Circles and Möbius Transformations
If is a circle and , then is also a circle.
Theorem 5.12 (Circles and Möbius Transformations)
Möbius transformations map circles to circles. Formally, if is a circle, then then is also a circle.
Proof
The group is generated by scalings, translations, and inversion. Scalings and translations map circles to circles. For inversion, a Euclidean circle with equation maps to a line when and otherwise to a Euclidean circle after completing the square. The line case is analogous.