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...
Episode 2: • How to Design the Perf...
=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:
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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Thank you for your support!
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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:
github.com/morpho-matters/mor...

[@teslainvestah5003]

Your animations are on the level of 3blue1brown's. You just animate what you're saying, no matter how hard it gets, and it makes your videos educational miracles. Love it.

[@llawliet666]

Ikr. I was thinking about it the whole way through the video. He deserves so many more subscribers.

[@katiejackson3900]

I think it's done using 3blue1browns python library manim, so it literally is 3blue1brown's level (though I assume there's some knack to using the library animations well). But the explaination and pacing are perfect.

[@khag.]

Morpho and Manim are two separate libraries, but the former was written with inspiration from the latter. I'm curious to know how much code from Manim was used in Morpho, if any. Is it a fork?

[@morphocular]

@@khag. I'm honored to have my animations compared to 3Blue1Brown's or to have Morpho compared to Manim, but as it turns out, Morpho is not based on Manim. I developed it almost completely independently (in fact, I have yet to learn Manim) originally just as a casual side project to mimic just a few 3b1b animations I liked. Though over time it wound up growing into a much bigger tool than I expected!

[@matthewboyer4212]

@@katiejackson3900 there is in fact some knack to using the animation library, much like python itself it's easy to learn but hard to master

[@AdrianHereToHelp]

Genuinely love the maths communication in this series; you have a wonderful talent for explaining things!

[@yash1152]

your pfp is super unique and nice.

[@AdrianHereToHelp]

@@yash1152 Thank you! I commissioned it from a close friend and I'm incredibly happy with it!

[@waddupbro]

Having the cardioid's couple wheel be a teardrop is oddly philosophical

[@wasp795]

a heart’s couple is a tear that sounds very deep

[@ND62511]

Here’s a fun fact; since the Nth harmonic of a complimentary wheel can be visually expressed as the pattern on the wheel repeating N times, we can use this to show that the straight line road is just another harmonic of the circular road example. It’s just the infinity-th harmonic.

[@chrisjackson5072]

Having watched the entire series, the biggest sign that you aren’t an engineer is that you haven’t used the words “no slip condition”. The other main difference in your approach is that you have largely avoided using vector operations. In engineering, there isn’t anything called the “orthogonal motion principle”. We would get that result from the no slip condition, where the wheel and the road at the contact point have zero relative velocity, and combine this with the formula for rigid body motion, v_b=v_a+w_B X ab. The results end up being the same, it’s just interesting.

[@MikeTheMan01]

No slip conditions are a god send in any transport phenomenon classes

[@kindlin]

@@MikeTheMan01 It's basically the fundamental boundary condition of fluid dynamics. Mabye I only think that as I'm an engineer.

[@chrisjackson5072]

@@kindlin how else would we calculate wall shear?

[@kindlin]

@@chrisjackson5072 No slip necessarily means wall. I just meant there are other boundary conditions. Like, 2 fluids moving relative to one another, you'll get turbulence at that boundary. That is a separate boundary condition, probably the other fundamental one of fluid dynamics. I don't think this is a formal term, I'm just thinking out loud.

[@diegodiaz6392]

2+2=4 (certified answer)

[@snowf00t]

After being in university for a couple of years, it seems necessary to make nice graphs that demonstrate your point, but I cannot even begin to imagine how long it might have taken to generate the animations that move so flawless, really great job and attention to detail!

[@NTVE404]

Which one

[@angeldude101]

Something that I think is really cool is how if you take the wheel-wheel equations, but then take the limit as one of the radii approaches infinity, then what you end up with is the wheel-road situation. It should theoretically be possible to express both forms with the same principles.

[@geekboy12357]

This was a great series! I hope you do a sequel series on gear design, it would be interesting to see the similarities and differences between gear design and this rolling wheel design!

[@filippomariachiappini1257]

This is fantastic work! I really appreciate all the effort you put in the series. Great Christmas gift, thank you.

[@timkw]

Thanks for the Subtitles, I like being able te read along!

