Inversive distance
= Inversive distance
{wiki}
For two <spheres> with positive <radii> $R_1,R_2$ and center separation $d$, put
$$
\chi=\frac{d^2-R_1^2-R_2^2}{2R_1R_2}.
$$
Externally disjoint <spheres> have $\chi>1$; strictly nested disjoint <spheres> have $\chi<-1$. For unoriented <spheres>, the invariant under <Möbius transformations> is $|\chi|$, provided both images remain finite <spheres>. The signed expression becomes invariant when the <spheres> carry consistently transported <orientations> and signed <radii>. Some authors use its negative.<Laguerre geometry in space forms and incircular nets by Alexander I. Bobenko and others>[Appendix B.1, Definition B.2 and Proposition B.6, page 66; that source uses the opposite sign]{ref}
The sign issue is visible without any general theory. Take unit <spheres> centered at $0$ and $3e_1$, so $\chi=7/2$. Under $x\mapsto x/|x|^2$, the first remains the unit <sphere>, while the second has center $3e_1/8$ and <radius> $1/8$, inside it. Substitution now gives $\chi=-7/2$. Thus the old positive-radius formula alone cannot be a signed conformal invariant. For disjoint unoriented <spheres>, $\operatorname{arcosh}|\chi|$ is another common distance convention.