Inversive distance

Inversive distance (usually denoted as δ) is a way of measuring the "distance" between two non-intersecting circles α and β. If α and β are inverted with respect to a circle centered at one of the limiting points of the pencil of α and β, then α and β will invert into concentric circles. If those concentric circles have radii a' and b', then the inversive distance is defined as

(\alpha,\beta) = \left| \ln \frac{a'}{b'} \right|.

In addition, if a and b are the radii of α and β (as opposed to their images), and c is the distance between their centers, then the inversive distance δ is given by

\cosh\delta = \left| \frac{a^2 %2B b^2 - c^2}{2ab} \right|.

See also

References