How to Design the Perfect Shaped Wheel for Any Given Road

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Last video, we looked at finding the ideal road for a square wheel to roll smoothly on, but what about other wheel shapes like polygons and ellipses? And what about the inverse problem: finding the ideal wheel to roll on any given road, such as a triangle wave?
Previous episode: • The Perfect Road for a...
=Chapters=
0:00 - Intro & Review
1:48 - Polygon Wheels
3:49 - Elliptical Wheel
5:30 - Focus-centered Ellipse
8:50 - Wheels From Roads
11:24 - How to Get a Closed Wheel
14:10 - The Many Wheels for a Sinewave
16:24 - The Wheel for a Triangle Wave Road
19:16 - The Wheel(s) for a Cycloid Road
20:24 - The Wheel for a Parabolic Road
20:58 - A Look Ahead and a Challenge
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Many of the ideas in this video came from, or were inspired by, "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). If you want a deeper dive, I encourage you to read it yourself. As far as math papers go, it's fairly easy to read:
web.mst.edu/~lmhall/Personal/...
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CREDITS
► The song at the beginning of this video is "Rubix Cube" and comes from Audionautix.com
===============================
Want to support future videos? Become a patron at / morphocular
Thank you for your support!
===============================
The animations in this video were mostly made with a homemade Python library called "Morpho".
I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:
github.com/morpho-matters/mor...

Опубликовано:

27 июн 2024

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Комментарии : 1,5 тыс.

@morphocular Год назад

As some of you have noted, the shape I've been calling a "sawtooth" in this video is actually what's usually called a "triangle wave". Sorry about that! Clearly I am not an engineer. EDIT: Also, I had no idea the pronunciation of "foci" was so contentious! My pronunciation is what I was taught growing up in the US, but evidently it's different elsewhere. Obviously the correct pronunciation is as follows: "GIF"

@Patrick462 Год назад

I've seen a saw, and it looks like your "sawtooth" shape. So there.

@UltraLuigi2401 Год назад

@@Patrick462 The actual sawtooth function has the function increase with a slope of 1, then jump down to 0 (infinite slope). A triangle wave has the decreasing slope be the negative of the increasing one. An actual saw would be somewhere in the middle, where one side of the "peaks" is steeper than the other, but not perpendicular to the length of the saw. Also, saws generally aren't even, so the "valleys" wouldn't all be along a single straight line.

@ldcent8482 Год назад

tbh I scrolled to the comments just when I heard that lol sorry about all us butthurt audiophiles D:

@ldcent8482 Год назад

@@UltraLuigi2401 I think a real saw's teeth side-on would look like three or four sawtooth waves out of phase. At least our cheapo push-cut blades do.

@WillemRuben Год назад

@Morphocular, would the issue of a wheel colliding with the road be solved if we only include wheels of which the axle has an uninterrupted line to every point of the wheels boundary? That is to say, that there are no wheel edges inbetween each other. I got a feeling that this was the problem with the cardoid shape, and will similarily not work with horseshoe shaped wheels.

@AbiGail-ok7fc Год назад

I'd say that a "smooth ride" also implies that a constant rotation frequency of the axle leads to a constant speed forward.

@Rudy97 Год назад

Yep, suspension can take care of up and down shaking but there is no suspension to absorb back and forth shaking.

@zrajm Год назад

Exactly! I wondered about this too! I'd love to see some analysis of the speed of the center of rotation as the wheel rolls over the road. Is the jaggyness of the motion different for the different shapes? - And how about the experience one would have if one actually built one of these wheels? The oval wheel move very quickly during part of the motion (just like a comet speeding up at it approaches the sun) -meaning that if i had an actual, physical wheel it's mass would accelerate/decelerate in different parts of the rotation cycle... I imagine slight acceleration would also occur with a square wheel? (Though would it? I'm not exactly sure.) but if that's the case the wheel does have a symmetrically placed mass around its rotation point, so maybe that wouldn't be so bad? (I'm just imagining what kind of experience I'd have if I had a bicycle [and fitting road] with these various wheels. How smooth would my rides be in terms of speed of forward motion? And in terms of how hard I'd have to push in different parts of the rotational cycle?)

