Summary of facts about Möbius maps

A Möbius transformation is a transformation of the Riemann sphere defined by one of the two formulas $$\frac{az+b}{cz+d} \qquad\text{or}\qquad \frac{a\bar z+b}{c\bar z+d}$$ where $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ is a $2\times2$ complex matrix of nonzero determinant. They are defined as continuous one-to-one maps of the Riemann sphere onto itself by natural rules of arithmetic applied to the point $\infty$ on the sphere. (See [1] p. 57 and p. 70.)

The former kind of Möbius transformations are angle-preserving, or alternatively conformal, while the latter kind (with $\bar z$) are angle-reversing or anticonformal. All Möbius transformations preserve the size of angles between curves; the difference lies in whether or not they also preserve the sense.

For a given matrix $m=\begin{bmatrix}a & b \\ c& d\end{bmatrix}$, we shall write the corresponding conformal transformation as $m(z)$, and the anticonformal one as $m(\bar z)$. By replacing $m$ by $\begin{bmatrix}ra & rb\\ rc& rd\end{bmatrix}$ for some complex number $r\neq0$, we do not change the corresponding Möbius transformation, and we can arrange that the determinant is 1.