[@Knightros]

Not enough KZbinrs have their own subtitles, it’s a shame, I like them

[@user-pr6ed3ri2k]

yee yeeread ttttuj7

[@manioqqqq]

@@Knightros 𝕤𝕒𝕞𝕖

[@viniciusfriasaleite8016]

Engineering student here! I really appreciated this series! Thank you! As my classes on mechanical elements didn't go much deep, I would love if you explored more about gear design

[@gammaboy4568]

I learned a lot more about gear design in the Kinematics course offered at my university. I did personally study quite a bit of it beforehand out of curiousty, but yeah... there's a lot that goes into the shape and design of gears, as well as designing gear trains. Even then, it was only a small portion of the course. If you do learn about gears, the focus will most likely be on power transmission rather than gear design itself. Gears are fairly standard, so understanding how to make them is a bit less important than knowing how to apply them. If you're an ME and you've taken a course on statics and dynamics, you might be getting a discussion on gears soon enough.

[@Bananabeacon]

Wow! You really are a mathematician. You don't just explain someone else's math, but create it yourself! Really inspiring!

[@notmyrealname5473]

nobody creates math dude....its just numbers

[@Bananabeacon]

@@notmyrealname5473 well you know what I mean right? He figured it out; Arranged the 'numbers' in the right way. By your logic a composer doesn't create music, because they are just notes!

[@snailcheeseyt]

@@notmyrealname5473 he creates equations and theorems happy?

[@notmyrealname5473]

@@snailcheeseyt no im not happy. this guy didnt invent no math!! he's just a youtuber.....

[@snailcheeseyt]

@@notmyrealname5473 well my statement holds true also, maybe do something cool or productive other than raging about some random dude who commented a slightly incorrect compliment

[@johnchessant3012]

The trick to finding self-coupling wheels was really clever! And you're right, it seems so obvious now that you told me

[@Doniazade]

This is my favorite video all year I think, perfectly explained without losing the audience. The concept of rolling shapes is very intuitive but this explores the concept in a very satisfying way. Extremely well done! ⭐

[@fede9003]

These videos are so amazing! They are easy to follow through, and the topic was really interesting. Thank you for this series!

[@alvarol.martinez5230]

This was delightful to watch! I especially loved the idea of how self-coupling wheels are in bijection with vertically symmetric graphs of periodic functions, such a gem

[@nerdsgalore5223]

This was an incredible series! I'm excited to see what else you have in store for the future!

[@General12th]

Hi Morph! I really appreciate you going through the _exact_ steps for the simple wheel example. It might seem obvious to walk through those steps, but I liked making sure I could follow along quantitatively, not just qualitatively.

[@WhattheHectogon]

Fantastic work---I was just thinking about how awesome I found your first couple videos in this series and lo and behold, you put out another! Thanks for showing off your teaching talents :)

[@strangerontheinternet7358]

thank you for this! I have been driving on a constantly changing shape in an endless void for so long, and this has helped me a lot!

[@JonKloske]

A wheelie detailed look at the topic. Well done!

[@nihil1]

This video is so comforting to watch.

[@potatogamerrrrr]

Super helpful, I’ve been needing to know how to do this for a while!

[@nomadow2423]

I joined here, having not seen the previous videos, and must applaud you: your animations, explanations, and even the tempo/pacing were perfect. What a fun topic! I loved the moment you revealed the stupid simple solution. It's moments like those that make math so enjoyable. Thank you for this video!

[@ramonhamm3885]

Amazing info! Well done video, thanks!

[@ZivShemesh]

I have been waiting for so long... Thank you so much for such an interesting series!

[@stevechrisman3185]

Fantastic illustrations !!!

[@TheAbyssCOC]

I love your videos and how you present everything. There is no clear comparison with 3b1b, but I would say you are on the same level. You do things differently and approach the problem differently but the content is still as amazing as his. Keep up the good work, we all appreciate it!

[@number1freshlemon56]

I watched the last two videos in this series yesterday, and then the third one is uploaded today? Fantastic.

[@iamtraditi4075]

Thank you for this series! It's been a great ride :)

[@stuchly1]

This was a wild ride to be sure but so much fun to watch!

[@colinbradley7361]

Love this series, sad I didn’t have it recommended on release but just got to watch it now. Great video.

[@gauthierruberti8065]

I loved this series!

[@alix9751]

This was an amazing series of video! I look forward to your content in the future, you're probably one of the best math youtuber out there. ^^

[@Benjax_95]

beautiful work, keep it up!!

[@guillermogil3391]

This series was AMAZING. Thank you

[@LucasdeKam]

Very nice and inspiring work!

[@gnosticagnostic9326]

I'm so impressed by all of the information you've gathered by reinventing the wheel.

[@silversoul21000]

superbly awesome video as always ! loved all the serie !

[@MikkoRantalainen]

Superb work!