@YourMJK Год назад

Is there even any other solution than a circle?

@Visch8826 Год назад

that's not rlly gonna work for any road other than flat cos the wheel will otherwise continuously decelate when it hits a bump

@igornoga5362 Год назад

From second wheel equation, if dx/dt is constant and dphi/dt is also constant, r has to be constant as well. So circle is the only solution.

@sarthaksharma4816 Год назад

I've always felt stupid with Maths since high school days. Still do. But things like these keeps the flame of curiosity going and help me study more. Thank You.

@Serizon_ Год назад

yeh as a indian (which i suppose according to your name you are also indian) we face it kinda more ( or i havent seen other nations schools) but like in high school we're taught see these are the formulas use these and only these only if an example dont use your creativity you dumb child just memorise it like i had a science paper in which i wrote a wonderful answer but she literally said that you're debating for the marks which i suppose your parents want and they're right about that but i wont give it to you cuz first my mood is off second you literally took a "wrong intepretation " even though what you say is correct my faith on schools is broken and it can never be sealed again man

@user-wm2bt5ng6n Год назад

Apple making proprietary roads for apple car

@LeoStaley 2 года назад

Hey, the next video is gonna be about gears! Yeah your videos are absolutely on par with 3b1b. I say this as an educational RU-vid junkie.

@parthibhayat Год назад

I for a while thought it was like 3b1b related lol. It was this good

@dokudenpa8207 Год назад

lol I was about to ask if the part at 18:55 is somehow related to the spur gear teeth profile

@arkdotgif Год назад

3b1b is definitely a channel to look up to but i don’t think comparing every maths channel with it is a good idea

@TonboIV Год назад

@@dokudenpa8207 18:55 IS a gear and rack, except that there are no lands, and the pressure angle is 45 degrees. A real rack will have a more vertical angle on the tooth faces (pressure angle) such as 20 degrees, and there will be flat sections (lands) separating the faces at the top and bottom. The principle of a spiral section rolling on an angled line will be the same though.

@citratune7830 Год назад

@Larry Brin Ffs its so annoying its like comparing every youtube creations channel to mark rober. Not everyone is 3b1b or mark rober and you shouldnt treat them like so.

@realcygnus Год назад

Priceless content ! Many years ago I was playing around with this(mostly just the regular polygon cases anyway), & there was VERY little information available about it, especially in one place. If there was, I never found it. I did eventually manage to solve those rather simple cases. But you took this lightyears beyond anything I ever even imagined, which is so cool. Really great work ! 👍

@morphocular Год назад

That's awesome! I'm really glad you got so much out of the video! Just thought I'd mention I've included the main source I used for this video in the description. If you're still looking for resources on this topic, definitely give it a look!

@dovos8572 Год назад

you can find more about this in the gear section of math. it is the problem of what shape the linear gear needs so that a given round gear can roll on it and transfer its energy efficiently.

@realcygnus Год назад

@@dovos8572 makes sense 👍

@Blox117 Год назад

the problem with any wheel that isnt a round circle, is that if they get out of sync with the road then it completely fails

@FRK_WasTaken Год назад

@@morphocular vgv v

@sf_board9924 Год назад

never have i been so interested in geometry in my life. this guy taught me more in one video about shapes than i have ever known

@guestive 2 месяца назад

Geometry Dash

@nate5862 Год назад

I'm so glad we finally have a method for making the driving experience bearable in Oklahoma, thank you.

@IONATVS Год назад

The final challenge, two wheels rolling against each other, is the classic problem of Gear design-except you can change the shapes of BOTH curves to make the “ride” as “smooth” (in this case, where there is always a contact point applying constant torque throughout the rotation, so not exactly the same problem, but a related one) as possible, with the most popular solution for in-plane gearing is the spur gear, with repeating teeth made from 3 sections: the walls which actually perform the contact are made from involute curves (the curve traced by the end of a string as it held taught and unwound from a circular spool), connected piecewise with other short curves (the tips of the teeth and trenches between them), generally computed numerically like the elliptical integrals, to avoid the problem you mentioned with triangular and cardioid wheels-two involute curves too close to each other would not be able to roll over each other in practice without help, but by having a trench and a point in a different road-wheel pair of shapes you can ensure there is always a point of contact on the involute curves and any contacts through the other parts transfer no torque.

