I will make this an answer although it is just a follow-up to the comment of LK

The recent names in this (but referring back to Bonk and Kleiner) are Stephen Buckley and David Herron, for proper spaces their one-point extension $\hat{X}$ is the one-point compactification, see pages 4 and 8 in 

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METRIC SPACE INVERSIONS, QUASIHYPERBOLIC DISTANCE, AND UNIFORM SPACES
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by SM Buckley - 2008 - Cited by 3 - Related articles
In a certain sense, inversion is dual to sphericalization. ... point compactification. All of the properties of dp mentioned above also hold for their con- ... 

https://eprints.nuim.ie/1610/1/BuckleyMetricSpace.pdf