# How to find a conformal map of the unit disk on a given simply-connected domain

By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site (Wayback Machine):

I am interested in finding such transformations for the simply-connected domains having natural origin: oak and maple leaves:

Is it possible to find and draw corresponding conformal maps?

Maybe there are some online instruments (like Wolframalpha or Maple) for doing such tasks.

The purpose of this activity is to obtain an attractive image for the cover of a textbook on univalent maps of the unit disk.

You may want to look at Don Marshall's Zipper algorithm: https://sites.math.washington.edu/~marshall/zipper.html

Added in Edit by T. Banakh. This Zipper algorithm yields the following image of the conformal map of the unit disk to an oak leaf.

Many thanks to Prof. Donald E. Marshall for producing this image (which I post here with his permission).

• At the moment the Marshall's Zipper algorithm turned out to be the most appropriate for my purposes. – Taras Banakh Nov 4 '18 at 19:07

The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/

It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.

• Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time? – Taras Banakh Oct 30 '18 at 18:41
• Sure thing! I should have a little bit of time later today. The main challenge is just converting an image to a mesh (but this isn't even so bad!) – yousuf soliman Oct 30 '18 at 18:55
• I just added a picture of the conformal map from the maple leaf into the disk! I used an extremely coarse mesh (and so the features near the boundary aren't particularly well resolved)... feel free to email me at ysoliman@caltech.edu if you have any questions about this / want to get this running yourself / or need any help rendering any figures! – yousuf soliman Oct 30 '18 at 20:51
• Thank you for the image, but this is far from what I wanted: the rectangular coordinate system is not natural for the unit disk. It does not allow to locate the image of the center and images of concentric circles (in order to see how they change their form approaching to the boundary) and images of radii (to see where they finally touch the boundary of the leaf). Maybe this program has a switch to the polar coordinate system? – Taras Banakh Oct 31 '18 at 6:03
• That's a very easy change. I can write some code to do this tomorrow or the day after! – yousuf soliman Oct 31 '18 at 6:04

These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.

• Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map). – Taras Banakh Oct 30 '18 at 17:51