We study nearest and farthest Voronoi diagrams of lines in three-dimensional Euclidean space, which partition space according to the closest and farthest input lines, respectively. For arbitrary lines, a gap remains between the known quadratic lower bound and near-cubic upper bounds for the complexity of the nearest diagram. We consider input lines chosen from one of two continuous families of straight lines on a smooth surface defined by a quadratic equation, with one line from each family passing through every point of the surface. Under the geometric assumptions stated in the paper, we prove quadratic upper bounds on the total number of vertices, edges, faces, and connected three-dimensional regions in both diagrams. We also construct quadratically many nearest vertices using lines from a single family on one fixed surface, establishing a tight quadratic worst-case bound. The farthest diagram has quadratic total complexity for every input satisfying our assumptions. Our proof uses a parametrization that arranges the input lines in a cyclic order and expresses tangency to a sphere by a polynomial equation of degree four. At a Voronoi vertex, the four tangent input lines give four distinct roots. The sign changes between these roots, together with the nearest or farthest condition, force two intervening parameter intervals to contain no other input parameters. Thus, only quadratically many choices of four lines can define a vertex, and each choice yields at most eight isolated candidate centers.
We study coresets for the continuous $k$-center problem in fixed-dimensional hyperbolic space $\mathbb H^D$, where centers may lie anywhere in the space. For an input set $P$ of $n$ points, we construct an unweighted coreset of size $(k/\varepsilon)^{O(kD)}$ in $O(nkD(k/\varepsilon)^{O(kD)})$ time, independent of the enclosing radius. Every exact optimal solution for the coreset is a $(1+\varepsilon)$-approximation for $P$. The same construction also yields a strong coreset after adjusting the accuracy parameter. Our method combines local Euclidean grids at bounded scales with radial shells at larger scales. Hyperbolic balls induce spherical caps on representative spheres, whose multiplicative enlargement can be controlled by additive hyperbolic distance errors. A recursive ball-coverage certificate then selects a radius-independent set of input witnesses. We establish lower bounds of $Ω_D(k\varepsilon^{-D})$ for $k\ge 8D$ and sufficiently small $\varepsilon$, and $2^{Ω(k)}$ for $D\ge2$ when $\varepsilon=Θ(1/k)$, ruling out bounds polynomial jointly in $k$ and $1/\varepsilon$. For $k$ sufficiently separated clusters, we obtain simultaneous additive and multiplicative weak coreset guarantees, with optimal worst-case size $Θ_D(k\varepsilon^{-(D-1)/2})$ when both errors equal $\varepsilon$. In contrast, even under the same separation condition, no radius-independent additive weak coreset bound exists in the Euclidean plane for $k=2$.