Straightening the square.

Resumen:

One interesting and very useful way of creating deformations in the plane is by ”straighten-ing ellipse fields”.  An ellipse field records how a function distorts angles at small scales; astraightening is a map from the plane to the plane solves a simple partial differential equa-tion: namely, all ellipses in the field must be mapped to circles. When it exists, the straight-ening map is essentially unique and can be used to solve various problems in geometry andtopology.We will look at a deceptively simple situation: outside a given square the ellipse field con-sists of circles and inside the square the ellipses are all parallel to the vertical side, with fixedratio K = major axis / minor axis.  In this case, there is a straightening map defined on theplane and the image of the square under the map is quite interesting! Now, here is a puzzle:what happens to this shape as K tends to infinity?

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