In space, the only conformal mappings are inversions, similarity transformations, and congruence transformations. Or, restated, every angle-preserving transformation is a sphere-preserving transformation.
Liouville's Conformality Theorem
See also
Conformal MappingExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Liouville's Conformality Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LiouvillesConformalityTheorem.html