Is there a **standard** construction of a metric on one-point compactification of a proper metric space?

**Comments:**

- A metric space is **proper** if all bounded closed sets are compact. 
- **Standard** means *found in literature*. 

**From the answers  and comments:**

Here is a simplification of the construction given [here][1] (thanks to Jonas for ref).
Let $d$ be the original metric. Fix a point $p$ and set $h(x)=1/(1+d(p,x))$. 
Then take the metric
$$\hat d(x,y)=\min\{d(x,y),\,h(x)+h(y)\},\ \ \ \  \hat d(\infty,x)=h(x).$$

A more complicated construction is given [here][2] (thanks to LK for ref), some call it "sphericalization".
One takes 
$$\bar d(x,y)=d(x,y)\cdot h(x)\cdot h(y),\ \ \ \  \bar d(\infty,x)=h(x).$$
The function $\bar d$ does not satisfies triangle inequality, but one can show that there is a metric $\rho$ such that $\tfrac14\cdot \bar d\le \rho\le \bar d$.


  [1]: http://www.jstor.org/pss/2047675
  [2]: http://front.math.ucdavis.edu/0006.5137