Exploring conformal mappings

Conformal mappings were a subject that I really enjoyed studying in graduate school. If you haven't heard of them before, they are mappings of the plane that preserve angles and orientations between curves, but not necessarily distances.

At first, the angle-preserving property seems extremely special, and it is very difficult to imagine that such mappings exist, besides some obvious and boring cases. However, you can see a variety of these mappings in the animation above. To highlight the fact that the mappings preserve angles between curves, I've animated horizontal and vertical lines that intersect each other on the left, and the image of those lines on the right. You can see that when each pair of horizontal and vertical lines intersects, the lines on the right also intersect at right angles (for the most part). Remarkably these examples seem to be related to functions that we are very familiar with from high school math class.

What are we looking at here exactly?

If you haven't studied functions of a complex variable, you might have a hard time understanding what you're seeing in the animation above.

Selecting an option from the dropdown menu chooses which function to graph. Initially the function selected is \(z^2\). If you're reading this, you're probably familiar with the parabola \(x^2\), and its U-shaped graph, but you might be wondering why the visual above for \(z^2\) doesn't look anything like a parabola.

The big difference in this case is that \(z^2\) is a function of a complex variable, and complex variables have a real and an imaginary part. Geometrically, what this means is that unlike the parabola, \(z^2\) maps a two-dimensional plane to a two-dimensional plane, and it is this higher-dimensional mapping that is being visualized above. The left side grid displays the input variable of the function and the right side displays the function's value.

For example, suppose that \(z = 1 + 2i\). If we just remember that \(i^2 = -1\) then it is quick to calculate that \(z^2 = -3 + 4i\). Viewing this geometrically, this means that \(z^2\) maps the coordinates \( (1,2) \) to \( (-3,4) \)

Conformal mappings and differentiable functions

The essential fact regarding the theory of conformal mappings in the plane is that they are precisely the differentiable (aka analytic) functions on the complex plane. So in other words, any differentiable function that is invertible, i.e. has non-zero derivative in its domain, defines a conformal mapping and conversely, any conformal mapping is given by a differentiable function.

Some of the functions above do have zero derivative at certain points, and hence do not define conformal mappings there. For example the function \( z^2 \) has zero derivative at the origin, and we can see that for the pair of lines crossing at the origin, the corresponding lines on the right do not cross at right angles, and instead meet head on.

In any case, by examining the differentiable complex functions that we know: square root, exponential, sine, cosine, polynomials, and combinations of these, we can discover a great variety of conformal mappings with this miraculous angle-preserving property. The examples shown above give a taste of some of the many possibilities.

Möbius transformations

In the next post, I'll explore a particular set of conformal mappings, known as Möbius transformations. We'll see how Möbius transformations generate a whole continuous family of conformal mappings, which when acting on the plane, make the points of the plane flow, like bubbles on the surface of a river.