Circle transformation

Möbius transformation

A transformation mapping circles onto circles. Considered as a point transformation, a Möbius transformation is a mapping of the extended Euclidean plane (i.e. the plane completed by adding a point at infinity), under which a circle or a straight line is mapped onto a circle or a straight line. In such cases one speakes of anallagmatic point geometry.

As a non-point transformation, a Möbius transformation is a particular case of a tangency transformation (or tangency circle transformation, or Lie circle transformation); the basic element is not a point but a circle. In that case one speaks of circular anallagmatic geometry.

References

[1]	P.S. Alexandroff [P.S. Aleksandrov] (ed.) et al. (ed.) , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian)