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3: Transformations

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    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.

    • 3.1: Basic Transformations of Complex Numbers
      In this section, we develop the following basic transformations of the plane, as well as some of their important features.
    • 3.2: Inversion
      Inversion offers a way to reflect points across a circle. This transformation plays a central role in visualizing the transformations of non-Euclidean geometry, and this section is the foundation of much of what follows.
    • 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.
    • 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).
    • 3.5: Möbius Transformations: A Closer Look
      To visualize Möbius transformations it is helpful to focus on fixed points and, in the case of two fixed points, on two families of clines with respect to these points.

    This page titled 3: Transformations is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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