This is a follow-up to a question I asked a year ago, which was helpfully answered by Anton Petrunin:

Fitting a mesh to a density function

I am trying to come up with a way to make a picture of an equilateral triangular mesh in the plane in which the density of triangles has elliptical level sets. For example, the following picture has a triangular mesh in which the density of triangles has *circular* level sets, which I obtained by taking the complex map $z\mapsto z^{2}$:

The picture below is closer to what I was looking for, but it's actually a total hack: I just took the complex map $z\mapsto \sin{z}$ and then physically *drew* a black ellipse in the center of the picture to suggest some kind of elliptical density:

If we zoom closer we can see that the level sets of the density are clearly not ellipses:

Does anyone have suggestions for better maps that might produce the kind of picture I'm looking for? I browsed through the "Dictionary of Conformal Mappings" at http://math.fullerton.edu/mathews/c2003/ConformalMapDictionary.1.html but haven't been able to find anything better than what I've got.