How to Design the Perfect Shaped Wheel for Any Given Road

[@morphocular]

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]

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

[@UltraLuigi2401]

@@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.

[@DuckPerc]

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

[@DuckPerc]

@@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]

@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]

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

[@Rudy97]

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

[@zrajm]

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]

Is there even any other solution than a circle?

[@Visch8826]

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]

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]

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_]

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

[@nate5862]

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

[@LeoStaley]

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

[@parthibhayat]

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

[@dokudenpa8207]

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

[@arkdotgif]

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]

@@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]

@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.

[@sf_board9924]

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]

Geometry Dash

[@realcygnus]

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]

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]

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]

@@dovos8572 makes sense 👍

[@Blox117]

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]

@@morphocular vgv v

[@IONATVS]

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]

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

[@paizdoto]

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?

[@thefullestcircle]

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]

oh hi fullest

[@smurphas6119]

this sounds genius

[@Arrow-Pointer]

10:52

[@thomasgyting3251]

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.

[@user-wm2bt5ng6n]

Apple making proprietary roads for apple car

[@kieranlovescats]

lol

[@Irish_Mann]

Why am I so invested in this? I will never use this, but I can't stop watching videos like these.

[@ethanos6868]

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

[@dcharliefox45]

No. I'm a nerd

[@Eacapple]

@@dcharliefox45same

[@TinyかわいいGamer]

Yes, except i watched this video before.

[@heha_discord]

Yes

[@user-gc8hw7lp7c]

Ya

[@jmiki89]

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.

[@zxuiji]

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

[@mathematicalmachinery7934]

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

[@Traxxya]

I have that number dyslexia thing, I forget the actual word for it but all in all, I hate math It confuses me so much, even when I think that I'm following along I suddenly get lost. And honestly, that doesn't change for this video, I'm still really lost and confused But the way you're talking about the math so simply doesn't make me feel bad about not understanding it! I actually really enjoy watching and listening to you go through the problem solving parts! Even if I don't actually know whats going on I think this is the first math related thing I've enjoyed since middle school, this is so awesome

[@sussusultimaniusdestructus5214]

I think it's called discalcula, but do correct me

[@NE0KRATOS]

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]

A quarter of the way there

[@fmga]

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

[@richardbullick7827]

@@fmga got 30,000 views in ten hours

[@johnjelatis2033]

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

[@gfoog3911]

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

[@gastonsolaril.237]

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

[@Rin8Kin]

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]

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

[@egebozdag9894]

@@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]

circle wheel has the least resistance

[@shy_dodecahedron]

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

[@Rin8Kin]

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

[@Thefuzzunderthechair]

Now do a circle!

[@dry_out444]

It's a straight road

[@King_Melon165]

@@dry_out444REALLY???

[@Bricksmappingandmore1445]

​@@dry_out444THAT IS SO SHOCKING

[@evanseifert8858]

@@dry_out444 The Joke -> Your head ->

[@BilliOnBoB]

@@dry_out444r/wooosh

[@anadice9489]

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."

[@christiangray7826]

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

[@1_1bman]

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!

[@Mythinull]

quite the interesting color choice for the cardioid. if you had chosen red or pink you could make the argument it does look like a heart, but you made it... a ballsack

[@duncanrobertson7472]

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]

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]

@@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]

@@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]

@@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]

@@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

[@AlryFireBlade]

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

[@LeoStaley]

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

[@ahbi-star]

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

[@droro8197]

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 🙏🏾

[@bernat8331]

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

[@MooImABunny]

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]

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]

@@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).

[@samueldeandrade8535]

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

[@SamLabbato]

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]

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]

likely not because of holonomy

[@lucasloh5726]

@@anselmschueler what’s holomony?

[@cephalosjr.1835]

@@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]

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.

[@leonardoteixeira3314]

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

[@rpyrat]

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

[@_P_a_o_l_o_]

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

[@PMX]

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]

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.]

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.

[@overanalyzed5258]

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.

[@Shendrift]

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

[@goldensparrow864]

I’d love to see a cok/ball road

[@yashrustogi2156]

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

[@_lunartemis]

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.

[@realdubfr]

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

[@versalgraphics]

that exists

[@TantalumPolytope]

@@versalgraphics whats it called?

[@versalgraphics]

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

[@rogofos]

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

[@sirreginaldfishingtonxvii6149]

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

[@lorenzodepaoli3642]

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

[@leoaso6984]

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 🤷

[@paulf3999]

Feels like Stan from South Park grew up, watched the shadocks and finally became a youtube mathematician.

[@cobrav3n0mx78]

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

[@finnberuldsen4624]

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

[@HelPfeffer]

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

[@alexiadamasceno1255]

circle wheel would work on a flat road

[@-CookieDev-]

i have no idea what you are talking about but i love how the shapes are cool and they role i am amused easily

[@EXA1024_]

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

[@morfie8209]

2:20 those eggs are way cooler

[@RailandOak]

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!

[@phoomgaming]

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

[@generalgentry8879]

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

[@andynz7]

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!

[@neverdaddy]

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

[@jdwg5]

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]

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

[@SoupEarthOfficial]

I watch this whenever I’m bored

[@spothix]

No way, a SES comment with no popularity (number lore fans arent actually into numbers!!!)

