Orthogonality principle

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

$(\alpha ,\beta )=|\ln \frac{a^{\prime }}{b^{\prime }}|.$

Alternatively, 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 may be calculated directly by the formula

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

See also

• Coaxal circles
• Inversive geometry

References

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