@jbay088 Год назад

Absolutely! And the road design problem is analogous to designing a rack-and-pinion gear system.

@paizdoto 2 месяца назад

cant you just do that by spacing both wheels so that they start at the same place on different items that are the same shape?

@jmiki89 Год назад

As for the questions at the end of the video: I didn't do the math, but I have a hunch, that the main difference would be that instead of the "vertical alignment property" it would be a "radial alignment property" meaning the axle and the contact point are collinear with the center of the road wheel. The other big change is, I think, that for the coordinate system of the road an other polar system would be useful instead of a cartesian one. Great video, btw.

@rpyrat Год назад

11:33 makes for a pretty fun screenshot when taken out of context

@Rin8Kin Год назад

One note though - they are THEORETICALLY ideal wheels in IDEAL experimental environment. Also it assumes that vehicle is PULLED by something along the road, while wheels just keep the vehicle horizontally stable. If you calculate what road will be ideal for DRIVING wheel, the shape of road will be different.

@melo3101 Год назад

Just to understand, how driving and pulling would differ from one another ?

@egebozdag9894 Год назад

@@melo3101 In pulling a force is applied to car. Driving involves wheels(car) applying power to the road. Which involves friction between surfaces and in the extreme cases (most of the roads in the video) wheel pushing the surface when it perfectly fits.

@Blox117 Год назад

circle wheel has the least resistance

@johnacetable7201 Год назад

Do you understand difference between math and physics/engineering? Because here we're talking about imaginary world where perfect objects can exist.

@Rin8Kin Год назад

@@johnacetable7201 "THEORETICALLY ideal wheels in IDEAL experimental environment" should have give you the clue i do understand it.

@thefullestcircle Год назад

I'm now wondering if you could take a shape with rotational symmetry, find its road (using the point of symmetry as an axis), then adjust the depth of the road until you get a wheel that doesn't have rotational symmetry around its axis anymore to find a "prime" version of the shape. Great video!

@daffa_fm4583 Год назад

oh hi fullest

@smurphas6119 Год назад

this sounds genius

@Arrow-Pointer Год назад

10:52

@thomasgyting3251 8 месяцев назад

It would always have rotational symmetry. As the road depth approaches negative infinity the wheel approaches a perfect circle, and as the road depth approaches zero the wheel becomes an infinitely elongated ellipse.

@mathematicalmachinery7934 Год назад

That 6:04 animation between “foci” and “focuses” got you a subscriber. That was cool.

@ethanos6868 Год назад

Anyone else watching this because it showed up on recommended even though it’s not anything to do with your normal content recommendations?

@dcharliefox45 Месяц назад

No. I'm a nerd

@Eacapple 10 дней назад

@@dcharliefox45same

@shilika1905 Месяц назад

This video was 22 minutes of pure joy

@1_1bman Год назад

i love the style here! honestly, it gets kind of repetitive seeing the same 3b1b visual style on tons of math videos, you're putting effort into giving it a cozier feeling more fit to your own style of teaching and i am noticing and appreciating that effort!

@NE0KRATOS 2 года назад

Omg, this video deserves millions of views, the maths and visuals are amazing! I wish you all the best and hope you’ll get the recognition you deserve! Less than 800 views right now is a crime! And when I started watching the video a few days ago it was less than half. Someone from the future please leave a comment when this video reaches 100.000 at least! Keep up the good work, I think we’ll see you among the big educational channels one day!

@gfoog3911 Год назад

A quarter of the way there

@fmga Год назад

@@gfoog3911 60%, looks like the algorithm is finally recommending this video!

@richardbullick7827 Год назад

@@fmga got 30,000 views in ten hours

@johnjelatis2033 Год назад

@@richardbullick7827 132k views total now, 15 hours later

@gfoog3911 Год назад

Nearly at 200%, I think it’s going viral

@anadice9489 Год назад

That "wheel" that came out around 11:00 got me picturing some sort of eldritch 5th-dimensional engineer. "I'm going to have to return this." "You asked for a wheel that works, it works." "It keeps breaking the minds of all the people trying to use it, which makes it not OSHA-compliant."

