Curves we (mostly) don't learn in high school (and applications)

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[@kattenelvis1778]

You can really see that this guy is an engineer since he keeps talking about the practical uses of math.

[@anteconfig5391]

I don't care what his reasons are. It's helpful for me to learn something if I can at least picture myself using that knowledge.

[@kattenelvis1778]

@@CrasterFamily He could be one, but he majored in Electrical Engineering

[@HighestRank]

Chef 11:49

[@5Djoe]

Practical Vs Theoretical strengths of math pls??

[@ignaciosavi7739]

Lol tru

[@jp4431]

When I was a teenager, I definitely discovered all sorts of curves via certain videos

[@poopfart65]

😳

[@juliaanimates9765]

actually, this is about math. not certain videos.

[@avory7938]

@@juliaanimates9765 no, it's about certain videos. Don't be fooled

[@cameo2277]

@@juliaanimates9765 don’t be fooled Julia

[@Callie_Cosmo]

@@juliaanimates9765 yeah Julia, this is secretly a sex education video, always gotta stay vigilant for things like this

[@raghavkamath]

*I mean who doesn't like them curves?*

[@datchnac4577]

That's why we got this natural instinct to study all of them!

[@randomdude9135]

@@datchnac4577 I like how Evolution and Mathematics are connected

[@deepasinghal4729]

Finally, after ALL THESE years, I now know why the hell the "curved line" on MS Paint moves haphazardly sometimes (being MS) on clicking the 4th time. It ruined so many of my creations and had to undo-redo the curve a million times - Bezier Curve. Researcher level Maths is required for arts, or a teacher who knows practical application rather than theoretical maths

[@chaotickreg7024]

Engineers are just artists that know math and have a job.

[@yomumma7803]

@@chaotickreg7024 wow, harsh bro lol

[@chaotickreg7024]

@@yomumma7803 lol you're right that was harsh. I respect artists as a backbone of every community, it just sucks that the art buying market is messed up and over saturated. Also plenty of artists can do math. I just think being able to combine practical engineering with design should automatically get you labelled as an artist of some sort.

[@AhmedMahmoud-tv9vw]

@@chaotickreg7024 At least engineers make useful stuff.

[@aurelia8028]

If you wanna learn more about how they're made and what other applications they have, go watch the video "The Beauty of Bezier Curves". Should be the top result.

[@anjaninator]

“Not applicable but definitely artistic”

[@ObjectsInMotion]

All my friends call me artistic! They say I'm so colorful I belong on the spectrum.

[@nonegiven3814]

His non applicable curves are hunted for in stocks charts.

[@genericusername4206]

@@ObjectsInMotion sameeee

[@oscarclereus5307]

0:22 Hey Vsause, Zach here

[@eggyrepublic]

When I learn math, I like to imagine everything to have some form of application, it's just that many of those applications haven't been discovered yet. After all, almost every math theorem were discovered purely for the sake of math and the applications found later.

[@NortheastGamer]

It's the year 5132 and a physicist has discovered that a strange tachyon particle is perfectly described by the batman curve authored by ancient bored highschoolers.

[@danin2013]

NortheastGamer lmao

[@yatexasnycaflnvnigga]

It just code make it develop and design that it they done this long time ago it was hard

[@yatexasnycaflnvnigga]

I was born in middle Get in calculus this is easy theory must theory get hardest in math ever cuvre or right scission

[@yatexasnycaflnvnigga]

It still think 3D we are thrid Dimension where not that far

[@HelplessGazellle]

More please. There isnt enough video content on non traditional school-math. You should make introductory type videos for different areas of math, like "what is topology?" Or "what is chaos theory?". You already go into a lot of that type of content, but it would be nice to have a good starting point into all of the different topics.

[@liebe1050]

2:39 Holy crap. In less than 30 seconds you answered a question I had for over a decade. Thanks!

[@techfreack8397]

@liebe please what was the question please

[@anshumanagrawal346]

What question?

[@NorseGraphic]

@@anshumanagrawal346 What are Bezier-curves? When that name came up, I knew what curves he was going to talk about, as in various 3D-programs you use these curves to give the shape you're looking for. The beauty with Bezier-curves is how it transitions from one type of curve into another type of curve without break-points or corners.

[@Djorgal]

As a maths teacher, I love Bézier curves. I use them all the time to create examples. You can easily use it to get beautiful smooth examples of functions with the values and derivatives you want. Very helpful for early calculus when you want your students to learn to read the slopes of tangents for instance.

