Conformal and Mobius Geometry

One more rung down the ladder

You have spent this whole rung learning Klein's one big idea: a geometry is whatever a transformation group leaves unchanged. You watched the hierarchy descend — projective on top seeing almost nothing but incidence and cross-ratio, then affine adding parallelism and ratios along a line, then similarity adding angle and shape, then rigid Euclidean geometry at the bottom guarding every distance. Each step down the ladder uses a smaller group and so keeps more invariants. This guide takes one last fork.

Here is the new question. The similarity group keeps two things at once: it keeps angles, and it keeps straightness (lines stay lines). What if we are greedy about one and generous about the other — insist on keeping angles, but agree to let go of straightness completely? A line would no longer have to stay a line. That single trade defines conformal geometry: the geometry whose transformations preserve angles but are allowed to bend the plane.

Inversion: the move that bends lines into circles

To build this geometry we need one genuinely new transformation — one no rung above could perform. It is inversion in a circle. Fix a circle of radius r centered at O. Inversion sends a point P to the point P' on ray OP such that the product of distances satisfies |OP| times |OP'| = r^2. Points just inside the circle fly far out; points far out come in close; points on the circle stay put. It turns the plane inside-out through the circle, like a reflection but across a curved mirror.

Inversion does two startling things. First, it is conformal: it preserves every angle (though it flips orientation, just like an ordinary mirror). Second, and this is the heart of the whole subject, it turns lines and circles into lines and circles — but it freely mixes the two. A line not through O inverts into a circle through O; a circle through O inverts into a line. Suddenly the rigid distinction between 'straight' and 'round' dissolves. From the inside of this geometry, a line is just a special kind of circle.

Mobius transformations: the group that runs the show

Now we name the group. Compose two inversions, or an inversion with a reflection, or stack several together, and you generate a rich family of maps of the inversive plane. The orientation-preserving ones — the ones built from an even number of inversions, so the flips cancel — are the celebrated Mobius transformations. They are the conformal symmetries of the inversive plane, and they form a group exactly the way Klein demands. This group sits one notch above similarity on the ladder: every similarity is a Mobius transformation, but most Mobius transformations are not similarities.

There is a beautifully compact way to write them, using complex numbers to label the points of the plane. If we call a point z, every Mobius transformation has the form below, where a, b, c, d are constants with ad - bc not zero. The condition ad - bc not zero is just the guarantee that the map is reversible — it does not collapse the plane.

f(z) = (a z + b) / (c z + d),   with   a d - b c != 0

  translation   z + b           (c = 0)
  scaling+turn  a z             (b = c = 0)
  inversion-ish 1 / z           (the genuinely new ingredient)

Every Mobius map is a product of these.  z = infinity is
an honest value: f sends z = infinity to a/c, and z = -d/c to infinity.

Read that formula against the ladder and the whole rung clicks into place. Set c = 0 and you are back to a similarity (z maps to a constant times z, plus a shift) — the rung above. The only genuinely new piece is the 1/z, the inversion, the one move that can curl a straight line into a circle. So conformal/Mobius geometry is precisely similarity geometry plus the single new power of inversion, and that one extra power costs you straightness: in this geometry 'is a straight line' is no longer a meaningful question to ask.

What survives, and the one number that measures everything

So what are the invariants of this new geometry — what can a Mobius-eyed observer still measure? Klein's discipline says: read them off the group. Distance is gone (inversion stretches it without mercy). Straightness is gone. But angles survive, because every map in the group is conformal — angle is the headline invariant of conformal geometry. And the class of objects called 'circles' (lines now included) is preserved as a whole: a circle always maps to a circle. An observer here can still ask 'do these two curves meet at a right angle?' and 'is this a circle?', but never 'how long is this?' or 'is this straight?'.

There is one more invariant, and it is the jewel of the subject: the cross-ratio of four points. You met it on the projective rung as the one number projection cannot destroy; here it returns in conformal dress. Given any four points, their cross-ratio is a single number, and every Mobius transformation leaves it unchanged. It is the conformal world's ruler — not of length, but of a subtler, projective-flavoured relationship four points have with one another. Crucially, four points are concyclic (lie on one circle, or one line) exactly when their cross-ratio is a real number; that is how the geometry detects its own circles.

1. Want to send three given points A, B, C to three target points anywhere you like? A Mobius transformation can always do it, and there is exactly one that does — three points pin the map down completely.
2. The standard trick: build the map that sends A, B, C to the easy targets 0, 1, infinity — write it straight from the cross-ratio formula, since the cross-ratio of (z, A, B, C) is the very value f(z).
3. Compose two such maps and you can drag any three points to any other three — which is why a Mobius map can carry any circle to any circle, and even the whole disk to the whole upper half-plane.

Why this matters, honestly

This is not an idle curiosity at the end of the ladder; it is a hinge between worlds. The two famous models of hyperbolic geometry you met earlier — the Poincare disk and the upper half-plane — are conformal models precisely because their straight lines are the boundary-perpendicular circular arcs that inversion knows how to draw, and their isometries (the rigid motions of the hyperbolic world) are exactly the Mobius transformations that fix the disk or the half-plane. The Erlangen viewpoint reveals that hyperbolic geometry is, at heart, a piece of Mobius geometry. Two rungs you thought were separate turn out to be one machine.

Let me be honest about two things, in the spirit of this whole rung. First, 'conformal' is a strictly local promise about angles at each point; it says nothing kind about distances, which inversion can distort enormously, so do not picture a Mobius map as a gentle rigid shuffle — picture it as a rubber sheet that keeps every tiny crossing angle while stretching the large-scale picture savagely. Second, the tidy story that 'a Mobius map sends circles to circles' is true only once you have adopted the inversive-plane convention that lines count as circles and infinity counts as a point; on the bare ordinary plane the statement genuinely has exceptions, and pretending otherwise would be a falsehood for the sake of neatness.

And so the ladder closes. Klein's 1872 program handed you a single, sober question to ask of any geometry — *what is the group, and what does it preserve?* Run it from the top and you get a guided tour: projective keeps incidence and cross-ratio; affine adds parallels; similarity adds angle and shape; Euclidean adds distance. Step off the main line and you find conformal/Mobius geometry, where you trade away straightness to gain the freedom of inversion, and angle becomes king. Each geometry is true to its own group — none more 'real' than another. That is the whole, liberating point of the Erlangen program, and you now hold the key it turns.