How to Design a Wheel That Rolls Smoothly Around Any Given Shape

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In previous videos, we looked at how to find the ideal road for any given wheel shape and vice-versa, but what about getting two wheels to roll smoothly around each other? Would two such wheels work as gears?

Episode 1:    • The Perfect Road for a Square Wheel a...  
Episode 2:    • How to Design the Perfect Shaped Whee...  

=Chapters=
0:00 - Intro
1:23 - Defining smooth rolling
2:30 - Sidenote about gears
3:16 - The Wheel-Coupling Equations
7:34 - Sanity check
9:24 - The partner for an ellipse
12:24 - The connection between ellipses and parabolas
13:23 - Finding self-coupling wheels
16:35 - The partner for a square
19:21 - A look back
20:20 - A fractal wheel??
20:47 - Brilliant ad


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I would also like to thank the user @BeekersSqueakers whose comment I think it was that taught me that a partner wheel can be generated by first generating a road and then generating a wheel on its underside. This comment was directly responsible for inspiring the technique shown in this video to easily generate self-coupling wheels, and dramatically simplified the second half of this video! So a seriously genuine thank you to @BeekersSqueakers and to all those who actually took up the call to answer my challenge problems in a comment! They can have a surprisingly big impact sometimes!


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For a deeper dive into the concepts explored in these videos, take a look at the paper "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). You can find it here:
https://web.mst.edu/~lmhall/Personal/...


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CREDITS

The music tracks used in this video are (in order of first appearance): "Rubix Cube", "Checkmate", "Ascending", "Orient", "Falling Snow"

The track "Rubix Cube" comes courtesy of Audionautix.com


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The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here:
https://github.com/morpho-matters/mor...