@droro8197 Год назад

Dude, that video was so cool. I never stop being amazed by the beauty of math and how complicated structures can arise from a very simple set of rules! Thank you for this content 🙏🏾

@duncanrobertson7472 Год назад

I'd be interested to see the equivalent for shapes of constant width, where the definition of a 'smooth ride' is having the top of the shape, rather than some fixed point, travel horizontally.

@blumoogle2901 Год назад

Doing the math for 2D objects of constant width, the ideal road seems to be is a straight line, and inversely for a straight line, you get not just a circle, but a constraint which implies a set containing all objects of constant width, and if you try to prevent clipping issues by adding extra constraints you get back an eclipse which has to be constrained to a circle to work and also some hideously complicated brain melting equations for the more complicated shapes which will work, sometimes.

@duncanrobertson7472 Год назад

@@blumoogle2901 Yeah, but what about shapes that aren't constant width, or a road that isn't flat, but restricted to the same smooth ride definition? E.g. what road would be required for a normal square for the top to be at a constant height?

@blumoogle2901 Год назад

@@duncanrobertson7472 Interesting question. I've not solved it, but starting intuitively, I'd start with regular polygons. A circle with the same radius as the distance from the centre of the polygon to the centre of each side fitted inside each shape. You'd then have triangles stick out over the fit circle. A cutout in the road with inflection points the same width but twice the height below the surface as that of the overlap triangle, connected with a brachistone curve should work to ensure that the highest point of the wheel stays at the same y coordinate throughout its motion. If you construct a piece-wise equation for the road in terms of these triangular overlaps and the radius of the fitting circle, you should get something pretty general. I'm not sure if the equation would be pretty though.

@SOTminecraft Год назад

@@duncanrobertson7472 I'd say you would have to replace the radius r with the diameter d passing thought the contact point (end ending at the antipodal point). In the case of a wheel with an axial point, the diameter can simply be the line passing thought the axial and contact point. If you don't care about having an axial point then d would simply be the difference between the height of the contact point and the point you want to have a constant height. Or hell, even more general: if you want to have a point (px, py) at a height h at a particular angle and your contact point is (cx, cy) (what ever the reference frame, the wheel's frame would be simpler) then d is just the translation you have to make relative to a contact point at height 0: d=h - (py - cy). Now ofc, when you don't have an axial point, the equation we have the rotation becomes invalid. So you'd have to figure out a new one, as the angle is needed to determine (py - cy).

@Aodhan2717 Год назад

@@duncanrobertson7472 intuitively, if you ‘rolled’ a square on a flat line, and traced along the highest point of the rolling square, the result would be a mirror image of the road you’re looking for

@Shendrift Год назад

I’d love to see wheels that intersect themselves, but still form a closed loop, like hypotrochoids.

@goldensparrow864 8 месяцев назад

I’d love to see a cok/ball road

@gastonsolaril.237 10 месяцев назад

This is actually incredibly useful for designing gear and pinion rack mechanisms with varying torques. Thank you VERY much!

@overanalyzed5258 Год назад

This is a fantastic quality video in both animation, demonstration, and explanation style. I particularly like the trial, exploration, feeling that arises from teating equations and getting unexpected results, then describing them.

@JammyRSCL Год назад

I love how at 1:43 it says ”Please stop”

@LeoStaley 2 года назад

I don't remember how I found this channel but I'm glad I turned on notifications, because this is fantastic.

@AlryFireBlade 9 месяцев назад

Some explanations where way over my head, but all these showcases where so satisfying to watch!

@Scyth3934 Год назад

Loving the production quality of this video.

@PMX Год назад

What if we add as an additional constraint for a "smooth ride" that the horizontal speed has to be constant? Does that limit the possible valid combinations to just a circle on a plane or are there other shape/road combinations that still work?

