Mobius transformations

Definition

Let a, b, c, d be complex numbers. We define a Mobius transformation from C (complex numbers) to C as

f(z) = (az+b) / (cz+d).

This family of transformations is also called fractional linear transformations. From the definition it is clear that c and d cannot both be zero. If c is zero we get

f(z) = (a/d)z + (b/d),

just the linear transformations. If c is non-zero then

f(z) = [(a/c)(cz+d) + b - (ad/c)] / (cz+d) = (a/c) + (b – ad/c) / (cz+d).

In both cases we see that if ad = bc the resulting transformation is constant. Therefore we always require ad - bc ≠ 0.

If c is non-zero f is not defined for z = - d/c. It is useful to add a new point "at infinity", denoted by ∞, and to define f(-d/c) = ∞ , f(∞) = a/c. If c is zero we define f(∞) = ∞ .

Algebraic Properties

• 1. Mobius transformations are 1-1 and onto.

  2. We construct an inverse to a Mobius transformation:

  3. w = (az+b) / (cz+d),

  4. czw+dw = az+b,

  5. z = (dw-b) / (-cw+a) = g(w).

  6. g is again a Mobius transformation (its coefficients satisfy the restriction). If c is zero then f and g are defined in all C and f(∞) = ∞ = g(∞) completes the claim. If c is non-zero then we check the “infinite” points, f(-d/c) = ∞ , g(∞) = -d/c, f(∞) = a/c, g(a/c) = ∞ .

1. Multiplying a, b, c, and d by a constant does not change the transformation.

   1. This is obvious, the constant cancels out. This shows that effectively there are three parameters that define the transformation.

2. Composition of Mobius transformations is a Mobius transformation.

Let f be defined using a, b, c, d, and g be defined by A, B, C, D. We compute the composition

g(f(z)) = (Af(z)+B) / (Cf(z)+D) =

= (A(az+b) / (cz+d)+B) / (C(az+b) / (cz+d)+D) =

= (Aaz+Ab+Bcz+Bd) / (Caz+Cb+Dcz+Dd) =

= ((Aa+Bc)z+(Ab+Bd)) / ((Ca+Dc)z+(Cb+Dd)).

We see that the composition is again a Mobius transformation. Moreover, if we let [f] be the 2x2 matrix with (a, b) in the first row and (c, d) in the second, and similarly for all Mobius transformations, we get

[gf] = [g][f],

where on the left we have composition of functions and on the right matrix multiplication. This defines an action of the group of invertible 2x2 matrices over C on C.

Geometric Properties

1. Exact 3-transitivity.

   1. Let f be a Mobius transformation, and suppose f(0)=x, f(1)=y, f(∞)=z, and that x,y,z are three different points. If z = ∞ we have seen that c = 0, and we can normalize so that d = 1. In this case b = x and a = y - b. If z is finite we can normalize so that c = 1, and then a = z. The other two conditions are b/d = x and (z+b) / (1+d) = y. If x = ∞ then d = 0 and b = y – z. If y = ∞ then d = -1 and b = -x. Otherwise we get b = dx, z+dx = y+dy, d = (z-y) / (y-x).

   2. In all cases we see that x,y,z define a unique Mobius transformation.

2. Invariance of lines and circles.

As we have seen a Mobius transformation is either a linear transformation (when c is zero), or it can be written as

f(z) = (a/c) + (b-ad/c) / (cz+d).

This can be written as a composition of a linear transformation g(z) = cz+d, followed by an inversion i(z) = 1/z, and then another linear transformation h(z) = (b-ad/c)z+(a/c). Linear transformations are composed of shifts, rotations and scale multiplications, so they transform lines to lines and circles to circles. Let us investigate the inversion i(z).

>> Click here for a geometric proof. <<

First we note that for any constant c

i(z) = ci(cz),

so we can first multiply by a constant when checking the image of the inversion. We will use this to rotate, when |c| = 1.

Let L be any line. Up to rotation we can assume L has the form L = x+it, with x a Real constant and t a Real parameter. The image i(L) consists of the points

1/(x+it) = (x-it)/(x²+t²).

For x = 0 this is just -i/t which is the same line with a different parametrization. Otherwise we look at the distance from 1/2x

|(x-it)/(x²+t²) - 1/2x| = [x/(x²+t²) - 1/2x]² + t²/(x²+t²)² =

= x²/(x²+t²)² + 1/4x² - 1/(x²+t²) + t²/(x²+t²)² =

= 1/4x².

The result is a circle centered at 1/2x with radius 1/2x.

Let C be a circle. Up to rotation we can assume C has the form C = x+re^it, with x and r positive Real constants and t a Real parameter. The image i(C) consists of the points

1/(x+re^it) = (x+re^-it)/(x²+r²+2xrcos(t)).

When x = r, C passes through zero, and i(C) is

(1/2x)(1+e^-it)/(1+cos(t)) =

= 1/2x - i(1/2x)sin(t)/(1+cos(t))),

Which is a parametrization of a line. Otherwise we look at the distance from x/(x²-r²)

|(x+re^-it)/(x²+r²+2xrcos(t)) - x/(x²-r²)| =

= [(x+rcos(t))/(x²+r²+2xrcos(t)) - x/(x²-r²)]² +

+ (rsin(t))²/(x²+r²+2xrcos(t))² =

= (x²+r²cos²(t)+2xrcos(t)+r²sin²(t))/(x²+r²+2xrcos(t))² +

+ x²/(x²-r²)² - 2(x²+xrcos(t))/[(x²-r²)(x²+r²+2xrcos(t))] =

= x²/(x²-r²)² + [1/(x²+r²+2xrcos(t))] [1-(2x²+2xrcos(t))/(x²-r²)] =

= x²/(x²-r²)² + [1/(x²+r²+2xrcos(t))] [(-x²-r²-2xrcos(t))/(x²-r²)] =

= x²/(x²-r²)² - 1/(x²-r²) = r²/(x²-r²)².

The result is a circle centered at x/(x²-r²) with a radius of r/(x²-r²).