MATLAB Plotting Coordinate Transformations in Matlab

[Lizard]

I was reading this article (https://www.nature.com/articles/srep40083) about designing a panoramic lens with transformation optics, and wanted to try to play around with modifying the coordinate transformations. I contacted one of the authors of the article, and she mentioned that she plotted the coordinate transformations in Matlab, creating this image (Figure 3 in the article):

However, I can't seem to figure out how the authors made these graphs.

These were the equations used:

What is the process for plotting coordinate transformations like these? I'm new to Matlab and don't really know where to start.

 

[Dr Transport]

Given the fact that you can't do overlays on polar coordinate plots in Matlab, I suspect that the author wrote a custom plot routine or calculated the x-y coordinate pairs at a very fine scale and plotted the points annotating in cylindrical coordinates.