The previous post was a visual introduction to bilinear transformations, a.k.a. Möbius transformations or fractional linear transformations. This post is a short follow-up focused more on calculation.

A bilinear transformation f has the form

where ad – bc ≠ 0.

Inverse

The inverse of f is given by

The transformation f is defined everywhere except at z = –d/c, and its inverse is defined everywhere except at w = a/c.

So f takes the complex plane minus one point to the complex plane minus one point. Or an elegant way of thinking about it is to think of f and g as functions on a sphere by adding a point at infinity. Then we say

Determining by three points

Bilinear transformations have three degrees of freedom. That is, you can pick three values in the domain and specify three places for them to go in the range. The unique bilinear transform sending z₁, z₂, and z₃ to w₁, w₂, and w₃ is given by

Plug in your constants and solve for w = f(z).

Update: Here are explicit formulas for the coefficients.

Example

For example, let’s look at the smiley face example from the previous post.

We’ll pick three points on the face and three places for them to go.

Let’s say we want the center of the face to stay put, mapping 0 to 0. Next let’s pick two places for the center of the eyes to go. These are at
±0.4+.2ji. Say we want the left eye to go down a little to -0.4 and the right eye to go up and over a little to 0.5 + 0.3i.

I used Mathematica to solve for the parameters.

    {z1, z2, z3} = {0, -2/5 + I/5, 2/5 + I/5}
    {w1, w2, w3} = {0, -2/5, 1/2 + 3 I/10}
    Solve[(w - w2) (w3 - w1)/((w - w1) (w3 - w2)) == 
          (z - z2) (z3 - z1)/((z - z1) (z3 - z2)), w]

This says the parameters are, in Python notation,

    a = -72 - 16j
    b = 0
    c = 30 - 35j
    d = -75

Using the code from the previous post we can verify that this transformation does what we designed it to do.

    print(mobius(0, a, b, c, d))
    print(mobius(-0.4 + 0.2j, a, b, c, d))
    print(mobius(0.4 + 0.2j, a, b, c, d))

and we can take a look at the result.

Related posts

• Circles to circles
• Schwarzian derivative
• Conformal mapping

This post expands on something I said in passing yesterday. I said in the body of the post that

… the image of a circle in the complex plane under a Möbius transformation is another circle.

and added in a footnote that

For this to always be true, you have to include a line as a special case of a circle, a circle of infinite radius if you like.

This post will illustrate these statements with Python code and plots. First, some code for drawing circles and other curves in the complex plane.

    from numpy import exp, pi, linspace
    import matplotlib.pyplot as plt

    θ = linspace(0, 2*pi, 200)

    def circle(radius, center):
        return center + radius*exp(1j*θ)

    def plot_curves(curves):
        for c in curves:
            plt.plot(c.real, c.imag)
        plt.axes().set_aspect(1)
        plt.show()
        plt.close()

Next, code for Möbius transformations, and the particular Möbius transformation we’ll use in our plots.

    def mobius(z, a, b, c, d):
        return (a*z + b)/(c*z + d)

    def m(curve):
        return mobius(curve, 1, 2, 3, 4)

Now we’ll plot three circles and their images under the Möbius transformation

m(z) = (z + 2)/(3z + 4)

with the following code.

    circles = [circle(1, 0), circle(2, 0), circle(2, 2)]
    plot_curves(circles)
    plot_curves([m(c) for c in circles])

This produces

and

Notice that the first circle, in blue, started out as the smallest circle and was contained inside the second circle, in orange. But in the image, the blue circle became the largest, and is no longer inside the orange circle. That is because our Möbius transformation has a singularity at -4/3, and things get turned inside-out around that point.

Next we’ll look at an example of lines being mapped to lines.

    line = linspace(-100, 100, 600)
    curves = [line, 1j*line - 4/3]
    plot_curves(curves)
    plot_curves([m(c) for c in curves])

This produces

and

These lines are mapped to lines because they both pass through the singularity at -4/3. The real axis, in blue, is mapped to itself. The line -4/3 + iy, is shifted over to have real part 1/3.

Finally, lets look at lines being mapped to circles. Since the inverse of a Möbius transformation is another Möbius transformation, this example also shows that circles can be mapped to lines.

    
    lines = [1j*line - 4, 1j*line + 4, line - 4j, line + 4j]
    plot_curves(lines)
    plot_curves([m(c) for c in lines])

This produces

and

Note that the circles don’t quite close. That’s because my line only runs from -100 to 100, not -∞ to ∞. The gap in the circles is at 1/3, because that’s the limit of our transformation (z + 2)/(3z + 4) as z goes to ±∞.

Smiley faces

To illustrate things further, I’d like to look at a smiley face and what happens to it under different Möbius transformations.

Here’s the code to draw the original face.

    dash = linspace(0.60, 0.90, 20)
    smile = 0.3*exp(1j*2*pi*dash) - 0.2j
    left_eye  = circle(0.1, -0.4+.2j)
    right_eye = circle(0.1,  0.4+.2j)
    face = [circle(1, 0), left_eye, smile, right_eye]

Next, let’s subject this face to the Möbius transformation with parameters (1, 0, 1, 3). The singularity is at -3, outside the face and fairly far away.

Next we’ll use parameters (1, 0, 1, -1+1j), which has a sigularity at 1 – i, closer to the face, and hence more distortion.

Now we use parameters (1, 0, 3, 1), putting the singularity at -1/3, inside the face.

Finally, we look at parameters (1, 0, 1, 0.4-0.2j), putting the singularity inside the left eye.

The next post explains how to pick the parameters of a Möbius transformation to make points go where you want.

There are many ways the basic derivative can be generalized: partial derivatives, directional derivatives, covariant derivatives, etc. These all reduce to the basic derivative under the right circumstances.

The Schwarzian derivative is not like that. It’s not a generalization of the familiar derivative but rather a differential operator analogous to a derivative. The Schwarzian derivative of a function f is defined [1] as

To understand the motivation behind such an arbitrary-looking definition, we need to first look at functions of the form

called Möbius transformations, or more descriptively, fractional linear transformations [2]. These transformations are very important in complex analysis and have a lot of interesting properties. For example, the image of a circle in the complex plane under a Möbius transformation is another circle [3].

Möbius transformations are to the Schwarzian derivative roughly what constants are to the ordinary derivative. A function is a Möbius transformation if and only if its Schwarzian derivative is zero.

Since the Schwarzian derivative is defined in terms of ordinary derivatives, adding a constant to a function doesn’t change its Schwarzian derivative. Furthermore, the Schwarzian derivative is defined in terms of the ratio of ordinary derivatives, so multiplying a function by a constant doesn’t change its Schwarzian derivative either.

Even more generally, applying a Möbius transformation to a function doesn’t change its Schwarzian derivative. That is, for a fractional linear transformation like g(z) above

for any function f. So you can pull Möbius transformations out of a Schwarzian derivative sorta like the way you can pull constants out of an ordinary derivative. The difference though is that instead of the Möbius transformation moving to the outside, it simply disappears.

You can think of the Schwarzian derivative as measuring how well a function can be approximated by a Möbius transformation. Schwarzian derivatives come up frequently in applications of complex analysis, such as conformal mapping.

More kinds of derivatives

• Complex derivatives
• Distributional derivative
• Fractional derivatives

[1] The Schwarzian derivative of a constant function is defined to be zero.

[2] Möbius transformations require ad – bc to not equal zero.

[3] For this to always be true, you have to include a line as a special case of a circle, a circle of infinite radius if you like. If you don’t like that definition, then you can rephrase the statement above as saying Möbius transformations map circles and lines to circles and lines.