Inversive distance

Inversive distance (usually denoted as δ) is a way of measuring the "distance" between two circles, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other.

It remains unchanged if the circles are inverted, or transformed by a Möbius transformation. One pair of circles can be transformed to another pair by a Möbius transformation if and only if both pairs have the same inversive distance.

For two circles in the Euclidean plane with radii $r$ and $R$, and distance $d$ between their centers, the inversive distance can be defined by the formula

$I=\left|{\frac {d^{2}-r^{2}-R^{2}}{2rR}}\right|.$

This formula gives a value of 1 for two circles that are tangent to each other, less than 1 for two circles that cross, and greater than one for two disjoint circles.

Some authors modify this formula by taking the hyperbolic cosine of the value given above, rather than the value itself. It is also possible to define the inversive distance for circles on a sphere, or for circles in the hyperbolic plane.

See also

• Coaxal circles
• Steiner's porism