Editing Inversive geometry (section)

== Anticonformal mapping property ==
The circle inversion map is anticonformal, which means that at every point it preserves angles and reverses orientation  (a map is called [[conformal map|conformal]] if it preserves ''oriented'' angles). Algebraically, a map is anticonformal if at every point the [[Jacobian matrix and determinant|Jacobian]] is a scalar times an [[orthogonal matrix]] with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point. This means that if ''J'' is the Jacobian, then <math>J \cdot J^T = k I</math> and <math>\det(J) = -\sqrt{k}.</math> Computing the Jacobian in the case {{nowrap|1=''z''<sub>''i''</sub> = ''x''<sub>''i''</sub>/{{norm|'''x'''}}<sup>2</sup>}}, where {{nowrap|1={{norm|'''x'''}}<sup>2</sup> = ''x''<sub>1</sub><sup>2</sup> + ... + ''x''<sub>''n''</sub><sup>2</sup>}} gives {{nowrap|1=''JJ''<sup>T</sup> = ''kI''}}, with {{nowrap|1=''k'' = 1/{{norm|'''x'''}}<sup>4n</sup>}}, and additionally det(''J'') is negative; hence the inversive map is anticonformal.

In the complex plane, the most obvious circle inversion map  (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking ''z'' to 1/''z''. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal.
In this case a [[homography]] is conformal while an [[anti-homography]] is anticonformal.