 Inversive distance

For two spheres with positive radii $R_1,R_2$ and center separation $d$, putExternally 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.^[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, $\mathrm{arcosh}|\chi |$ is another common distance convention.

Inversive distance is a mathematical concept used primarily in the fields of geometry and complex analysis. It is often employed in the context of circles or spherical geometry and is defined in relation to circles. The inversive distance between two circles is defined as the reciprocal of the distance between their respective centers, adjusted for the radii of the circles.