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Let $\phi = \dfrac{1}{r} ∘ T_{-O_C}$, $O_C$ the center of $C$, $T_{-O_C}$ is a translation.

I want to show that the inversion of a circle $C \in \mathbb{C}$ can be written as: $$\iota_C = \phi ∘ \iota ∘ \phi^{-1}$$

I have to show 3 things:

  • $\forall z \in C$ , $\iota_C (z) \in C$

  • $\iota_C$ exchanges the inside and the outside of C

  • $\iota_C$ globally preserves the circles and the lines perpendicular to $C$

The first properties are easy to show but for the third property, I don't see why after a homotheticity of $1/r$, and after a translation, a line $l$ perpendicular to $C$ remains the same

aribaldi
  • 1,400
  • Also it is not clear what is a translation because they don't exist in hyperbolic geometry. – Kii Jun 26 '16 at 08:30
  • @Kii But some property states that any inverstion of a circle C ca be written as: $\iota_C = \Phi  ∘ \iota  ∘ \Phi$ where $\Phi$ is the composition oh a transmation and of a homothecity and where $\iota$ sends z on 1/z – aribaldi Jun 26 '16 at 08:36
  • Are you familiar with https://en.wikipedia.org/wiki/M%C3%B6bius_transformation ? – Kii Jun 26 '16 at 08:40

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