[@The_wyvern_wynnedow]

THANK YOU FOR MY FAVORITE SERIES ON KZbin!!!!!!

[@tedrickhernandez66]

Amazing video! Subscribed!

[@warlock2507]

The animations here are incredible! It’s just great how easily I can follow along :D

[@videoinformer]

Just the title and 2-second thumbnail animation revealed how fascinating this video would be! Extremely cool, even in concept, let alone in all the details!

[@constexprDuck]

This series of videos was amazing to warch! Thanks a lot

[@mohamedb737]

your handling of this problem was truely masterful. Simply elegant math and that's beautiful!

[@trattoretrattore8228]

I'm in love with this video. Also, this time, I managed to understand the differential equation without issue, you're great at explaining

[@TiagoTiagoT]

17:53 The partner of the heart is a teardrop... Damn....

[@bmw_de]

Thank you for the series, I really enjoyed it! I have one thing to say about gears, they are actually not slipping, but perfectly rolling on each other. Their contact point is also always perpendicular to each other. Exploring the special shape of spur gears and the mathematical origin would really be a perfect fit for the series!

[@bandana_girl6507]

Well, involute gears perfectly roll. The more important thing is that the line of action stays constant (or at least some reasonable approximation of constant)

[@bpark10001]

@@bandana_girl6507 & Jacques: NO THEY DON'T! Involute (& all other toothform gears) have contact path that IS NOT along the line of centers (except for the ONE point on the side of each tooth that crosses the pitch circle, for circular gears, or the pitch curve, for non-circular gears). The teeth slip as well as roll except at that point. That's why power gears require an oil bath for lubrication, whereas bearings require only sealed-in grease. What is "perfect" are the angular rotation rates, in that the rotation (of the toothed structures is indistinguishable from pitch-line rollers in frictional contact. This is referred to as "conjugate action". Involute toothform has additional property that conjugate action is maintained despite center-distance change.

[@Rudmin]

This is actually incorrect. Gears can only experience pure rolling for a single instant when their contact point crosses the pitch circle

[@bmw_de]

@@Rudmin no, this is not true. The whole point of gears is that they only experience rolling, not slipping. The contact is perpendicular to the tooth at every moment

[@Rudmin]

@@bmw_de that’s a very common misconception that is widely held, but @Morphocular did their homework and was correct about gears. All conjugate action gear forms (except for circles) experience a combination of rolling and sliding at the contact point. Your cars transmission actually relies on this sliding action to keep an oil film between gear teeth. It would not last nearly as long as it does with pure rolling. If you do the math (like this video), pure rolling is impossible for constant velocity gears at any point not on the pitch circle. It’s a direct consequence of the shared instantaneous velocity relationship required for pure rolling that was found near the start of this video.

[@Fenizrael]

This was a great video. I didn’t expect I’d be caring about the math of rolling objects or the cool ways you could implement it but here I am.

[@Enzi_Meteori_902]

finally the anticipated part

[@lukelinigaming]

Beautiful video as always

[@aditya95sriram]

Glad to have discovered your channel! 🙂

[@morphocular]

Thanks so much!

[@levia.phillips4311]

It is currently 2:40 AM and I’m laying on the floor of my kitchen and SOMEHOW this video of ALL OF KZbin has brought me comfort. Thank you :)

[@renesperb]

Very interesting problems and very well presented.

[@Vextrove]

This series is wonderful. I would watch it again

[@NickAndriadze]

Boy was this wheel-road series a ton of fun to watch... Even if the complex trigonometry that goes into discovering, getting and using these formulas that is kind of out of my reach currently, it was still nonetheless a ton of fun and taught me a fair bit as well.

[@5eurosenelsuelo]

Such an interesting series. I look forward to the continuation where we get to see real gears. The applications of math to the real world are always interesting and help people to realize of how important it is.

[@clarysshow3253]

The part where you talked about sidenote about gears, I am glad that you added this part as well. People explaining rolling motion or gears sometimes doesn't fully show the picture how gears are a bit more complicated than just two circles rolling

[@TheArmyofWin]

Before you described the rotationally symmetric road to generate self-coupling wheels I was saying “should be simple enough if you just set r(t) = ρ(t+Q*π) where Q is some rational number; but using the previous solution to generate wheels on both sides is extremely clever!

[@vincentwhite938]

Nice concept.

[@adammichna5175]

This is really well done!

