Inversive distance: Difference between revisions
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'''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 [[ |
'''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 |
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:<math>(\alpha,\beta) = \left| \ln \frac{a'}{b'} \right|.</math> |
:<math>(\alpha,\beta) = \left| \ln \frac{a'}{b'} \right|.</math> |
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Revision as of 04:24, 22 December 2013
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
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
See also
References
- Coxeter, H. S. M. (1967). Geometry Revisited. Washington: MAA. ISBN 0-88385-619-0.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help)
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