Week 39

This week we have covered 2.5, 2.6 and most of 2.7.  In 2.5 the important result is that there exists an essentially unique distance function characterizing betweenness and congruence of segments, and it can be expressed in terms of the cross-ratio.  The message in the short 2.6 is that the Euclidean angle measure does the same thing for angles, and hence can be used as angle measure also in the upper halfplane model for hyperbolic geometry.  Chapter 2.7 introduces the more symmetric Poincaré disk model, and all the geometry is transported from the upper half plane by a fractional linear transformation identifying them.  We got as far as transferring the cross-ratio definition of  the metric to D, and next time we will use it to deduce explicit formulas for metrics both in D and H.

Note: there is an annoying misprint in the expression for the compex number  ß' on page 51.  It should be c-b+(a+d)i, and not c-b-(a+d)i.

Problems for Tuesday 30.9:  2.5.2, 2.6.1, 2.6.2, 2.7.1, 2.7.3.  You should also familiarize yourself with the formulas in 2.7.8.