@d.l.7416 Год назад

Keeping the speed of the axle constant is easy, but keeping it constant to other potential speeds does limit the shape. There are three "speeds" I could thing of: speed the axle moves at (dx/dt), rotational speed (dθ/dt), and speed moved along the surface (as in measuring distance along the surface) Each can easily be constant on its own, but in combination there are limitations. If rotational speed and horizontal speed are both constant, dx/dt and dθ/dt are constant, so r is also constant so its a circle with the axle at the center. If we then add the surface speed a circle still works. For constant surface speed and horizontal speed, we need that surface length / horizontal length is a constant (since d(surface length)/d(horizontal length) is constant). That means the road must be made of lines with a slope ±some constant. So the triangle wave as a road works, and theres a bunch more. So the wheels are made of parts of logarithmic spirals with the same base, r = b^θ For constant surface speed and rotational speed, we first see that "distance along surface" is the same as "distance along the shape" because there is no slipping involved. So we need arc length / θ to be constant, and the only shape that works is a circle that passes through the origin, r=sinθ. So if we build our wheel out of parts of this it works. The corresponding road is made of parts of semicircles.

@KIWI_DUDE. Год назад

Sound to me like you could have any wheel you want, but the axel has to be in the middle of the wheel. No focus point.

@leoaso6984 Год назад

4:30 "They line up pretty well, but if I zoom in, you can see they don't coincide perfectly" Me, an engineer: I don't see the problem here 🤷

@generalgentry8879 Год назад

I love both the explanations and the animations hand in hand with each other

@16rosati 5 месяцев назад

These graphics are incredible. Amazing work

@HelPfeffer Год назад

5:20 For a moment I thought "What about a circle? 🤔" XD

@alexiadamasceno1255 9 месяцев назад

circle wheel would work on a flat road

@zxuiji Год назад

2:31, Never thought I'd say this but I never wanna see a pair of testicles roll again

@KingPenndragon Год назад

Absolutely love these kinds of videos, randomly stumbled upon this today and I'm beaming lol

@yashrustogi2156 Год назад

firstly I would like to wish you well and to say a huge thank you for uploading these videos as they have been an invaluable resource to

@MooImABunny Год назад

Slight correction on 6:34 Kepler's laws say the planet's orbit is precisely an ellipse. Newtonian mechanics agree that if you only have two spherically symmetric objects (which is a fair assumption) then this rule keeps holding exactly, with the caviat that the more massive object also spins in an elliptical, counter trajectory. The first complication comes from adding more objects, which, when you consider the fact that the planets make up only a 1/1000 of the solar system's mass and are pretty far apart, it's still a pretty fair assumption to ignore this. The second complication, which honestly applies to Mercury only, is relativity, which is still a tiny effect for most objects in the solar system. But technically, Kepler's (i think 1st?) rule states that planets orbit the sun at exactly an elliptical trajectory

@dovos8572 Год назад

well they don't have exactly eliptical trajectory because the trajectory of earth is slowly rotating around the sun too. that means that the earth isn't exactly at the same point where it started each year.

@MooImABunny Год назад

@@dovos8572 yeah sure, it only takes 112,000 years. I didn't say the orbits are exact ellipses, though for the earth it's a pretty darn good estimate for human time scales. I said that Kepler's laws don't take into account those other forces, so it says the orbit is a perfect eternal ellipse. Once you add other pulls (and maybe GR but I think its contribution would be comedically small here) you find an orbit that precesses every 112 ky, and maybe changes in other ways as well (More precisely it takes the trajectory this much time to finish a precession cycle, meaning after 112 ky it comes back to as it was).

@Thefuzzunderthechair Год назад

Now do a circle!

@alexandernalo8247 2 месяца назад

It's a straight road

@King_Melon165 2 месяца назад

@@alexandernalo8247REALLY???

@brickinatorstopmotions1445 Месяц назад

@@alexandernalo8247THAT IS SO SHOCKING

@evanseifert8858 Месяц назад

@@alexandernalo8247 The Joke -> Your head ->

@BilliOnBoB Месяц назад

@@alexandernalo8247r/wooosh

@danelyn.1374 Год назад

this is a beautiful problem and I'm really happy to see it done both ways after seeing the previous video to this... really beautiful!

@_P_a_o_l_o_ Год назад

This topic is so refreshing. Thank you for your videos!

@F17A Год назад

2:25 balls

@VoidGravitational 3 месяца назад

😳

@SamLabbato Год назад

Could you take this to a 3d space?, instead of just a 2-way road, could you develop a weird road shape that could be driven on from any angle and turned on at any point? I imagine you'd lose the ability to contact with every point of the road at a time like you see here and would have to rely on multiple points or geometric shapes balancing the weird wheel shape.

