A map built from four trustworthy pieces
The first two guides of this rung told you that every holomorphic map with non-zero derivative is a conformal map — it preserves angles — and that this fails exactly at the critical points where f'(z) = 0. That is wonderful in principle, but it leaves a practical hunger: which concrete maps should we actually reach for when we want to move one region onto another? The most useful explicit family of all is the Mobius transformation (also called a linear-fractional transformation), and the lovely thing is that it is assembled from operations you already met in the very first rung.
A Mobius transformation is any map of the form T(z) = (a z + b) / (c z + d), where a, b, c, d are complex constants and we insist that a d - b c is not zero. That last condition is the whole point: if a d - b c were zero the numerator and denominator would be proportional, T would collapse to a single constant value, and there would be nothing to map. The quantity a d - b c is exactly the determinant of the 2-by-2 matrix [a, b; c, d], and demanding it be non-zero is demanding that this matrix be invertible.
Why should one trust a quotient like this? Differentiate it: the quotient rule gives T'(z) = (a d - b c) / (c z + d)^2. Because a d - b c is non-zero, T'(z) is never zero anywhere it is defined — so by the very results of guides 1 and 2, T has no critical points and is conformal at every point of its domain. There is one place to be careful: T is undefined where c z + d = 0, namely at z = -d / c, which we will tidy up in a moment.
Translate, scale-and-rotate, invert — and that is all
Here is the secret that makes Mobius maps so easy to understand: every one of them is a short recipe built from three kinds of move you already trust. A translation z -> z + b slides the plane; a rotation-and-scaling z -> a z spins and stretches it; and the inversion z -> 1/z turns the plane inside out about the unit circle. The first two are visibly tame. The inversion is the only ingredient with any drama, and it is the one that does all the interesting bending.
When c is not zero you can check by patient algebra that T factors into exactly these moves performed in order: first add d/c, then invert, then scale by -(a d - b c)/c^2, then add a/c. (When c = 0 the map is simply the affine z -> (a/c... ) — more plainly, T(z) = (a/d) z + (b/d), a pure rotation-scaling followed by a translation, with no inversion at all.) Since each piece is conformal and each carries lines-and-circles to lines-and-circles — as we are about to see for inversion — the composite must do the same. This decomposition is not just a curiosity; it is the cleanest proof technique in the whole subject.
T(z) = (a z + b) / (c z + d), a d - b c != 0 when c != 0, build it in four steps: z -> z + d/c (translate) -> 1 / (z + d/c) (invert) -> -(ad - bc)/c^2 * .. (scale & rotate) -> .. + a/c (translate) = T(z)
The point at infinity makes it a clean bijection
We left a loose end: T blows up at z = -d/c, and nothing in the ordinary plane is sent to where T 'wants to' send infinity. The fix you met back in the foundations rung was to add a single point at infinity and work on the Riemann sphere — the plane wrapped up into a globe with one extra point at the top. On that sphere a Mobius map becomes a flawless one-to-one correspondence: we simply declare that T(-d/c) = infinity and T(infinity) = a/c (or, when c = 0, that infinity maps to infinity).
With those two extra rules, T becomes a bijection of the sphere onto itself, and so does its inverse — which, satisfyingly, is itself another Mobius map, T-inverse(w) = (d w - b) / (-c w + a). The collection of all Mobius transformations is therefore closed under composition and inversion: it forms a group, the Mobius group. The matrix [a, b; c, d] is not just bookkeeping — composing two Mobius maps corresponds exactly to multiplying their matrices, which is why the determinant condition and the group structure mirror invertible matrices so perfectly.
Why every line-and-circle stays a line-and-circle
Now for the headline property. Back in the foundations rung you learned to treat lines and circles as one family rather than two: a straight line is just a circle that happens to pass through the point at infinity, a circle of infinite radius. Once you adopt that generous viewpoint, the magic statement is clean: every Mobius transformation maps the family of all lines-and-circles onto itself. A line might become a genuine circle, a circle might flatten into a line — but you will never get anything jagged, never a parabola or an ellipse or a torn edge.
The proof rides entirely on the decomposition from earlier. Translations and rotation-scalings obviously send lines-and-circles to lines-and-circles — sliding, spinning, and uniformly stretching a circle or a line keeps it a circle or a line. So the only thing left to check is the inversion z -> 1/z. And here is the satisfying part: a single algebraic identity shows that inversion shuffles the four cases among themselves. A circle not through the origin goes to a circle; a circle through the origin (which passes near the trouble point 0) opens up into a line; a line not through the origin curls into a circle; a line through the origin maps to a line. Compose any number of these, and the circle-preserving property survives intact.
Fixed points, and a first famous map
How rigid is a Mobius map? Ask which points it leaves alone. A fixed point satisfies T(z) = z, that is (a z + b)/(c z + d) = z, which clears to the quadratic c z^2 + (d - a) z - b = 0. A quadratic has at most two roots — so a Mobius map other than the identity can fix at most two points of the sphere. That single fact has a powerful consequence we will exploit next guide: if a Mobius map holds three distinct points still, it must be the identity, and so three points determine a Mobius map completely.
Let's meet a celebrated example to make this concrete. The Cayley transform T(z) = (z - i) / (z + i) takes the upper half-plane (all z with positive imaginary part) and maps it exactly onto the open unit disk. You can feel why it works: the real axis is a line — a circle through infinity — and the Cayley map carries it to the unit circle. Plug in z = 0 to get -i/i = -1, z = 1 to get (1-i)/(1+i) = -i, z = -1 to get -(1+i)/(1-i) = i: three points of the real axis land neatly on the unit circle, pinning the image down.
This is the seed of transplanting a problem: the disk and the half-plane are two of the friendliest regions in analysis, and the Cayley map (a Mobius transformation, so conformal and circle-preserving) lets you carry a question freely between them. Guide 5 of this rung is devoted to exactly that half-plane-to-disk passage, and the cross-ratio of guide 4 will let you cook up such maps to order. For now, hold onto the picture: a quotient of two linear pieces, never tearing, gliding lines-and-circles around the sphere while preserving every angle.