Poincare ball model ($d{s}^2=4|dx{|}^2/(1-|x{|}^2{)}^2$)

The unit ball in Euclidean space with the displayed Riemannian metric models hyperbolic space of curvature minus one. Its ideal boundary is the unit sphere, and its geodesics are diameters or circle arcs orthogonal to that sphere. The metric is invariant under the conformal ball automorphisms.

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 $(|dz{|}^2+d{t}^2)/t^2$; composing these extensions constructs the general case. Ball-origin stabilizers are ordinary rotations, represented on the boundary by $SU(2)$ modulo its central signs.