@Yoel202 Год назад

I imagine that in a 3D road with those characteristics you would make cube wheel riding on its tips to take the most advantage of it.

@anselmschueler Год назад

likely not because of holonomy

@lucasloh5726 Год назад

@@anselmschueler what’s holomony?

@cephalosjr.1835 Год назад

@@lucasloh5726 In very oversimplified terms, holonomy is when you lose the data of an object by transporting that object along a closed loop. For instance, transporting vectors along a triangle on a sphere can alter their direction based on the size of the triangle.

@matts3178 Год назад

Picture a log lying on the ground aligned north-south. (You can use a pencil as a makeshift log.) Roll the log one log-length east (rolling normally) and then "roll" it south. It's now standing on its end. Now, "roll" it south then "roll" it east. First it's on its end, and then it's on its side, now aligned east-west. In both cases the log is in the same spot but contacting the ground in a completely different way. When you allow the extra degree of freedom in the form of movement in an extra dimension, your shape can end up above any given spot in an infinite number of orientations. To roll an arbitrary non-uniform 3d shape on a perfect road plane, you would need that plane to have infinitely many shapes at the same time.

@Andrew-McCormick Год назад

There’s 20-some minutes of my life I’ll never get back, and yet I don’t mind too much. Thanks!

@leonardoteixeira3314 Год назад

tears droped from my eyes with this video... just keep doing it. Thanks

@skull1495 Год назад

6:35 : A hungry sum operator is floating around in the upper left-hand corner

@morphocular Год назад

I was wondering if anyone would ever notice :)

@Vex_The_Vexillologist 7 месяцев назад

@@morphoculari did!

@_lunartemis Год назад

I'm not a math person but I like what I see here. Hopefully someone can make a program where you can draw your own shape and it calculates the perfect road for that shape.

@TotFr1 Год назад

It seems like it would be easy but I have no coding experience

@versalgraphics Год назад

that exists

@TantalumPolytope Год назад

@@versalgraphics whats it called?

@versalgraphics Год назад

@@TantalumPolytope Vsauce used it in his video on the brachistochrone, can't remember the name though

@hanR6 Год назад

i love watching these just because its interesting, i dont understand most of it because of a bunch of formulas and blah blah but its still entertaining to watch or just to have it on in the background

@christiangray7826 Год назад

First video like this I’ve seen, and I correctly predicted the “sawtooth” wheel for the zigzag road. Nice!

@phoomsgamingvid8943 Год назад

At 10:49 Morphocular: **uses bot to create elipse** The bot: Take this Music: **stops** Morphocular: Wait, what?

@TheBarcelonianRepublic 17 дней назад

This is not an ellipse.

@the747videoer Год назад

that "sawtooth" wave is actually a triangle wave. also fun fact, parabolas are what transportation engineers use for changes in vertical alignment on roads. so parabola shaped roads are real and every time you go over a crest or sag in the road, that section is pretty close to a parabola

@Baer1990 Год назад

The animations are fire! Very nice video to watch

@andreaalflavendett Год назад

Wow, that video is amazing!

@timothymclean Год назад

19:10: I nominate the two-petal collection of logarithmic spiral segments as the weirdest wheel in this video.

@rexperverziff Год назад

3:23 MAGIK TRIANGLE: Clips through stuff. ATK: 69 DEF: 69

@cvjlmaker9890 Год назад

This is some 3 brown 1 blue type stuff, amazing!

@RailandOak Год назад

This is incredible and I can't wait for the next video! I have looked for this type of information for YEARS. Having (regretfully) never taken trig or calculus I hardly even know how to search for such information. Being the simple person that I am, I would assume that for a wheel to follow a road that is a circle you would need to have a radial alignment property instead of a vertical alignment property. I am quite certain it is much more complicated than that but that's about as far as my smooth brain could get me! ha!

@fritzyberger Год назад

The issue with the triangle is very similar to gears and their required backlash. It is always awesome to learn a little mathematics

@EXA1024_ Год назад

im really interested to see more shapes like the one at 0:17 great video!

