The number four points agree on

Inversion in a circle is a reflection. It fixes the circle pointwise, it exchanges inside with outside, and it reverses the sense of every angle — an anticlockwise turn becomes a clockwise one. So it is not a motion, and the natural thing to do with a reflection is to compose two of them.

Two inversions compose to a map that preserves orientation, carries circles-and-lines to circles-and-lines, and preserves angles exactly. Those maps are the Möbius transformations, and they are the group this whole ladder has been working inside without naming.

What the composition is #

Write points of the plane as complex numbers. Inversion in the unit circle about the origin is

$$z \mapsto \frac{1}{\bar z},$$

which is the conjugate of $1/z$: the conjugation is the reflection, and $1/z$ is the orientation-preserving part. Writing it that way is what reading a point of the plane as a complex number buys — the geometry of the map becomes the arithmetic of one division. Composing two inversions conjugates twice, so the conjugations cancel and what is left is a composition of maps of the form $z \mapsto (az+b)/(cz+d)$ with $ad - bc \ne 0$.

That is the general Möbius transformation, and every one of them arises this way. The group they form is generated by inversions in circles together with reflections in lines — which is what makes inversion, rather than the formula, the right place to start.

Three facts follow at once and all three are worth stating.

A Möbius map is determined by where it sends three points. Three points and three images is six real conditions on the four complex coefficients, which have three degrees of freedom after scaling; the counts match, and the map is unique.

Every Möbius map has one or two fixed points. Solving $z = (az+b)/(cz+d)$ is a quadratic, so there are two roots or a repeated one. That is a strong constraint: a map of the plane with three fixed points and this form is the identity.

They act on the sphere, not on the plane. The map $z \mapsto 1/z$ sends the origin nowhere, unless a point at infinity is added — and adding exactly one point at infinity turns the plane into a sphere. On the sphere every Möbius map is a bijection with no exceptions, which is the reason the natural home of this group is a sphere rather than a plane.

The invariant #

A group is understood by what it leaves alone, and this one leaves alone a number computed from four points.

For four points $z_1, z_2, z_3, z_4$, the cross-ratio is

$$(z_1, z_2; z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_2 - z_3)(z_1 - z_4)}.$$

It is unchanged by every Möbius transformation. The proof is a computation and it is short: each factor of the form $z - w$ picks up the same denominator under the map, and the four denominators cancel in pairs.

What the figure measures is the sharper statement, which distinguishes the group from the larger one containing the reflections. A single inversion conjugates the cross-ratio, and two inversions therefore restore it exactly. The imaginary part flips sign and flips back — visible in the numbers under the figure rather than argued.

Distances do not survive. Angles do, and so does the cross-ratio, and the two facts are related: an angle is a ratio and a length is not, while the cross-ratio is built entirely from ratios of differences, and a ratio of differences is the largest thing that can survive a map which multiplies distances by different factors in different places.

What being real means #

The cross-ratio is a complex number, and its being real is a geometric condition.

Four points lie on one circle or one line exactly when their cross-ratio is real.

That is the criterion, and it is checked in the figure on four points chosen on a circle — not the four that are drawn, which are deliberately not concyclic so that the conjugation has something to do.

The reason is one line once the invariance is available. Any three points lie on a unique circle; a Möbius map carries that circle to a line, since some Möbius map sends any given point to infinity. So the question is the fourth point on the circle through the other three becomes is the fourth image on the line through the other three, and four points on a line have a real cross-ratio because every difference is real.

The classification #

A Möbius map has one or two fixed points, and what it does between them is decided by a single number.

Move the fixed points to $0$ and $\infty$ by another Möbius map — always possible, and the whole point of having a group — and the map becomes $z \mapsto \lambda z$ for some complex $\lambda$. Four cases, and every Möbius map is one of them.

The number $\lambda$ is called the multiplier, and it is itself an invariant of the map: it does not depend on which Möbius map was used to move the fixed points, because two such maps differ by something fixing $0$ and $\infty$, which multiplies $\lambda$ by nothing at all.

Elliptic, when $|\lambda| = 1$ and $\lambda \ne 1$. The map is a rotation about the two fixed points; orbits are circles. A composition of two inversions in circles that cross is of this kind, and the angle of rotation is twice the angle at which they cross.

Hyperbolic, when $\lambda$ is real and positive. Points flow from one fixed point to the other along circular arcs. Two inversions in disjoint circles compose to this.

Loxodromic, when $\lambda$ is neither. Orbits spiral from one fixed point to the other.

Parabolic, the degenerate case with one fixed point. Two inversions in circles that are tangent compose to this, and orbits run along circles all tangent at the fixed point.

So the classification of the maps is the classification of pairs of circles — crossing, disjoint or tangent — which is a satisfying place for a ladder about inversion to arrive at. The map is two mirrors, and the type of the map is the relationship between the mirrors. It is the same accounting that makes two reflections in the plane a rotation when the lines meet and a translation when they are parallel — and the same accounting that classifies the symmetries of a solid, where a composition of reflections is named by how the mirrors sit.

Three points, and the coordinate they fix #

The statement that a Möbius map is determined by three points has a use that is worth spelling out, because it is how the cross-ratio is found rather than merely verified.

