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Inversive Distance


The inversive distance is the natural logarithm of the ratio of two concentric circles into which the given circles can be inverted. Let c be the distance between the centers of two nonintersecting circles of radii a and b<a. Then the inversive distance is

 delta=cosh^(-1)|(a^2+b^2-c^2)/(2ab)|
(1)

(Coxeter and Greitzer 1967).

The inversive distance between the Soddy circles is given by

 delta=2cosh^(-1)2,
(2)

and the circumcircle and incircle of a triangle with circumradius R and inradius r are at inversive distance

 delta=2sinh^(-1)(1/2sqrt(r/R))
(3)

(Coxeter and Greitzer 1967, pp. 130-131).


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References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 123-124 and 127-131, 1967.

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Inversive Distance

Cite this as:

Weisstein, Eric W. "Inversive Distance." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InversiveDistance.html

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