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
[PDF] METRIC SPACE INVERSIONS, QUASIHYPERBOLIC DISTANCE, AND UNIFORM SPACES File Format: PDF/Adobe Acrobat - Quick View 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- ...
