1. Circle Inversion

In this first part, before we even talk about complex numbers, we're going to build up some intuition for a fancy transformation of the plane called "inversion". This will be extremely useful in the next part, when we build on this to derive Möbius maps.

Background: Interactive Intuition

When solving 2D geometry problems, usually we want to find some transformation of the plane that makes our problem easier to solve.

Most people are familiar with linear transformations, like rotation, reflection, or scaling (zooming in/out). Play around with the interactive toy below:

Circle Inversion is another transformation of the plane, but it's nonlinear, which means it does some wild things. Drag the slider below to see what it looks like:

In the visualisation above, inversion distorts each of the shapes, except for the circle, which only changes size. This is a key property of inversion: circles map to circles (or lines). We'll explore this more later.

Notice that points close to the centre get mapped very far away, whereas points near the dotted circle stay near it.

Also notice how the lines of the cartesian grid change: the gridlines meet at right angles, and after inverting, although the gridlines have become curved, they still meet at right angles! (Imagine zooming into a meeting point until the curved lines become basically straight)

You can see the angle at the meeting point is also preserved for the edges of the triangle meeting at a vertex (it's 60 degrees initially, and still 60 degrees after you slide the slider all the way to the right). This angle preservation property turns out to be true for any two (sufficiently smooth) curves that meet at a point.

Let's state this neatly:

If two curves meet at an angle $\alpha$, then after inversion, the two inverted curves will still meet at an angle $\alpha$. ¹

This property, of preserving meeting angles of curves, is called being a conformal map. I think it's pretty cool - even though inversion is this crazy operation that flips a plane inside-out, it still preserves angles!
Summarized: Inversion is a conformal map.

The Formal Definition

Defn. (Inversion)

Let $C$ be a reference circle with centre $O$ and positive radius $r$.

We define inversion about $C$ to be the transformation of the plane that does the following:

• for any point $A$, we send $A$ to the point $A^*$ lying on ray $OA$ such that length $OA^* = \frac{r^2}{OA}$.

You can visualise this as follows: in the diagram above, as $A$ moves closer and closer to $O$ along the coloured line, the point $A^*$ will slide rapidly further and further away along the coloured line.
This alludes to an issue: our definition does not work when $A=O$, since $A^*$ would have to be infinitely far away. To fix this, we add a special point at infinity to the plane, denoted $P_\infty$, and define that $O$ is sent to $P_\infty$ and vice versa. You can think of this as saying "$\frac{r^2}{0} = \infty$" and "$\frac{r^2}{\infty} = 0$".

Just like reflection, inversion is self-inverse - if you do it twice, the second transformation undoes the first one. In other words, inversion swaps pairs of points (e.g. in the diagram above, $A$ gets sent to $A^*$, and $A^*$ gets sent to $A$).

Where Do Circles Go?

As we mentioned earlier, it's an interesting fact that inversion sends circles to circles. Let's try to look at this in more detail.

As you can see, it looks like inversion sends circles to circles! However, sometimes something interesting happens...

Defn. (cline)

Instead of saying "circles and lines" all the time, we define a cline to be a circle or a line.

By now, hopefully your intuition agrees that: a line is just a circle with infinite radius. Every ordinary line passes through our special "point at infinity", $P_\infty$, and no circle passes through it. In other words, a cline is a line if and only if it passes through $P_\infty$.

Now we can phrase our observation as:

Inversion sends clines to clines.

This property is what makes inversion so powerful for solving problems - inversion lets us turn circles (which are tricky to deal with) into lines (which are easy to deal with).

It might not seem like it right now, but recall that the end goal of this book is to teach you about Möbius maps, which are a topic in complex analysis. So far, we've only introduced inversion. But I promise that in the next part, all the intuition you will build in this part, will be 100000% worth it.

Example Problem

To try and get our heads around some of the key properties of inversion that we've discussed so far, let's use inversion to destroy this maths olympiad problem from EGMO.

Let $\triangle ABC$ be a right triangle with $\angle C = 90^{\circ}$ and let $X$ and $Y$ be points in the interiors of $CA$ and $CB$, respectively. Construct four circles passing through $C$, centred at $A, B, X, Y$ . Prove that the four points lying on at exactly two of these four circles are concyclic (i.e. lie on a common circle).

Key Properties: Pop Quiz

Here's a little test of everything we've learned so far - there are also a couple of unseen statements, so if you're not sure about any of the statements, go back to the interactive displays and experiment away!

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Hopefully you now love inversion as much as I do. In the next part, we're going to neatly describe inversion using the complex plane, and use it to derive Möbius maps.

1. This is hard to prove using pure Euclidean geometry, since we need the notion of "tangent lines" to curves, meaning we need the notion of a 2D derivative. This is where complex analysis is useful!↩