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*. 

**A possible construction:**

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
$$\hat d(x,y)=\min\{d(x,y),\\,h(x)+h(y)\},\ \ \ \  \hat d(\infty,x)=h(x)$$

 - Is this a *standard* construction? 
 - Does it have a *name*? 


  [1]: http://www.jstor.org/pss/2047675