms.  We will now study this geometry further and develop concepts like distance and angle measure and trigonometric identities relating them, as well as arc length and area.

The theme this week has been the study of geometric properties of M?bius transformations.  The goal is to obtain a small number of "normal forms" which makes the geometry transparent and represent all M?bius transformations up to conjugacy.  This will be concluded next time, and then we are ready to prove that we have an example of Hyperbolic geometry.

Note that there is a misprint at the end of the second paragraph, page 34: the appropriate interval sould be (0,¦°/2], and not (0,¦°/2).

Exercises for Sep.16: 2.3.1, 4, 5, 6

This week we have discussed general properties of M?bius transformations and relations with the cross-ratio of four points. We finished section 2.2 and will start on the classification of real M?bius transformations (2.3) Tuesday.

Exercises for Tuesday 9.9: 2.2.7, 2.2.9, 2.2.10.

We have now finished the introductory chapter on axiomatic geometry and started om chapter 2, where we construct and analyze an important model for the hyperbolic plane: the Poincar¨¦ upper halfplane. The points and lines here are in exact correspondence with points and chords in an open disk in R². The correspondence goes via orthogonal and spherical projections, so we had to study the latter in some detail.  Next we want to define a "group of congruences" for this geometry, and the idea is to find a suitable group of transformations of the extended complex plane using complex function theory, and then restrict to those preserving the upper half plane.

Exercises for September 2: 2.1.2 and 2.1.4.

I plan to write a brief progress report every week, and here is the first one.  In the beginning we study Hilbert's set of axioms for plane geometry.  This will not play a big part later in the course, and we don't develop it very far, but I believe that anybody who studies geometry should have at least some understanding of how things can be built up axiomatically. 

We came as far as to present briefly the congruence axioms, but today I also went through a proof of some important properties of geometries with only incidence and betweenness.  Detailed notes can be found under the link "Solutions" to the left on this page.

I plan to finish this introduction and, hopefully, start on Hyperbolic geometry next time.

An important change: starting August 26, the Tuesday lectures will be moved to Room 108 in NHA.