inversive geometry Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all th ...

, the inversive distance is a way of measuring the "

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

" between two

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

s, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other.

Properties

The inversive distance remains unchanged if the circles are inverted, or transformed by a

Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...

. 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 In geometry, the Beckman–Quarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all ...

holds true for the inversive distance: if a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

of the set of circles in the inversive plane preserves the inversive distance between pairs of circles at some chosen fixed distance

\delta

, then it must be a Möbius transformation that preserves all inversive distances..

Distance formula

For two circles in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

with radii

r

and

R

, and distance

d

between their centers, the inversive distance can be defined by the formula. :

I=\frac.

This formula gives: *a value greater than 1 for two disjoint circles, *a value of 1 for two circles that are tangent to each other and both outside each other, *a value between −1 and 1 for two circles that intersect, **a value of 0 for two circles that intersect each other at

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s , *a value of −1 for two circles that are tangent to each other, one inside of the other, *and a value less than −1 when one circle contains the other. (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 In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The ...

of the value given above, rather than the value itself. That is, rather than using the number

I

as the inversive distance, the distance is instead defined as the number

\delta

obeying the equation :

\delta=\operatorname( I).

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 In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. Although the definiti ...

. That is, if three circles belong to a common pencil, then (using

\delta

in place of

I

as the inversive distance) one of their three pairwise distances will be the sum of the other two..

In other geometries

It is also possible to define the inversive distance for circles on a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, or for circles in the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

Applications

Steiner chains

Steiner chain In geometry, a Steiner chain is a set of circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where is finite and each circle in the chain is tangent to the previous and next circles in the 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

whose numerator is the number of circles in the chain and whose denominator is the number of times it wraps around. All chains for the same two circles have the same value of

p

. If the inversive distance between the two circles (after taking the inverse hyperbolic cosine) is

\delta

, then

p

can be found by the formula :

p=\frac.

Conversely, every two disjoint circles for which this formula gives a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

will support a Steiner chain. More generally, an arbitrary pair of disjoint circles can be approximated arbitrarily closely by pairs of circles that support Steiner chains whose

p

values are rational approximations to the value of this formula for the given two circles.

Circle packings

The inversive distance has been used to define the concept of an inversive-distance

circle packing In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated '' packing de ...

: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...

) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the

circle packing theorem The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in gen ...

, in which specified pairs of circles are tangent to each other. 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 In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cros ...

and set of Euclidean or hyperbolic inversive distances. This rigidity property can be generalized broadly, to Euclidean or hyperbolic metrics on triangulated

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s with

angular defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the defe ...

s at their vertices. However, for manifolds with spherical geometry, these packings are no longer unique.. In turn, inversive-distance circle packings have been used to construct approximations to

conformal mapping In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...

References

External links

*{{mathworld, title=Inversive Distance, urlname=InversiveDistance Inversive geometry