In inversive geometry, the inversive distance is a way of measuring the "distance" between two circles, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other.
The inversive distance remains unchanged if the circles are inverted, or transformed by a Möbius transformation. One pair of circles can be transformed to another pair by a Möbius transformation if and only if both pairs have the same inversive distance.
An analogue of the Beckman–Quarles theorem holds true for the inversive distance: if a bijection of the set of circles in the inversive plane preserves the inversive distance between pairs of circles at some chosen fixed distance
\delta
For two circles in the Euclidean plane with radii
r
R
d
I= | d2-r2-R2 |
2rR |
.
(Some authors define the absolute inversive distance as the absolute value of the inversive distance.)
Some authors modify this formula by taking the inverse hyperbolic cosine of the value given above, rather than the value itself. That is, rather than using the number
I
\delta
\delta=\operatorname{arcosh}(I).
\delta
I
It is also possible to define the inversive distance for circles on a sphere, or for circles in the hyperbolic plane.[2]
A Steiner chain for two disjoint circles is a finite cyclic sequence of additional circles, each of which is tangent to the two given circles and to its two neighbors in the chain.Steiner's porism states that if two circles have a Steiner chain, they have infinitely many such chains.The chain is allowed to wrap more than once around the two circles, and can be characterized by a rational number
p
p
\delta
p
p= | \pi |
\sin-1\tanh(\delta/2) |
.
p
The inversive distance has been used to define the concept of an inversive-distance circle packing: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a planar graph) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the circle packing theorem, in which specified pairs of circles are tangent to each other.[2] [4] Although less is known about the existence of inversive distance circle packings than for tangent circle packings, it is known that, when they exist, they can be uniquely specified (up to Möbius transformations) by a given maximal planar graph and set of Euclidean or hyperbolic inversive distances. This rigidity property can be generalized broadly, to Euclidean or hyperbolic metrics on triangulated manifolds with angular defects at their vertices.[5] However, for manifolds with spherical geometry, these packings are no longer unique.[6] In turn, inversive-distance circle packings have been used to construct approximations to conformal mappings.[2]