On a side note, the log-polar coordinate system (https://en.wikipedia.org/wiki/Log-polar_coordinates) also uses this kind of transformation. It has various applications, e.g. in image processing where the density of squares in the grid correlates with the density of light receptors on the retina.
Another derivation of c * e^z: http://mesecons.net/random/EXPlanation.pdf
The other video on how to make the planet behave somewhat like a sphere: https://www.youtube.com/watch?v=ztAg643gJBA
How to get this game:
For Windows:
Download this and run minetest.exe in the "bin" directory:
https://github.com/Jeija/spheretest/releases/download/release/spheretest_release.zip
For all other platforms, download and compile the game from GitHub:
https://github.com/Jeija/spheretest
Add satellite mod, skybox and demo map manually:
https://github.com/Jeija/spheretest/releases/tag/release
Minetest Forum Topic: https://forum.minetest.net/viewtopic.php?f=14&t=15643
If you have any questions / corrections / suggestions, please let me know in the comments!
Thanks to 3blue1brown for the amazing manim library which was used for the math animations in this video: https://www.youtube.com/3blue1brown