[@Raye938]

I've got minimal subscriptions but you found your place among them. Nice series!

[@benjaminrichards6501]

I like your funny words magic man! Jokes aside this was a fascinating watch, very well animated and educational.

[@paperboy1790]

Watched this video at 5am after an all nighter and started crying. 10/10 best video I've seen in a long time

[@npm3605]

Amazing! Thanks a lot.

[@Aerobrake]

20:09 no, thank you for being amazing and sharing this infortmation with people who love mathematics! Merry Christmas and happy holidays, Morphocular!

[@santiagorocha5595]

I think that this series is AMAZING 10/10 😀😃😀

[@1.4142]

Very inspiring, a lot of functions to try this with.

[@gastonsolaril.237]

Past video about the "wheels for different roads" is very useful for gear and rack/pinion design in mechanics. This one is a good way of tackling cam-and-follower mechanisms. So all of these videos you're releasing, are excellent for engineering machinery!

[@yash1152]

16:04 the following part of animations was super beautiful

[@realcygnus]

Really amazing ! 👍

[@jfeast5469]

2:10 That's the kind of stuff that makes the maths so cool! Great videos!

[@Ihab.A]

Another great invaluable math video animation!

[@supu8599]

Waiting for it

[@ribozyme2899]

12:02 This is actually a really pretty consequence of the definition of an ellipse. For points on an ellipse, the sum of the distances to the two foci is constant. But if you turn that around, you can turn two ellipses of the same shape around each other, and the contact point will have the sum of the distances to the two foci/axles constant.

[@squirrel_szn]

its videos like this that reinvigorate my love for math

[@clangauss4155]

Well done.

[@TheMemesofDestruction]

Love it! Thank you! ^.^

[@jakobr_]

Great video and great series! I was hooked the whole way through. You’re a great presenter! If you ever do continue this series it would be cool if you could explore a “realistic” constraint that the wheel has to “work” in real life, that is, dealing with collisions on all points of the shape at all times so that they don’t “clip” into each other.

[@bernat8331]

The derivative of the function needs to be continuous

[@HeavyMetalMouse]

@@bernat8331 That seems like a sufficient condition, but not a necessary one, as we see non-clipping wheels/roads that have discontinuous derivatives (polygonal wheels with sides 4 or more, for example). The actual condition seems like it should be related, but weaker than that.

[@chri-k]

@@HeavyMetalMouse actually, a perfect polygonal wheel would teleport at corners, which is also not realistic. So the condition is correct. And it *has* to teleport, the math used here cannot handle a stationary point of contact. and in the case of real physics, a perfect corner cannot exist in the first place

[@ZomB1986]

Don't forget a realistic wheel also needs to have constant velocity. (otherwise the wine glasses on top of the car would still tumble). Unfortunately, with the constraints of smooth rolling and constant velocity, the only solution is a plain old round wheel. It's the reason gears necessarily need to slip.

[@alexvalentim1418]

Fantastic series, this is very useful for people that want to play with 3D printing

[@craziestdavid15]

Really appreciate that you bring this to my regular life. It reminds me my interest to math.

[@78Mathius]

Love your work

[@ghl1123]

Awesome!

[@petatirrumator3005]

You deserve a lot more subs.

[@alhusain]

Video of the year

[@kwongkc.samuel]

Excellent video.

[@somehow_sane]

I have never seen a fractal wheel before. Neat!

[@user-kq3xn3jp7m]

yay, more wheels!

[@mghyy2846]

Great video, thank You

[@epixa2004]

Excellent work! I suggest discussing the issue of local strength and the limitations in the geometry due to the production of the teeth in real cases, say of gears vs chain in bicycles! Great stuff!

[@sgtravz]

Thanks!

[@PriyadarshiPrashant]

hey wheel master thank you for your amazing weird fantastic pleasing and overall very informative videos

[@NoahK216]

The king of rolling is back

[@realcrazj1399]

when i heard he got a sponser for this i fing that as an amazing achivement for youtube espcially with 48.5k subscribers. well done man

[@astro_cat030]

learning what interests you and not forced is really what makes education fun. Even tho i dont understand sum of the video (the whole literally) it was fun to watch.

[@samuelthygerson6009]

This series of odd closed shapes and not closed shapes really cool to learn about.

[@nicholasmoffett4327]

thanks for the ride! it sure wasn't a bumpy one 😎

[@kikiblipblop3047]

Perfect

[@ilysmaxx]

very cool and makes you think