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For a nonempty finite set $\{p_1,\dots ,p_m\}$ in finite-dimensional hyperbolic space, the continuous function $R(x)={\max }_{j}d(x,p_j)$ attains its minimum. Indeed, a bounded candidate radius confines all possible improving centers to a closed bounded ball about $p_1$. Such a ball is compact in the Poincare ball model, since its points remain away from the ideal boundary. Minimizing $R$ on it supplies a closed ball of smallest enclosing radius. More generally this argument works for any nonempty bounded set in a proper metric space when the supremum-distance function is finite.

On the unit three-ball $B^3$, the distance in the Poincare ball model is the infimum of the lengthsover piecewise smooth paths joining the two points. Equivalently the hyperbolic metric as a metric-space distance isIt has curvature-minus-one normalization; in particular $d(0,x)=2\mathrm{artanh}|x|$.

For a nonempty finite set $p_1,\dots ,p_m$, define $R(x)={\max }_{j}d(x,p_j)$. The triangle inequality gives $|R(x)-R(y)|\le d(x,y)$, so $R$ is continuous. Take $R_0=R(p_1)$. Any improving center satisfies $d(x,p_1)\le R(x)\le R_0$, and thus lies in the closed hyperbolic ball $K=\overline{B}(p_1,R_0)$.

This ball is compact: an isometry sending $p_1$ to the origin identifies it with the Euclidean closed ball of radius $\tanh (R_0/2)<1$, and the Euclidean and hyperbolic topologies agree inside $B^3$. The continuous function $R$ attains a minimum on $K$. Centers outside $K$ have $R>R_0$, so this is also the global minimum. Its associated closed ball proves a minimum enclosing ball in hyperbolic space exists. For the empty finite set, any center and radius zero suffice.

The Poincare ball model has ideal boundary the Riemann sphere. To construct the hyperbolic extension of a Möbius transformation, conjugate the ball to the upper half-space model with coordinates $(z,t)\in \mathbb{C}\times (0,\infty )$ and metric $(|dz{|}^2+d{t}^2)/t^2$. The generators of the Möbius group extend asandTranslations and similarities plainly preserve the metric; Euclidean inversion rescales both the squared Euclidean line element and $t^2$ by the same factor, proving the last case. Composition gives the unique orientation-preserving hyperbolic extension. Equivalently, for a determinant-one matrix its upper half-space formula is

An isometry fixing the ball origin acts on its tangent space by an orientation-preserving orthogonal transformation; radial geodesics then show that the whole map is that ordinary rotation. Conversely every rotation preserves the ball metric. On the sphere these are exactly the projective special-unitary mapsthe image of $SU(2)$ modulo its central signs.

A hyperbolic line is determined by its unordered pair of ideal endpoints. It is invariant precisely when that pair is preserved as a set. A nonidentity parabolic map has one fixed point, and its square also has one, so it cannot fix or interchange two distinct endpoints. Every nonparabolic nonidentity map has two fixed points, and their joining line is invariant.

More precisely, normalize the latter map to $z\mapsto \lambda z$. Unless $\lambda =-1$, any invariant pair must consist of zero and infinity: a swapped pair would be fixed pointwise by the square, whose only fixed points remain zero and infinity when ${\lambda }^2\ne 1$. This proves uniqueness for every nonidentity map except an elliptic half-turn. For $\lambda =-1$, every pair $\{z,-z\}$ is also invariant, giving infinitely many additional lines, besides the fixed-point axis. The identity preserves every line. ThusHere the uniqueness statement includes hyperbolic translations, genuine loxodromic maps and elliptic rotations through angles other than a half-turn.