Möbius scaling generated by translations and reciprocal inversion
= Möbius scaling generated by translations and reciprocal inversion
{c}
{title2=$T_s I T_{-1/s} I T_s I(z)=-s^2z$}
Let $T_b(z)=z+b$ and $I(z)=1/z$ on the <Riemann sphere>. For $s\ne0$, direct composition gives the displayed identity. Every nonzero complex scaling $z\mapsto\lambda z$ is obtained by choosing $s^2=-\lambda$. Since translations, scalings and reciprocal inversion generate the <Möbius group>, translations and reciprocal inversion alone also generate it.