[@royelhajj2612]

In the first curve ( 0:12 ) add some coefficients before x ,y ,xy ... (with sliders) the result is truely fascinating

[@RC32Smiths01]

Curves are an interesting concept. You definitely have to utilize more techniques to accomplish them, but they are wonderful! Cheers man

[@eddiemorrone870]

Thank you for explaining Bezier curves. I only vaguely remembered how they work. This clears it up.

[@hershmysson]

As a Graphic designer I've used Bezier Curves for years and years, but seeing the process behind them is marvellous, it's so cool.

[@eriktempelman2097]

Nice video. You can add that the structural benefit of a catenary arch is that it is either in pure tension (hanging down) or pure compression (standing up). So, no bending. This makes these shapes extremely efficient.

[@paladin1147]

Incredible content man, loving all these videos you have been putting up lately. I really appreciate the work you do here.

[@789radek]

What a great video. I always thought someone should talk about interesting curves in general as a youtube video, and the animations make the video a perfect execution of that idea:)

[@Colony28]

In Barcelona, they are building a cathedral currently (so far, it took just a century), authored by Antoni Gaudi. Gaudi had some amazing design ideas, one of which was hanging strings upside down, and putting weights in specific places. Like this, he built an upside down string model with the optimal arches for weight distribution - catenary curves. This string model can be seen there and I was totally taken away by that. I highly recommend visiting and seeing Sagrada Familia in person.

[@elhombre8636]

Hey Zach, love your videos! As a mechanical engineer who works on aerodynamic optimization I wanted to add another use of the Bezier curves (and parametric curves in general). In shape optimization (say we want to optimize the shape of an airfoil to get minimum drag) we often use Bezier curves to parameterize the shape and we move the control points to optimize it. We do that because if we optimized the shape by moving each node of the geometry, the resulting shape would not be smooth and pretty much impossible to manufacture. So there you go, have a great day!

[@AJ-et3vf]

Great content! I particularly love the Bezier curves! Fascinating and useful

[@robertschlesinger1342]

Excellent video. There is a great story about a challenge question published by Bernoulli. The challenge was delivered to Newton upon arriving home after a long day as Master of the Mint. He worked on the problem that night and sent the solution to Bernoulli without signing the paper. Bernoulli knew it was the work of Newton stating that one can discern the work of the lion by its paw print. The curve, as I recall, was a catenary.

[@dsdy1205]

Actually it was the brachistochrone

[@eliyasne9695]

12:30 *Everything* travelers along geodesics in GR as long as there is no force acting on it. (In GR gravity is more like a fictitious force that is a result of the spacetime geometry)

[@RG-pr5xx]

This was an amazing video. It reminded me of your conic sections video, which was equally incredible.

[@roygalaasen]

The first thing I thought when I saw the weierstrauss function was “hey! This must be useful in procedural generated worlds somehow”. But then probably not. It is intriguing enough that I want to hear more about it!

[@robertschlesinger1342]

Excellent and worthwhile video on curves of various types. A must see for everyone to view, especially mathematics and science students.

[@zaidyounas1602]

what i really like about this channel is that it gives us information about the practical applications of everything.

[@spaceCowboy924]

Geodesics are also widely used in flight trajectories as he shortest distance between two points on an oblate earth

[@digitig]

Not *so* much, because the route that takes minimum time or fuel is rarely a geodesic. Aircraft go out of their way to avoid headwinds and to get tailwinds, and there are elaborate tracking and monitoring systems to help them do that. Sure, the geodesic is still in there, but as the track they're deliberately deviating from, not the track they're flying.

[@cl0p38]

Jokes on you, everyone knows the earth is an hyperbola 🙄

[@jaykaku7646]

I absolutely love this channel coz of it's content, I literally clapped at the end 😭 Great Job!!!

[@roygalaasen]

I wonder how he manages to be so productive yet still so fresh

[@jaykaku7646]

@@roygalaasen I know right!! I hope school taught us Maths in this way, I am so sure students would go an extra step to learn , just tell them the applications. I had learnt calculus the Physics way, I mean calculus was introduced to me in Physics class and taught with applications.

[@kicivil8173]

I always stayed away from maths. some how I managed to crack all maths exam during my engineering and now after watching videos of your and other I thought we haven't learned anything or taught anything like this or this way. Keep it up.

