  The main purpose of this paper is to investigate the properties of a mapping
which is required to be roughly bilipschitz with respect to the Apollonian
metric (roughly Apollonian bilipschitz) of its domain. We prove that under
these mappings the uniformity, $\varphi$-uniformity and $\delta$-hyperbolicity
(in the sense of Gromov with respect to quasihyperbolic metric) of proper
domains of $\mathbb{R}^n$ are invariant. As applications, we give four
equivalent conditions for a quasiconformal mapping which is defined on a
uniform domain to be roughly Apollonian bilipschitz, and we conclude that
$\varphi$-uniformity is invariant under quasim\"obius mappings.
