# Distance function in an asymptotically hyperbolic space

The distance function in the Poincare disk model of hyperbolic space is given by

$$d(p, q) = \cosh\left(1 + 2\frac{\|p - q\|^2}{(1- \|p\|^2)(1-\|q\|^2)}\right)$$

where $$p, q$$ are two points in the Poincare disk and $$\|\cdot\|$$ denotes the Euclidean norm.

What is the distance function for a asymptotically hyperbolic space? While I can find expressions for the metric, I can't find an explicit distance function.