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''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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Alexandroff [P.S. Aleksandrov] (ed.)  et al. (ed.) , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1969)  (Translated from Russian)</TD></TR></table>