I have a square S, and I want to convert it to the unit disc D.


The [Riemann mapping theorem](http://en.wikipedia.org/wiki/Riemann_mapping_theorem) says that I can do this 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](http://www.flickr.com/photos/sbprzd/362529354): 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:


$$Distortion(d,s) = area(d(s)) / area(m(d(s))) - 1$$


So, if $s$ is mapped to a square, then $d(s)$ is a square, $m(d(s))=d(s)$, and $Distortion(d,s)=0$. On the contrary, if $s$ is a very distorted shape, probably $m(d(s))$ will be quite small, and $Distortion(d,s)$ will be large.


For each mapping $d$, it is possible to define its maximal distortion for squares of a certain size x:


$$MaxDistortion(d,x) = max[over\ all\ squares\ s\ of\ size\ x\ in\ S]{Distortion(d,s)}.$$


similarly: 


$$AvgDistortion(d,x) = avg[over\ all\ squares\ s\ of\ size\ x\ in\ S]{Distortion(d,s)}.$$


My questions are:


* **For a given Riemann mapping d and square-size x, how can I find $MaxDistortion(d,x)$ and/or $AvgDistortion(d,x)$?**
* **Of all Riemannian mappings from S to D, how can I find the Riemann mapping with the smallest MaxDistortion/AvgDistortion (for a given square-size x)?**


If a similar question has been studied, even with different definitions of distortion, I will also be happy to know.



What happens to small squares in Riemann mapping?
1. Question: In general it seems difficult to find max and mean distortion as a function of $x$. For the square is looks doable.
The following papers describe numerical code to find the Riemann mapping for arbitrary simple curves (described by landmark points).


	* Sergey Kushnarev: Teichons: Soliton-like Geodesics on Universal Teichmüller Space,
	Experimental Mathematics, Volume 18, Number 3 / 2009
	* E. Sharon and D. Mumford, 2D-Shape Analysis using Conformal Mapping (pdf), Proceedings IEEE Conference on Computer Vision and Pattern Recognition, Washington, DC, June 2004. [(pdf)](http://www.dam.brown.edu/people/eitans/publications/SharonMumford-ShapeCVPR04.pdf)
2. Question: The Riemann mapping is unique up to composition with an Moebius-Transformation in $SU(1,1)$, (which fixes the unit circle) $z\mapsto \frac{az + b}{cz+d}$. The effect of the Moebius transfomation on areas is easily computable.