[@illford6921]

I mean honestly all I learnt is that maths I thought wasn't useful usually explains maths that is useful

[@whyarewestillhere2244]

Just a few minutes into this video made me pause it and I started working on a Matlab script to reverse the Bezier curve calculation. It takes a curve and calculates the control points needed to get that curve back. With simple curves it works quite well. For example I made a perfect heart with an order 20 Bezier curve. Added a nice simulation with all the moving lines etc. It looks pretty awesome.

[@MikeAshelby]

I liked the engineering approach to these curves. I've always thought of the bezier curve as being the curve that solves the parametric cubic for 2 tangents (point and slope), but that's a much more mathsy way of thinking of it. I remember coding different ways to interpolate sets of points in pov-ray before it got anything clever to do it. Fun to try your own ways (I liked connecting sets of points with just quadratics: get some crazy results!)

[@andjelatatarovic8309]

This is hilarious! But I also feel it opened my mind as to the array of curves that exist! Thank you!

[@zenmonke]

just one word: amazing. i love your vids. please keep em coming

[@iamsauras]

Thank you Zach! This was a wonderful wonderful video. ❤️✨

[@yBazo82]

This channel is increasing in quality

[@Andrew90046zero]

That was the best explanation for the photoshop pen tool I have ever heard, thank you very much for that Also, my favorite curve is Perlin Noise :3

[@energyeve2152]

This is awesome! Thanks for sharing

[@ventriloquistmagician4735]

physics phd student here. It's nice to see such a practical video.

[@mikekramer7737]

Thanks. First curve can be plotted in Jupyter Lab with a contour plot (import numpy as np and matplotlib as plt) delta = 0.005 xrange = np.arange(-2, 10, delta) yrange = np.arange(-2, 10, delta) X, Y = np.meshgrid(xrange,yrange) F = np.sin(np.sin(X)+np.cos(Y)) G = np.cos(np.sin(X*Y)+np.cos(X)) fig = plt.figure(num=1,figsize=(12,12),facecolor = (0.9,0.9,0.9)) ax = fig.add_subplot(111) ax.contour((F - G),0) ax.set_title('Implicit Function') plt.tight_layout plt.show()

[@farhanmadilfaraz]

Your are doing a great job, please make a video on how do you draw figures, curves or other animations that you playing with. Thanks

[@topilinkala1594]

You might want to check this parametric set of curves: x = 2cos(t), y=2cos(nt) that creates n-degree polynomial functions that have nice local behaviour. I put the 2:s there because it makes the polynomials nicer as then the cofficient of the highest x-term is always 1.

[@aydmu]

Your channel is awesome. keep it up!

[@pi4313]

the infinite complexity of curvature is astounding - good video

[@rafaellisboa8493]

very cool video man, I love these

[@TheRuler89]

Thank you very much for the video and the recommendation of "The Secret Life of Chaos". A few years ago my friend told me about a video that talks about the chaos theory with the help of a beamer. I searched for the video for a long time and never found it. Now I have finally seen it, thank you very much.

[@LucasDuarte-jg5kr]

Continue the serie!!!! Awesome!!

[@scottekoontz]

That first formula is so cool. Typed it into a graphing app and you can explore it like a fractal.

[@Bigfoot_With_Internet_Access]

Hmm yes, these curves do seem pretty cool

[@tifahefendijagaming9606]

I didnt know bigfoots were into math

[@dondake3409]

why are you everywhere?

[@MAJORVBEZ]

lmao where u to next? Ryan's Toy Review??

[@Avinashkumar-ec9ov]

By the videos on which I see you you definitely seem like an engineer

[@math_the_why_behind]

Agreed!

[@neskhymontoya2096]

Your animations and pictures come in handy for me, in order to get a better understanding on The degree I'm signed "Electrical Engineering" here in Central America. Is one of the toughest degrees to get through here, roughly 5% of the students graduate from it. I encourage you to create a channel were you solve challenging aplication problems on Furier Series, Electromagnetism and electrict circuits. I'm subscribed to your channel, one further suggestion, I dont know if it is up to you, to set subtitles in "english" instead of "english autogenerated" customizable straight from youtube by the users or for those who view, it helps all foreigners whom tune your channel overseas and like me, have learnt english as a second language.

[@bjarnew9441]

Actually, and this is a really stunning result, The nowhere differentiable continuous curves over bounded intervalls are dense in the continous functions. That means, for every continuous function we find a nowhere differentiable function, similar to the Weierstrass funciton, that is as near to the function as we want. This is remarkable, because the set of nowhere differantiable functions seems to be so small, but it actually is big enough, that we only need to understand them in order to understand every continuous function..

