Orthogonality principle

2013-05-21T00:54:18Z

108.221.45.116: /* Example */

'''Inversive distance''' (usually denoted as ''δ'') is a way of measuring the "[[distance]]" between two non-intersecting [[circle]]s ''α'' and ''β''. If ''α'' and ''β'' are [[inversive geometry|inverted]] with respect to a circle centered at one of the [[Limiting point (geometry)|limiting points]] of the [[Apollonian circles#Pencils of circles|pencil of ''α'' and ''β'']], then ''α'' and ''β'' will invert into concentric circles. If those concentric circles have radii ''a<nowiki>'</nowiki>'' and ''b<nowiki>'</nowiki>'', then the inversive distance is defined as
:<math>(\alpha,\beta) = \left| \ln \frac{a'}{b'} \right|.</math>

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
:<math>\cosh\delta = \left| \frac{a^2 + b^2 - c^2}{2ab} \right|.</math>

== See also ==
*[[Coaxal circles]]
*[[Inversive geometry]]

== References ==
*{{cite book |title=Geometry Revisited |last=Coxeter |first=H. S. M. |coauthors=S. L. Greitzer |year=1967 |publisher=[[Mathematical Association of America|MAA]] |location=[[Washington, D.C.|Washington]] |isbn=0-88385-619-0 }}

[[Category:Inversive geometry]]

{{elementary-geometry-stub}}

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Orthogonality principle