Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 14

Note for problems 12, 13 on p. 76:
The book doesn't explicitly define Steiner circles of the second kind. So here's an explicit defintion:
Let T be a Mobius transformation that fixes distinct points p and q. Let S be the transformation S(z) = (z-p)/(z-q).  A Steiner circle of the second kind for T is any cline C such that S(C) is a circle centered at the origin.

One of the goals of the following  problems is for you to see whether you read pages 67-70 "actively" or "passively" and how well you understood what you read.

1. On page 69, the covering transformation S^(-1) is used to lift T to a transformation R. Show that this satisfies the definition of lift on p. 31-32. Also, state which letters on page 69 correspond to each of the letters D, R, S, f, g in the definition of lift.
2. Prove that R as defined on p. 69 fixes 0 and infinity.
3. Prove (derive) the equation at the very bottom of p. 69 (fill in details for the argument given in the book).
4. Let T be a Mobius transformation that fixes distinct points p and q. Prove that (Tz, z, p, q) is constant (i.e., independent of z).