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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 preserved 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 az̄ + 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∪{∞} = Ĉ, 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 geometry is the study of those figures in Ĉ that are preserved by these inversive
transformations. We went on to describe some minutia regarding Möbius transformations, and that every Möbius transformation is inversive, and that they therefore
map generalized circles to generalized circles, and preserve magnitude and orientation of angles. We went on to describe the matrix representation of a Möbius
transformation, and how there is a homomorphism from Gl2 (C) onto the group of
all Möbius transformations. . . we went on to describe the kernel of this homomorphism as well. Finally, we proved that all inversive transformations are of the form
M (z) or M (z̄) for some Möbius transformation M .
(4) Section 5.4: The Fundamental Theorem of Inversive Geometry. We discussed the
manner in which there exists a unique Möbius Transformation that takes any 3
points in Ĉ to any other 3 points in Ĉ, and how such a Möbius transformation 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 specifically 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!