MATH 614 EXAM # 1 REVIEW SHEET

Hi everyone, here is a sheet reviewing some of the important concepts from what we’ve done so far. Enjoy!

(1) Section 5.1: Inversion. We developed the concept of inversion in (or through) a circle. We found that this operation is self-inverse, and that it generalizes reflection over a line. We also found that what we now call generalized circles (circles or lines) get mapped to generalized circles (depending on where each is located relative to the circle of inversion. In addition, we proved that magnitudes of angles are pre- served under inversion, and their orientation possibly reversed. We also developed a formula for inversion through the unit circle, although this formula will be made easier in subsequent sections. (2) Section 5.2: Extending the Plane. We briefly discussed some basic and foundational information about C–in particular, we discussed how the argument of a complex number changes in complex multiplication. We proved that every isometry of C is of the form az + b or az¯ + b, where a, and b are complex, and |a| = 1. A pivotal observation is how to decompose these functions as compositions of reflections. We then discussed the scaling function z 7→ kz for k > 0 (in particular, k is real), and how this, too, is a composition of reflections (inversions). We came up with a formula for inversion in any circle in C as well. The section goes on to describe the manner in which linear and reciprocal functions send generalized circles to generalized circles, and preserve angles. After this, we discussed the extended plane: C∪{∞} = Cˆ, providing a convenience in language concerning what we can now call a generalized circle, and everything’s (the functions we have studied) relationship to ∞. Finally, we discussed the Riemann Sphere, and stereographic projection, and found a formula for this and its inverse. Again: stereographic projection sends generalized circles to generalized circles, and preserves magnitudes of angles. (3) Section 5.3: Inversive Geometry. We defined the group of inversive transformations as those functions that are compositions of (generalized) inversions. Inversive ge- ometry is the study of those figures in Cˆ that are preserved by these inversive transformations. We went on to describe some minutia regarding M¨obiustransfor- mations, and that every M¨obiustransformation is inversive, and that they therefore map generalized circles to generalized circles, and preserve magnitude and orien- tation of angles. We went on to describe the matrix representation of a M¨obius transformation, and how there is a homomorphism from Gl2(C) onto the group of all M¨obiustransformations. . . we went on to describe the kernel of this homomor- phism as well. Finally, we proved that all inversive transformations are of the form M(z) or M(¯z) for some M¨obiustransformation M. 1 2 MATH 614 EXAM # 1 REVIEW SHEET

(4) Section 5.4: The Fundamental Theorem of Inversive Geometry. We discussed the manner in which there exists a unique M¨obiusTransformation that takes any 3 points in Cˆ to any other 3 points in Cˆ, and how such a M¨obiustransformation is unique. We closed the section with a discussion as to the manner in which one would actually construct such a function. (5) General suggestions: I suggest that you study for this exam, and that you specif- ically study the homework questions and issues we’ve gone over in class, as the exam will likely feature a lot of this material. Good luck!