  We show that in the Klein projective ball model of hyperbolic space, the
hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power
diagram, requiring however algebraic arithmetic. By considering the
lesser-known Beltrami hemisphere model of hyperbolic geometry, we overcome the
arithmetic limitations of Klein construction. Finally, we characterize the
bisectors and geodesics in the other Poincar\' e upper half-space, the
Poincar\'e ball, and the Lorentz hyperboloid models, and discusses on
degenerate cases for which the dual hyperbolic Delaunay complex is not a
triangulation.
