Editing Möbius transformation (section)

=== Preservation of angles and generalized circles ===
From this decomposition, we see that Möbius transformations carry over all non-trivial properties of [[circle inversion]]. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and [[isometries]] (translation, reflection, rotation), which trivially preserve angles.

Furthermore, Möbius transformations map [[generalized circle]]s to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center.