[@LowtechLLC]

Thanks! I just learned how the Bezier curve works!

[@bpark10001]

3D Bezier curves are used for roller coaster tracks. Involute curve (not mentioned) is used for gear-tooth curves (to get uniform motion).

[@iangolsby8471]

Aren't the art (einstein, yoga, etc.) curves just fourier transforms? Those are super applicable to a lot of things like audio engineering and control theory! 3b1b has a super awesome video about them (it might even be a series I can't remember)

[@Tuxfanturnip]

The bit about calculus of variations reminds me of Story of Your Life, the short story the movie Arrival was based on, in which aliens perceive time as an optimized solution to such a problem rather than a series of events determined by causal rules.... it's a really interesting way to think about the world!

[@chrslb]

Awesome stuff!

[@dimaryk11]

Didn't even notice how fast the video flew by. Interesting stuff!

[@LethalChicken77]

When you said "you've probably never seen the (lemniscate) like this" there was immediately a chicken nuggets ad. Yeah I haven't seen it like that before.

[@Konstantinos340]

i am an engineer too and to be honest in university we only focused on what we needed to get the job done i know theres so much i dont know and even more i havent looked at. This video has reminded thats i have a lot of material to read up on. Theres so much to read but if i didnt need it for assignemnts i just glossed or ignored it completely

[@shirokyu_9223]

Im still in highschool and I already learned beziér curves, etc. before watching the video. Its definitely useful in physics. Its also artistic. It actually works like electrons and the lines are the trajectory of a particle.

[@dillonzhang978]

Ah yes: From a 90 - year old Electronics Academic Journal article, while in the Los Angeles cloud, in searching for a True Tesla Bifilar TTB equation for inductance, I'm happy to say that I've found it; and also now have all the instruments to measure those coil parametrics. This particular EQN is about a paragraph to a page long and takes about 50 pages to explain. It has the sin and cosine geometry of two different sized wires, "twined" together and coiled in a spiral (just like Birkeland double layers). I'm almost certain that Tesla himself had to have reviewed this article, since he was the teacher of many. Including people in my own family, some with patents (unpatented). If I were to post the equation (here), y'all would see; and it does take up a page. This equation describes two different size wires wrapped around one another, connected in series at each of one end, and meant to conduct back and forth in complement to itself; whereas the measure of calculated inductance would be measured across the outer unconnected ends (2 of 4 total). If anyone in mathematics were to examine this article, my conclusion was: "prove in practice as a living example of the anti-derivative formulation."

[@mathpluspuzzles]

thank you for a beautiful video.

[@quosswimblik4489]

Curves I invented. Curves and structures using a second imaginary number plane. Line equations approaching the equations of an intercepting line. The triangular radial shift curve. The cycloid cord and skidded cord even to the point of working with infinitesimals and super geometry. I love curves.

[@mahdipourahmad3995]

I was a little skeptical to watch this video when I saw the title, but it proved me wrong and it was a great video!

[@chadx8269]

Excellent presentation.

[@michaeldamolsen]

@12:05 - That is not a spiral, it is a helix :p Nice guided tour of some interesting curves though, well done!

[@lloydislearning]

As a ex engineering student and currently a design student, this is awesome.

[@poorman-trending]

Internally, adobe only uses quadratic curves - it’s just that for the curve that looks cubic, one of the control points it at the same location as one of the end points. Mathematically though, that reduces to the same thing as a cubic curve. This is also true for straight lines - both control points are at the same locations as the end points.

[@flounderflounder6833]

Please do more of these

[@kenj4136]

I would watch a whole series dedicated to these less commonly used functions. Super interesting

[@matthewboyd8689]

Put a ball at any point on that slide and it will take the same amount of time. That blew my mind when Adam Savage showed it off in his shop. (I believe it was on his KZbin channel)

[@John-zq6pt]

Do you have other recommended readings to gain this kinda insight on these kinds of topics and the things you talk about on your channel? Also some information on what prerequisites are necessary to understand the stuff would be nice

[@cearnicus]

I looked up Bezier curves a long, long time ago and pretty much every site just gave the resulting polynomial without explaining the underlying idea. All I could think then was "... 'kay ..." But here as soon as the blue and green points started moving at 2:40, I understood where this was going. _Glowers at those sites,_ see this? _This is how you explain Bezier curves!_

[@sivonparansun]

I liked where you started doing applications

[@alirazi9198]

I had to do design and communication graphics and we had to draw all of these but i didn't have a clue how they work mathematically really interesting

[@alejandroduque772]

Mechanical engineer here. We are seeing bezier curves to find the parametric equation of a cam ,and thus its shape, given that we need to pass through certain points at a given part of the rotation, satisfying things like velocity or acceleration continuity

[@henrymhp]

"But as far as I know, I am just kidding" - That was the first time someone earned an instant sub from me before watching the content.

