Finding the area of this ellipse written in complex polar form.

Question

I was reading the Chapter $4$ of Hubbard's Teichmüller Theory where he begins to introduce quasiconformal mappings. Here he writes a linear tranformation $T: \mathbb{C} \rightarrow \mathbb{C}$ as $T(u)=au+b\bar{u}$ and then notes that the determinant is $$\det{T}= |a|^{2}-|b|^{2}$$

He justifies this by saying that if we let $u=re^{i\theta}$, $a:=|a|e^{i\alpha}$, $b:=|b|e^{i\beta}$, then expanding $|T(u)|=|au+b\bar{u}|=1$, we get an ellipse with minor axis of semi-length $\frac{1}{|a|+|b|}$ and major axis $\frac{1}{||a|-|b||}$ and then noting that the ratio of the unit circle to the area of this ellipse is $|a|^{2}-|b|^{2}$.

Anyways the actual equation for the ellipse is as follows:

$$\bigg|(|a|+|b|)\cos\big(\theta+\frac{\alpha-\beta}{2}\big)+i(|a|-|b|)\sin\big(\theta+\frac{\alpha-\beta}{2}\big) \bigg|=\frac{1}{r}$$

I was wondering why the $r$ doesn't come up in the area or the semi-length of the major and minor axis? (Sorry, if this is a dumb question)

Answer 1

The $r$ comes in as the semimajor/semiminor axes' lengths when you expand $|T(re^{i\theta})|=1$, i.e., the minimum $r_\min$ is the semiminor axis length and $r_\max$ is the semimajor axis length.

Answer 2

The author is determining all $u$ that satisfy the equation $|T(u)|=1$. When converted to polar coordinates, the solution is the set of all $(r,\theta)$ that satisfy: $$ \bigg|(|a|+|b|)\cos\big(\theta+\frac{\alpha-\beta}{2}\big)+i(|a|-|b|)\sin\big(\theta+\frac{\alpha-\beta}{2}\big) \bigg|=\frac{1}{r}.\tag1 $$ So the equation (1) is a condition that $r$ and $\theta$ have to satisfy.

To see that (1) represents an ellipse, here is the argument in more detail: Squaring both sides of (1) and multiplying by $r^2$, (1) is equivalent to $$ (|a|+|b|)^2\left(r\cos\big(\theta+\frac{\alpha-\beta}{2}\big)\right)^2+(|a|-|b|)^2\left(r\sin\big(\theta+\frac{\alpha-\beta}{2}\big)\right)^2=1.\tag2 $$ We can convert (2) from polar coordinates back to orthogonal axes $(u,v)$ using the transformation $u:=r\cos\big(\theta+\frac{\alpha-\beta}{2}\big)$ and $v:=r\sin\big(\theta+\frac{\alpha-\beta}{2}\big)$, yielding $$ (|a|+|b|)^2u^2+(\big||a|-|b|\big|)^2v^2=1\tag3 $$ (note the $(u,v)$ plane is rotated from the original $(x,y)$ plane). But equation (3) is the equation of an ellipse, whose semi-major and semi-minor axis lengths you can read off.

In summary, you're misinterpreting the generic polar-coordinates variable $r$ as a property of the ellipse. Rather, $r$ is a variable in a particular coordinate system. The variable $r$ doesn't come up in the major and minor axes of the ellipse any more than $x$ or $y$ come up in the radius of the circle defined by $x^2+y^2=c^2$.