Given any three distinct points, there is exactly one Möbius map sending them to $0$, $1$ and $\infty$. Apply it to a fourth point and the image is a number — and that number is the cross-ratio of the four, in the order the three were listed.

So the cross-ratio is not an ad hoc formula. It is the coordinate that three points impose on the rest of the sphere, and its invariance is then immediate: two configurations related by a Möbius map impose the same coordinate, because the maps to $0, 1, \infty$ differ by exactly that map.

Read that way, the criterion of the previous section is also immediate. The three chosen points lie on a unique circle, and the map sends that circle to the line through $0$, $1$ and $\infty$ — which is the real axis. The fourth point is on the circle exactly when its image is on the real axis, which is exactly when the cross-ratio is real. The whole argument is one sentence once the coordinate is in place.

That is a common shape and worth recognising: an invariant is often best defined as the coordinate a normalisation produces, and its invariance is then a triviality rather than a computation. Deriving it from the formula, as the section above did, checks it; deriving it from the normalisation explains it.

What the group does not do #

Two things a reader will expect and should not.

It does not preserve centres. The image of a circle is a circle, and the image of its centre is not the image circle’s centre. That is the first thing everybody gets wrong about inversion and it survives into the composition.

It does not preserve betweenness. Three points on a line have a middle one; their images under a Möbius map lie on a circle, where “middle” has no meaning at all. Any statement of the form this point lies between those two is destroyed by the group, and every argument in this ladder has had to avoid making one.

It does not preserve size or shape in any usual sense. A Möbius map can take a small circle to an enormous one and a nearly straight arc to a nearly closed one. What it preserves is angles at points, which is a purely local statement — the map is conformal, and conformality says nothing at all about how a picture looks at any scale larger than a point.

The right way to hold both is that a Möbius map is a rigid motion of the sphere’s conformal structure and of nothing else. On the sphere the group is exactly the set of conformal bijections, which is a theorem and a good one — the same statement that makes a map of the globe unable to be both angle-true and area-true a fact about surfaces rather than about cartography: there is nothing else, and the six-dimensional group of Möbius maps is the whole symmetry available.

Where the group turns up #

Three places, and they are worth collecting because the group looks like a curiosity until they are put side by side.

Non-Euclidean geometry. Restrict the Möbius maps to those carrying the unit disc to itself, and what is left is the isometry group of the hyperbolic plane in its disc model — where inversion in a circle orthogonal to the boundary is a reflection, and a hyperbolic line is an arc of such a circle. A model in which the parallel postulate fails is built out of exactly this map.

Complex analysis. The Möbius maps are the conformal bijections of the sphere to itself, and they are therefore the only changes of coordinate a complex analyst may make without changing what is being studied. Every classification of anything on the sphere is stated up to this group for that reason.

Numbers. The maps with whole-number coefficients and determinant one form a group acting on the upper half-plane, and its orbits are where continued fractions and the mediant construction live. That connection is not an analogy: the same matrices act in both places.

Where this leaves the ladder #

The rung above is Koebe’s theorem: every triangulation of a sphere is the tangency pattern of a circle packing, and the packing is unique up to Möbius transformations.

That qualification is why this rung had to come first. A statement of the form “unique up to a group” is worth nothing until the group is understood, and the group here is exactly the one two inversions generate. A circle packing is as canonical as a picture of a graph can be, and the freedom that remains is six real parameters — three complex coefficients up to scale — which is precisely the freedom used throughout this ladder to move a configuration somewhere convenient.

What the pictures cannot show #

The four points are not concyclic, deliberately. Four points on a circle have a real cross-ratio, and the conjugation an inversion applies to it would then be invisible. The concyclic criterion is checked separately, on points that are not drawn.

The multiplier is not computed anywhere. The classification’s four cases are distinguished by a complex number attached to a map, and no figure on this page extracts one; the ring figure illustrates the elliptic case and reports the closure rather than the angle.

The classification is described and not drawn. Four types of orbit would be four figures, and each would show the orbit of a single starting point rather than the map.

No figure shows the group. A group of maps is not a thing that can be drawn; what can be drawn is one map’s effect on one configuration, which is what all three panels of the first figure do.

The cross-ratio depends on the order of the four points. Permuting them gives six values in general, related by $\lambda \mapsto 1/\lambda$ and $\lambda \mapsto 1-\lambda$; the figure computes one of the six, in a fixed order, and being real is the one property all six share or none does.

And the sphere is absent. Every claim here is cleanest on the sphere and every figure is in the plane, with the point at infinity present in the arithmetic and nowhere in the picture.

What is worth carrying away #

A transformation is characterised by what it leaves alone, and the useful invariant is usually not the one that comes to mind first.

Inversion preserves no distance, no area, no straightness and no centre. It preserves angle and the class of circles-and-lines, which sounds like very little — and out of that little comes a number, the cross-ratio, that four points carry with them through every map in the group. It is worth comparing with what the two rungs below extracted from the same map: a count of eight and an equation between four curvatures. All three are consequences of the same two preserved things, and each is found by asking the same question — what survives? The invariant was found by asking what could possibly survive, and the answer was a ratio of ratios, because a ratio of ratios is what remains when everything is scaled by different amounts in different places.