I have a square $S$, and I want to convert it to the unit disc $D$.
The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion.
Specifically, if S contains small sub-squares, they will be mapped into sub-shapes of C that are not squares. The amount of distortion depends on the specific mapping selected, and also on the placement of the small square inside S (see, for example, this nice illustration: the distortion is minimal near the center, and maximal near the corners).
The amount of distortion can be quantified in the following way: Let $s$ be a small sub-square contained in $S$. Let $d(s)$ be its image under the conformal mapping ($d(s)$ is contained in $D$). Let $m(d(s))$ be the maximum-area square that is contained in $d(s)$. Define:
$distortion(s) = area(d(s)) / area(m(d(s))) - 1$
So, if $s$ is mapped to a square, then $d(s)$ is a square, $m(d(s))=d(s)$, and $distortion(s)=0$.
I would like to know:
- What is the maximum distortion of a square of a certain size?
- What is the average distortion, over all squares of a certain size?
- If we divide $S$ into a k-by-k grid of $k^2$ sub-squares, what is the average distortion over all $k^2$ sub-squares?