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.
[1] 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
, then it must be a Möbius transformation that preserves all inversive distances.
obeying the equation Although transforming the inversive distance in this way makes the distance formula more complicated, and prevents its application to crossing pairs of circles, it has the advantage that (like the usual distance for points on a line) the distance becomes additive for circles in a pencil of circles.
The chain is allowed to wrap more than once around the two circles, and can be characterized by a rational number
More generally, an arbitrary pair of disjoint circles can be approximated arbitrarily closely by pairs of circles that support Steiner chains whose
[2] 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.
[1][6] 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.
[7] However, for manifolds with spherical geometry, these packings are no longer unique.
[8] In turn, inversive-distance circle packings have been used to construct approximations to conformal mappings.