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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Want to support future videos? Become a patron at / morphocular
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...

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30 июн 2024

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Комментарии : 629

@teslainvestah5003 Год назад

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 Год назад

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

@katiejackson3900 Год назад

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. Год назад

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 Год назад

@@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 Год назад

@@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

@waddupbro Год назад

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

@wasp795 3 месяца назад

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

@AdrianHereToHelp Год назад

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

@yash1152 Год назад

your pfp is super unique and nice.

@AdrianHereToHelp Год назад

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

@ND62511 Год назад

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 Год назад

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 Год назад

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

@kindlin Год назад

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

@chrisjackson5072 Год назад

@@kindlin how else would we calculate wall shear?

@kindlin Год назад

@@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 Год назад

2+2=4 (certified answer)

@angeldude101 Год назад

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.

@snowf00t Год назад

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 6 месяцев назад

Which one

@filippomariachiappini1257 Год назад

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

@geekboy12357 Год назад

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!

@viniciusfriasaleite8016 Год назад

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 Год назад

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 Год назад

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

@notmyrealname5473 Год назад

nobody creates math dude....its just numbers

@Bananabeacon Год назад

@@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 Год назад

@@notmyrealname5473 he creates equations and theorems happy?

@notmyrealname5473 Год назад

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

@snailcheeseyt Год назад

@@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

@timkw Год назад

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

@Knightros Год назад

Not enough RU-vidrs have their own subtitles, it’s a shame, I like them

@user-pr6ed3ri2k Год назад

yee yeeread ttttuj7

@manioqqqq Год назад

@@Knightros 𝕤𝕒𝕞𝕖

@Doniazade Год назад

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 Год назад

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

@General12th Год назад

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.

@johnchessant3012 Год назад

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

@alvarol.martinez5230 Год назад

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

@WhattheHectogon Год назад

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 :)

@ribozyme2899 10 месяцев назад

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.

@bmw_de Год назад

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 Год назад

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 Год назад

@@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 11 месяцев назад

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

@bmw_de 11 месяцев назад

@@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 11 месяцев назад

@@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.

@nomadow2423 Год назад

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!

@nerdsgalore5223 Год назад

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

@strangerontheinternet7358 Год назад

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!

@TheAbyssCOC Год назад

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!

@jakobr_ Год назад

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 Год назад

The derivative of the function needs to be continuous

@HeavyMetalMouse Год назад

@@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 Год назад

@@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 Год назад

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.

@TheArmyofWin Год назад

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!

@colinbradley7361 Год назад

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

@alix9751 Год назад

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. ^^

@ZivShemesh Год назад

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

@number1freshlemon56 Год назад

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

@levia.phillips4311 Год назад

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

@gnosticagnostic9326 Год назад

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

@NickAndriadze Год назад

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.

@JonKloske Год назад

A wheelie detailed look at the topic. Well done!

@nihil1 Год назад

This video is so comforting to watch.

@guillermogil3391 Год назад

This series was AMAZING. Thank you

@warlock2507 Год назад

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

@clarysshow3253 Год назад

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

@5eurosenelsuelo Год назад

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.

@iamtraditi4075 Год назад

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

@potatogamerrrrr 11 месяцев назад

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

@silversoul21000 Год назад

superbly awesome video as always ! loved all the serie !

@mohamedb737 Год назад

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

@trattoretrattore8228 10 месяцев назад

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

@Fenizrael Год назад

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.

@videoinformer Год назад

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!

@Aerobrake Год назад

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

@gastonsolaril.237 10 месяцев назад

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!

@constexprDuck Год назад

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

@ramonhamm3885 8 месяцев назад

Amazing info! Well done video, thanks!

@stevechrisman3185 Год назад

Fantastic illustrations !!!

@epixa2004 Год назад

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!

@gauthierruberti8065 Год назад

I loved this series!

@Raye938 Год назад

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

@paperboy1790 11 месяцев назад

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

@The_wyvern_wynnedow Год назад

THANK YOU FOR MY FAVORITE SERIES ON RU-vid!!!!!!

@LucasdeKam Год назад

Very nice and inspiring work!

@stuchly1 Месяц назад

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

@Benjax_95 8 месяцев назад

beautiful work, keep it up!!

@lukelinigaming Год назад

Beautiful video as always

@aditya95sriram 2 месяца назад

Glad to have discovered your channel! 🙂

@morphocular 2 месяца назад

Thanks so much!

@MikkoRantalainen 9 месяцев назад

Superb work!

@santiagorocha5595 Год назад

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

@squirrel_szn Год назад

its videos like this that reinvigorate my love for math

@tedrickhernandez66 Год назад

Amazing video! Subscribed!

@craziestdavid15 Год назад

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

@TiagoTiagoT Год назад

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

@Vextrove Год назад

This series is wonderful. I would watch it again

@alexvalentim1418 9 месяцев назад

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

@astro_cat030 Год назад

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.

