  In this paper, we introduce a new metric $\tilde{c}$ which is associated with
the domain boundary for a Ptolemy space $(X,d)$. Moreover, we study the
inclusion relation of the $\tilde{c}$ metric balls and some related hyperbolic
type metric balls in subdomains of $\mathbb{R}^n$. In addition, we study
distortion properties of M\"obius transformations with respect to the
$\tilde{c}$ metric in the unit ball and the quasiconformality of bilipschitz
mappings in $\tilde{c}$ metric.
