= Hyperbolic extension of a Möbius transformation
A <Möbius transformation> of the ideal two-sphere extends uniquely to an orientation-preserving <isometry> of three-dimensional <hyperbolic space>. In upper half-space, translations, complex similarities, and $z\mapsto-1/z$ extend respectively to $(z,t)\mapsto(z+b,t)$, $(\lambda z,|\lambda|t)$, and $(-\overline z,t)/(|z|^2+t^2)$. Each preserves $(|dz|^2+dt^2)/t^2$; composing these extensions constructs the general case. Ball-origin stabilizers are ordinary rotations, represented on the boundary by $SU(2)$ modulo its central signs.
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