@daltanionwaves Год назад

72,000 subscribers off of 10 videos... What?... It shows how starved we are for concise geometry content, explained in an intuitive visual way, without cutting corners on the math. Brilliant!! This is a great public service. Much appreciated 👍

@renesperb 9 месяцев назад

Very interesting problems and very well presented.

@yash1152 Год назад

16:04 the following part of animations was super beautiful

@chezz444 Год назад

Great video! It seems like the n-th wheel harmonic as n tends to infinity would tend towards the "rolling on the ground" case, with the source wheel being scaled smaller and smaller while the partner wheel gets scaled up to become the "ground". It'd be interesting to see if we can prove that the ground case is just a limiting case of two wheels?

@ValkyRiver Год назад

13:15 what happens if you push the conic sections even further and get rolling hyperbolae where the axle traces a path?

@dshmoish Год назад

I was wondering just the same. Since we have self-coupled ellipses tracing a circle (of positive curvature), and self-coupled parabolas tracing a straight line (zero curvature), there just have to be self-coupled hyperbolas tracing a negative-curvature path. Would be interesting to see an animation of that!

@nanamacapagal8342 Год назад

@@dshmoish I can already see the hyperbolas tracing a semicircle in the INSIDE direction rather than the outside direction. No proof just yet but it feels intuitive that it should work that way I might go through the math and try and figure out different kinds of inside-wheel systems

@YourMom-wi5to Год назад

@@nanamacapagal8342 yes I completely agree that’s exactly what I was thinking the ellipse was circle and moving straight with positive curve in a self-coupled tracking

@Zakru Год назад

No, thank YOU for making such fascinating videos about an interesting topic. Videos like these simultaneously both satisfy and fuel my hunger for math. Honestly, math is a big part of why I'm studying in university. I've been told by employers and colleagues alike that formal papers aren't a necessity to succeed in the programming biz (although I do prefer having the safety net), but I could not live without muh math. RU-vid math explainers highlight exactly the reason why I love this stuff.

@benjaminrichards6501 Год назад

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

@jfeast5469 Год назад

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

@adammichna5175 Год назад

This is really well done!

@ebco756 Год назад

Awesome video! The mystery spiral for the not-cardioid partner to the square is most likely the involute of a circle. That’s also the type of curve that is used in the shape of gear teeth-small world!

@gammaboy4568 Год назад

I tried looking into the shape a bit, and I've noticed one critical issue... a circle's involute is not defined at a radius smaller than that of the base circle which defines it. The scale of the involute reflects the scale of the circle, so a circle with a radius of 0 would not create an involute. It also forms a normal point to the circle surface (mimicking the "unwinding" motion often referenced when talking about Involutes). This spiral appears to repeat into the origin, and does not ever contact. However, its shape does feel familiar.

@realcrazj1399 Год назад

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

@realcygnus Год назад

Really amazing ! 👍

@sploopst6868 Год назад

something about the ellipses rolling along one other on the sine wave really reminds me of a Jacob's Ladder - the wooden blocks are more lozenge/rounded rod shaped in profile, but maybe there's something similar going on there

@1.4142 Год назад

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

@vincentwhite938 Год назад

Nice concept.

@npm3605 Год назад

Amazing! Thanks a lot.

@petatirrumator3005 Год назад

You deserve a lot more subs.

@samuelthygerson6009 Год назад

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

@TheMemesofDestruction Год назад

Love it! Thank you! ^.^

@Enzi_Meteori_902 Год назад

finally the anticipated part

@PriyadarshiPrashant Год назад

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

@78Mathius Год назад

Love your work

@enbyennui Год назад

My guess before watching was that it would be a cardioid, certainly much more interesting to be close but incorrect than either totally correct or totally wrong. Great video!

@Ihab.A 10 месяцев назад

Another great invaluable math video animation!

@julianying98 Год назад

The more I see, the more I think there's a principle for governing all of these things, because patterns appear here and there, often enough to make me think it's not a coincidence... Great video! Coming knowing nothing, left with much more.

@aras_aras_aras_aras Год назад

The sawtooth wave at 16:09 is not physically realizable, as the wheels overlap with each other, which is visible in some frames of the video. This would also happen with the road of shape square wave at 16:20, but the rising and falling edges of the square wave have some finite slope, so the wheels seem to be not overlapping. So, I am curious about the mathematical condition for a physically realizable wheel shape that doesn't overlap with the other wheel 🤔

@nerdiconium1365 Год назад

This is just a conjecture on my part, but based on previous info in the series (mainly the “roll~pivot” theorem), any road/wheel with corners (any non smooth part) must have corners with angles greater than 90 degrees in order to prevent intersection (Note: definitely a necessary condition for non-intersection, but possibly not sufficient, as in there may be wheels that don’t have acute corners but still end up intersecting their roads).

@japanpanda2179 Год назад

The problem is that the lines aren't infinitely thin. This would work IRL because there are no lines to overlap.

@SheepUndefined Год назад

As someone who dropped outta school bc of math, it's been kind of nice to get back into it on my own terms. I don't understand a lot of this stuff still, but I knew right off the bat that polar coordinates would be involved somehow, and that makes me really happy, even if I got lost immediately after. ^^;