This is a study of the generalized point pair function. Given a domain \(G\subseteq {\mathbb R}^n\), the Euclidean distance to the boundary of \(G\) is denoted, for all points \(x\in G\), by \(d_G(x) = \inf_{z\in\partial G} |x-z|\). The generalized point pair function \(p_{G}^{\alpha}: G\times G \rightarrow [0,1)\) is defined by \[p_{G}^{\alpha}(x,y)=\frac{|x-y|}{\sqrt{|x-y|^2+\alpha d_G(x)d_G(y)}}\] The particular case in which \(\alpha = 4\) has been already studied in [J. Chen et al., Ann. Acad. Sci. Fenn., Math. 40, No. 2, 683–709 (2015; Zbl 1374.30069)].
It is shown that \(p_{G}^{\alpha}\) is a quasi-metric for all values of \(\alpha > 0\) and this metric is compared to several established hyperbolic-type metrics, such as the \(j^*\)-metric, the triangular ratio metric, and the hyperbolic metric. Most of the inequalities between \(p_{G}^{\alpha}\) and these other metrics have the best possible constants in terms of \(\alpha\). Another topic of research pursued in this paper is the distortion of \(p_{G}^{\alpha}\) under conformal and quasiregular mappings.