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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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...
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.
Ikr. I was thinking about it the whole way through the video. He deserves so many more subscribers.
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.
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?
@@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!
@@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
Genuinely love the maths communication in this series; you have a wonderful talent for explaining things!
your pfp is super unique and nice.
@@yash1152 Thank you! I commissioned it from a close friend and I'm incredibly happy with it!
Having the cardioid's couple wheel be a teardrop is oddly philosophical
a heart’s couple is a tear that sounds very deep
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.
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!
Which one
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.
No slip conditions are a god send in any transport phenomenon classes
@@MikeTheMan01 It's basically the fundamental boundary condition of fluid dynamics. Mabye I only think that as I'm an engineer.
@@kindlin how else would we calculate wall shear?
@@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.
2+2=4 (certified answer)
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.
This is fantastic work! I really appreciate all the effort you put in the series. Great Christmas gift, thank you.
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!
Wow! You really are a mathematician. You don't just explain someone else's math, but create it yourself! Really inspiring!
nobody creates math dude....its just numbers
@@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!
@@notmyrealname5473 he creates equations and theorems happy?
@@snailcheeseyt no im not happy. this guy didnt invent no math!! he's just a youtuber.....
@@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
Thanks for the Subtitles, I like being able te read along!
Not enough KZheadrs have their own subtitles, it’s a shame, I like them
yee yeeread ttttuj7
@@Knightros 𝕤𝕒𝕞𝕖
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
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.
The trick to finding self-coupling wheels was really clever! And you're right, it seems so obvious now that you told me
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! ⭐
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!
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)
@@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.
This is actually incorrect. Gears can only experience pure rolling for a single instant when their contact point crosses the pitch circle
@@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
@@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.
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.
17:53 The partner of the heart is a teardrop... Damn....
These videos are so amazing! They are easy to follow through, and the topic was really interesting. Thank you for this series!
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.
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.
The derivative of the function needs to be continuous
@@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.
@@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
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.
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!
16:04 the following part of animations was super beautiful
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
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!
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 :)
It is currently 2:40 AM and I’m laying on the floor of my kitchen and SOMEHOW this video of ALL OF KZhead has brought me comfort. Thank you :)
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!
This was an incredible series! I'm excited to see what else you have in store for the future!
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!
Watched this video at 5am after an all nighter and started crying. 10/10 best video I've seen in a long time
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!
I'm so impressed by all of the information you've gathered by reinventing the wheel.
Love this series, sad I didn’t have it recommended on release but just got to watch it now. Great video.
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 👍
This series was AMAZING. Thank you
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.
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. ^^
I have been waiting for so long... Thank you so much for such an interesting series!
The animations here are incredible! It’s just great how easily I can follow along :D
This video is so comforting to watch.
A wheelie detailed look at the topic. Well done!
your handling of this problem was truely masterful. Simply elegant math and that's beautiful!
superbly awesome video as always ! loved all the serie !
Thank you for this series! It's been a great ride :)
I watched the last two videos in this series yesterday, and then the third one is uploaded today? Fantastic.
Super helpful, I’ve been needing to know how to do this for a while!
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
This series of videos was amazing to warch! Thanks a lot
Amazing info! Well done video, thanks!
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.
Idc about the teaching, this is just so satisfying
I'm in love with this video. Also, this time, I managed to understand the differential equation without issue, you're great at explaining
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.
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!
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 🤔
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).
The problem is that the lines aren't infinitely thin. This would work IRL because there are no lines to overlap.
Very nice and inspiring work!
20:09 no, thank you for being amazing and sharing this infortmation with people who love mathematics! Merry Christmas and happy holidays, Morphocular!
Fantastic illustrations !!!
This series is wonderful. I would watch it again
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!
13:15 what happens if you push the conic sections even further and get rolling hyperbolae where the axle traces a path?
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!
@@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
@@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
beautiful work, keep it up!!
I've got minimal subscriptions but you found your place among them. Nice series!
I loved this series!
There is one issue here, specifically with the fractal ones, and more-properly, any shapes that seem to go beyond 180 degrees while touching the road. The issue is that when your wheel rolls along it must naturally phase through the road in order for parts that were touching to no longer. If you look carefully at the trailing and leading edges of the fractal you can see this happen, although its quite small. I think this is just an inate issue with the kind of shapes, again, ones in which the curve goes beyond 180 at any time, not by ANY means an issue with you math. All in all this is absolutly spectacular
its videos like this that reinvigorate my love for math
Glad to have discovered your channel! 🙂
Thanks so much!
THANK YOU FOR MY FAVORITE SERIES ON KZhead!!!!!!
Superb work!
Beautiful video as always
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?
The king of rolling is back
Very interesting problems and very well presented.
Amazing video! Subscribed!
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
I think that this series is AMAZING 10/10 😀😃😀
Very inspiring, a lot of functions to try this with.
Really appreciate that you bring this to my regular life. It reminds me my interest to math.
This is really well done!
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.
Fantastic series, this is very useful for people that want to play with 3D printing
2:10 That's the kind of stuff that makes the maths so cool! Great videos!
I like your funny words magic man! Jokes aside this was a fascinating watch, very well animated and educational.
i have a question, morphocular. I just watched your video and wondered, is there any example of a stationary wheel and moving wheel, such that the moving wheels axel path has its own shape? for example, a circle and a circle is an example, since the axel point is moving in a circular path. But what about jagged shapes? or maybe how many examples are there? Is there a formula for the number of bumps on stationary and moving wheel?
I think for certain symmetric shapes if you find a stable axle, you can offset it by a bit to get it to trace out its path. Probably gets more complicated for nonsymetric wheels
14:00 Funny coincidence, that the rabbit-like shape has a fox-like complement (they look like rabbit and fox head looking to the left). Unless this was an intentional easter egg in editing. In that case, nice job!
you have reignited the love for math that i lost in highschool. thank you
Incredible video. I’ve loved this series. The part with the base harmonic for the square wheel was screaming the first main bulb of the Mandelbrot set, or maybe Euler’s golden ratio spiral?. Is there any relation to either of these here?
i came here from a video of someone making the silliest gear shapes they could imagine, and now i know how they made them go smoothly :) i love maths
(in order to get them to transmit force, they simply added teeth to the outside of each shape)
Never thought I’d get a fascinating lesson on wheels, physics and problem solving from Jiminy Cricket.
Another great invaluable math video animation!
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. KZhead math explainers highlight exactly the reason why I love this stuff.
Really amazing ! 👍
hey wheel master thank you for your amazing weird fantastic pleasing and overall very informative videos
finally the anticipated part
The big difference between these shapes and gears is that gears are defined by conjugate action. Meaning that if one gear turns with a constant speed, then so does its partner. This ensures no vibrations when transmitting power. With the exception of circles (called capstan wheels when they are used as gears) it’s impossible to have pure rolling contact and conjugate action. While it’s true that sliding is helpful for the lubrication of high quality ground steel gears, it’s definitely not helpful for cheap plastic gears. The other source of unwanted vibration in gears is variation in the pressure angle. This is one of the reason that involute gears are now the norm. They maintain a constant pressure angle. Still it would be interesting to make a transmission from a set of 3 of your rolling “gears” such that the first and last one experience conjugate action but the middle transmitting gear rotates with a non-constant speed.
This series of odd closed shapes and not closed shapes really cool to learn about.
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
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!
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.
Love it! Thank you! ^.^
its one of those channels "i dont know what are u talkin about but its entertaining"