3: Transformations
Transformations will be the focus of this chapter. They are functions first and foremost, often used to push objects from one place in a space to a more convenient place, but transformations do much more. They will be used to define different geometries, and we will think of a transformation in terms of the sorts of objects (and functions) that are unaffected by it.
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- 3.3: The Extended Plane
- Consider again inversion about the circle C given by |z−z_0|=r, and observe that points close to z_0 get mapped to points in the plane far away from z_0. In fact, a sequence of points in complex numbers whose limit is z_0 will be inverted to a sequence of points whose magnitudes go to ∞. Conversely, any sequence of points in complex numbers having magnitudes marching off to ∞ will be inverted to a sequence of points whose limit is z_0.
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- 3.4: Möbius Transformations
- If we compose two Möbius transformations, the result is another Möbius transformation. Since Möbius transformations are composed of inversions, they will embrace the finer qualities of inversions. For instance, since inversion preserves clines, so do Möbius transformations, and since inversion preserves angle magnitudes, Möbius transformations preserve angles (as an even number of inversions).