Let
Γ
\Gamma
be a cyclic subgroup of
P
S
L
2
(
C
)
PSL_2({\mathbb C})
generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of
Γ
\Gamma
on
H
R
3
{{\mathbb H}^3_{\mathbb R}}
are the complements of configurations of half-balls centered on the plane at infinity
∂
H
R
3
{\partial }{{\mathbb H}^3_{\mathbb R}}
. Jørgensen (On cyclic groups of Möbius transformations, Math. Scand. 33 (1973), 250–260) proved that the boundary of the intersection of the Ford fundamental domain with
∂
H
R
3
{\partial }{{\mathbb H}^3_{\mathbb R}}
always consists of either two, four, or six circular arcs and stated that an arbitrarily large number of hemispheres could contribute faces to the Ford domain in the interior of
H
R
3
{{\mathbb H}^3_{\mathbb R}}
. We give new proofs of Jørgensen’s results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of
H
R
3
{{\mathbb H}^3_{\mathbb R}}
, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.