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 inclusion relations of \(\tilde{c}\) metric balls and some related hyperbolic type metric balls in subdomains of \({\mathbb {R}}^n.\) In addition, we study distortion properties of the \(\tilde{c}\) metric under Möbius transformations of the unit ball and quasiconformality of bilipschitz mappings in the \(\tilde{c}\) metric.