@morfie8209 Год назад

2:20 those eggs are way cooler

@lorenzodepaoli3642 Год назад

7:38 I hope you don't mind me screenshotting this part to show to my math teacher, it's just so mathematically perfect

@rogofos Год назад

I did not expect to burst out laughing in a math video but this "wheel" (that was supposed to be an ellipse) really got me

@reidflemingworldstoughestm1394 Год назад

That parabola relationship is interesting. I found out that as you continuously change the value for B for the parabola Ax^2+Bx+C, the vertex of the parabola traces out the parabola -Ax^2+C.

@praloedra6738 9 месяцев назад

What...?

@cobrav3n0mx78 Год назад

Here i am having failed calculus h a r d but still watching 22min of wizardry maths about wheels.

@robdenteuling3270 Год назад

Amazing video! Subscribed and looking forward to the next one. Keep up the good work!

@finnberuldsen4624 Год назад

This is one of the coolest videos I've seen and i hope i never forget it.

@ALFAGamer92 Год назад

why did you draw rooling balls at 2:20?

@sirreginaldfishingtonxvii6149 Год назад

I don't much like maths, not at all, but this was very enjoyable. Great video mate!

@Markusgebvor Год назад

Amazing work. Thanks a lot.

@acecabezon Год назад

Wow, videos like this are why I love RU-vid!

@mockemperor953 Год назад

15:15 This is also interesting because this is how gear ratios work

@chris1to1pher 11 месяцев назад

w ratio

@maxybg 11 месяцев назад

19:49 I feel like the video editor painted the cardioid with a skin tone on purpose

@nielsniels5008 Год назад

Thank you for this video. I really liked it. It was really basic and I didn't have to think because I just ignored the details of the math. This is perfect RU-vid content, just wheels going weeeeee

@bernat8331 Год назад

Underrated channel. Keep doing this and you will have great success

@PixelBytesPixelArtist Год назад

The logarithmic spiral is also the involute of a circle. The involute is used in engineering to produce the most effective gears. You basically just reinvented the rack and pinion.

@wernerfritsch6436 Год назад

20:47 If I remember correctly y=1/4 is the focus of the nomal parabula. The parabula reflects all vertical lines into that point.

@charliedulol Год назад

most genuine "wait what" i've heard in a long time.

@Broockle Год назад

Love these animations 😃

@idontknowledge-real Год назад

10:41 Ah yes the casual sanity check

@roboltamy Год назад

Awesome video! I was wondering: does the flower fit the sawtooth without bumping into it? It seemed like it might intersect it as it gets towards the points.

@AdrianHereToHelp Год назад

One of the questions I was interested in after the first video was whether these solutions were unique. I never would have guessed the solution would be axle height! (though in hindsight it seems so obvious) Loving these videos so far; just a great bit of maths communication with cool and interesting applications.

@samueldeandrade8535 9 месяцев назад

I'm glad I found this channel again. For what I watched in my life, this is the best math channel.

@cheetah219 Год назад

Curious to see what a road looks like when considering more than one wheel on a road. I would imagine very similar to what you're showing but when looking at applying the formulas, we obviously use 2 or 4 wheels on vehicles the most (bikes, cars, motorcycles). So taking this idea and expanding this to 3D (length, width and height) for more than one wheel. Very transferable but turning the idea onto application

@andynz7 Год назад

What is this mystical construction at 6:12?!! I've haven't seen foci determined like this, what is this? Teach us more about elliptical foci!

@energyeve2152 Год назад

Great animations and great content. Thank you for sharing!

@neverdaddy 9 дней назад

This is the type of videos that make me interested in science and math . ❤ really brings out the curiosity to learn

@karolakkolo123 2 года назад

Just one thing I'd like to point out. For the elliptical wheel, you said how the elliptic integral has no nice closed form. Well, arguably, the standard trigonometric functions don't either. But we accept sines and cosines as elementry functions. In my opinion, we should accept elliptic integrals and elliptic functions as elementary functions as well. They have so many parallels between trigonometric and hyperbolic functions that it's a sin imo that they are not usually included in the elementary function set. After all, that definition is arbitrary to a certain extent Edit: the only explanation I can find for why they are excluded is the fact that elliptical functions generalize both the circular and hyperbolic functions, and so their derivatives and integrals are harder to compute or see. Also, besides the elliptical sine and cosine (sn and cn), we also have a dn function. This makes up for a total of 12 elliptical functions, two for each combination of the letters s,c,n,d in their name. Anyways, it would be interesting to see a video on elliptical funcs if that's possible!

