mg.metric geometry - Metric on one-point compactification

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 (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 (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 $tfrac14cdot bar dle rhole bar d$.

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