[@reidflemingworldstoughestm1394]

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]

What...?

[@shilika1905]

This video was 22 minutes of pure joy

[@ALFAGamer92]

why did you draw rooling balls at 2:20?

[@BudgetCat164]

Bro stop being so dirty minded it’s just a random shape he made

[@alextheawesomeone4662]

@@BudgetCat164 id say it looks like one of those bread rolls my mom makes

[@paulholleger8538]

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.

[@the747videoer]

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

[@alexmiron3296]

Swear the god , some of this stuff is so easy to see i cant describe.

[@NORMAN-HATES-YOU]

Visual...

[@skull1495]

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

[@morphocular]

I was wondering if anyone would ever notice :)

[@Vex_The_Vexillologist]

​@@morphoculari did!

[@16rosati]

These graphics are incredible. Amazing work

[@maxybg]

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

[@acecabezon]

Wow, videos like this are why I love KZbin!

[@fritzyberger]

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

[@TheDukeOfThunder]

I like your funny words, magic man.

[@karolakkolo123]

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]

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]

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

[@angeldude101]

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]

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]

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.

[@danelyn.1374]

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!

[@JammyRSCL]

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

[@randomcooldud]

WAIT IT DOES 🪑🪑🪑

[@luizhenriquegarcia3186]

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

[@BeekersSqueakers]

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)

[@Scyth3934]

Loving the production quality of this video.

[@Egg-ehg]

Imma pretend that I grew smarter

[@gurgleglug]

Waresq sqpot

[@anujbangad3973]

wassup smarty?

[@Deficard]

what-

[@anchlama2786]

Egg

[@klaasbil8459]

6:01 I never saw this procedure with two intersecting circles to find the foci of an ellips. What a neat thing!

[@PixelBytesPixelArtist]

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.

[@astro_cat030]

Ah yes more interesting youtube content that makes my day and be satisfied.

[@rexperverziff]

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

[@ecernosoft3096]

I’d love to see a program where you could draw a wheel and place its axel and have it generate a road for you.

[@timothymclean]

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

[@KingPenndragon]

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

[@JoJoDo]

I kind of have a few problems with some things: 'Smooth' (as defined in the first video) isn't exactly smooth, as the axle point is clearly accelerating and decelerating constantly, so a normal car engine wouldn't work. Also, there are multiple shapes for a given underground, though right? And I don't mean underground height, but primarily axle position. It changes the underground massively, so why wouldn't that work reversed?

[@shlak]

well rotation means a normal car engine would work as rate of rotation is constant.

[@JoJoDo]

@@shlak I don't think that's true? It might be... I don't know actually. I would like to see graphs of rotation over time and axle speed over time.

[@d.l.7416]

The actual speed at which everything moves is arbitrary, you could make the horizontal speed constant, or you could make the rotational speed constant. (it depends on what exactly θ(t) and x(t) are, all the video does is look at relations between them) If you want to do both at once only a circle would work, since dθ/dt and dx/dt are both constant so r must also be constant.

[@JoJoDo]

@@d.l.7416 that's what I wanted to hear...

[@karelpgbr]

Normally I detest maths, but videos like these intrigue me, I kind of understand it, but also very don’t haha! Great video mate

[@GibusWearingMann]

I'm curious what road/wheel combinations satisfy the additional property that horizontal speed and rotation speed are both constants, in addition to the classic circle wheel/flat road. Edit: I checked the comments of the other video and apparently only circular wheels do that!

[@dovos8572]

not exactly. well depends on the definition but round gears satisfy it too if the n in 2pi/n is big enough. these calculations gets used indirectly when calculating linear gear shapes for given round gears.

[@staplif]

I cannot say that I even understood half of this but it certainly was entertaining!

[@cheetah219]

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

[@aidenorr9536]

Alright, this is the first video I watched of this channel, and yet somehow you made geometry not repetitive and boring. Like, in school, I would be constantly writing down formulas, (mostly polygons that could be divided into different ones), and it got boring, really fast. Just constant “(pi*r^2)/2, (3.14*x^2)/2, then solve it and add to others. At least we got calculators. But then you come along and teach me a part of probably high school math, within thirty seconds, and my honors algebra topics brain went, “wow that’s cool.”

[@kipchickensout]

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

[@thatrandomoctoling]

That little “evil Sigma notation” floating about in the top left corner at 6:36 is just so funny to me

[@bonkser]

4:35 *you showed a sine wave *but it's not the same *boy! what a shame!

[@AshrellStudios753]

DANDILION

[@charliedulol]

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

[@sdspivey]

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

[@bastienpabiot3678]

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

[@christophersavignon4191]

@@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.

[@acreman1259]

the length of the road can be proportioned by the circumference of the circle

[@blinded6502]

21:32 If you match your reference frame to that of a line between two axils, you'll see, that these two wheels act like two gears. And now, suddenly, this problem has a real life significance.

[@Andrew-McCormick]

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

[@idontknowledge-real]

10:41 Ah yes the casual sanity check

[@janewilson8029]

anyone else here not for the math but the satisfying of the shapes going smoothly across the surface