Let
D
D
be a proper subdomain of
R
n
{R^n}
and
k
D
{k_D}
the quasihyperbolic metric defined by the conformal metric tensor
d
s
¯
2
=
dist
(
x
,
∂
D
)
−
2
d
s
2
d{\overline s ^2} = \operatorname {dist} {(x,\partial D)^{ - 2}}d{s^2}
. The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for
k
D
{k_D}
; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in
R
n
{R^n}
, in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.