@morphocular 2 года назад

That's a fair point. I think it's standard practice in the math world to trash talk elliptic integrals, so I thought it'd be funny to make my reaction deliberately over the top.

@karolakkolo123 2 года назад

@@morphocular ah I see. It was funny! But I just felt like I had to make that comment

@angeldude101 2 года назад

Given how much I disliked integrals for the amount of magic in them, the fact that trigonometric integrals would be as hard as elliptic ones if they weren't treated as elementary feels oddly fitting. Granted, looking at the wikipedia page for elementary functions, it includes the exponential function and compositions, which would still make sine and cosine "elementary" anyways.

@TheOneMaddin Год назад

While I would like to agree, I see some point why they are not elementary. As far as I know, there are not 2 or 3 elliptic functions, but infinitely many, right? Because they involve a parameter. But please correct me. Second, we have to draw the line somewhere. We could als make erf elementary. And sinc, and integral sine etc... but that would defy the purpose of "elementary"-ness

@evanev7 Год назад

You can form cosine and sine from complex exponentials, which afaik you can't do with elliptic integrals. While I'm not completely against accepting them as elementary, cosine and sine do feel more elementary as you can derive them only from exp.

@jdwg5 Год назад

Given your last sentiment about other definitions of "smooth ride", I would be curious to see what would change if you set the defining characteristic is a smooth ride to be a constant axle velocity as opposed to a fixed axel "height" (not relative to the road)

@victortitov1740 Год назад

you'll get gear profiles that are used in actual mechanisms

@middyyyy Год назад

i appreciate your effort and our comment interest in wheels and roads. thank you so much for sharing this.

@erikchung129 Год назад

Really good video! Thanks for the in-depth explanation!

@sdspivey Год назад

For a smooth ride, I would also require the wheel to spin at a consistent rate.

@bastienpabiot3678 Год назад

This is too much constraint and only circles would work in this case

@christophersavignon4191 Год назад

@@bastienpabiot3678 Which is a big reason you don't ever see wheels that aren't circular. I also don't think it's too much constraint to have the conditions for a smooth ride actually be the conditions for a smooth ride. Yes, the experiment becomes pointless (or academical) if you hold it to viable standards. That's not a problem with the standards though.

@kipchickensout Год назад

18:58 interesting how my mind exactly came up with this shape when the sawtooth road was first shown edit: ayy also predicted the parabola

@idosiegler1194 Год назад

Thank you so much for all these tutorials bro. So much valuable knowledge

@luizhenriquegarcia3186 Год назад

Kudos man. You kept it very simple and helped make the first steps in soft soft. Very Helpfull! Thanks!

@allan8910 Год назад

20:09 looks painful

@BeekersSqueakers Год назад

I just realized you could use this same concept for create unique gear sets. Select a weird geometry for the first gear, determine the ideal "road", then use the inverse of the road to determine the geometry of the second gear. If match them up in the animation above and below the road, they would always contact each other at every point of their profile. (Caveat being the case of the triangle example)

@paulholleger8538 Год назад

Super interesting video! When you posed your question at the end about another way to measure a "smooth ride", my mind immediately went to constant horizontal velocity over the road, rather than constant axle height above the road. For circular wheels with a center axle, this is super uninteresting. But for some of the other shapes and axle placements, it could be fascinating. Maybe. I'm not sure if the solutions would be interesting or just annoying, but the problem intrigues me.

@cellokid5104 Год назад

I really enjoyed this video

@Wagon_Lord Год назад

15:50 Ok now raise the road just a little bit more... I MUST KNOW

@notHepterice Год назад

I think it just collapse on itself

@timothyvandyke9511 Год назад

12:22 does the fact that the ellipse's axel isn't centered have anything to do with it? I notice every shape that has been spit out so far DOES seem to have centered axels