[@rameshkvaw542]

Just loved it

[@duckymomo7935]

Weiertrass isn’t application but demonstrates that we can’t rely on assumptions or intuition because it turns out most curves are miserable But most common and natural curves (polynomials, trig and exponential) are nice and we really got lucky that they are in fact analytic.

[@cameronball3998]

Dang I wasn't ready for the video to end so abruptly, I could've watched another half hour of weird curves

[@yalmanmfp]

i m cnc operator and in vector format i use curves too much because every dot u use in design cnc stop a little bit like 0.1 second its matter for sensite work for a laser cnc its burn the metarial little much but if u use curves fine u got the same shape and less stop which means better cut. hopefully cad and cam programs do the job now at least they help :)

[@TheF1leader]

Well... this video deserves like just for a sole reason. It had taught me how to draw bezier curve in a single animation.

[@hbarudi]

The Lemniscate curve only appears when you study the polar coordinates usually in any math class you take after Algebra 2. The other curves may appear when you learn implicit differentiation in Calculus.

[@nickhodgskin]

Really interesting video!!! Thanks :D

[@eriktempelman2097]

Look up the Limacon curve. With the right parameters it nicely models the cross section of the human belly :-) . The cardioid is a special case of the Limacon.

[@miltonbother8743]

This channel is great because you recommend books, and that's great

[@yatikasera3780]

great work

[@S.G.Wallner]

Shedding tears from the beauty.

[@miro.s]

Very nice you showed cosh shape of a rope. I've seen in many books, one of them about mechanics, but also one video on youtube where people claim incorrectly it is parabola.

[@und0]

Anyone else recognized the bezier curve from css curve from css transitions/animations? I really like that css feature, it can make or break an animation :)

[@christianmarquez3143]

1:57 - Einstein curve! Nice! So cool- thanks for sharing. :)

[@TheBaronOfBromley]

Zach, this is really helping me with my esoteric journey. I don't identify as a mathematical person and there is no professional or academic context for these topics in my life. Rather I experience myself as a musician, visionary and contrarian. So it may be surprising to find that watching your expositions of geometric concepts and technical processes brings me a sense of great joy and peace. There is something beautifully fundamental here which lies beyond the present grasp of my conscious intellect and resonates on a much deeper level. As with great poetry or fine art, the core truth is felt rather than understood. It's a privilege to say that I am in love with the allegorical power of these presentations. Thank you for your work!

[@esu7116]

That intro though perfect 😂😂👌

[@fentonflatt297]

at 7:53 he puts a picture of a curve that is not the brachistochrone, because it is also known as the tautochrone curve which means same time. it is called this because whatever point on the curve you start on, the time it takes you to get to point b is constant. the curve he puts in the video goes under point b, and if you start under it, you cant go up and reach it.

[@JalebJay]

9:18 you scared me into thinking my computer screen froze D:

[@waterpidez6732]

Bruh;hhhgddh

[@katicauwu7204]

Here are some random curves: Curve 1: x=cos(y^y)+cos((y^4y)-3)+cos(y^-4) Curve 2: x=sin(y^8)+cos(xy^9) Curve 3: x=sin(xy) Curve 4: x=cos(y)+cos(y^7)+sin(x^4)+sin(x^-5)

[@pipertripp]

Good stuff. The weierstrass curve was wild.

[@harleyspeedthrust4013]

10:35 Isn't that the Lagrangian?

[@michaeldamolsen]

Yes it is.

[@1495978707]

Yes, he probably didn’t say as much because it would make it seem complicated and obscure

[@maxonmendel5757]

Which one?

[@michaeldamolsen]

@@maxonmendel5757 Typically when the name "Lagrangian" is used alone, it refers to the Lagrangian function of Lagrangian mechanics, which is kinetic energy minus potential energy. So in this case, the formula ½mv² = mgh. For a brief overview, check the wikipedia article on Lagrangian mechanics; the part around the equation L=T-V directly addresses the formula mentioned here. If instead you meant "which Lagrangian function" (out of many possible), this one is the one describing a free falling test particle in a central force gravitational field; a very good approximation of orbital mechanics, or simply dropping stuff.

[@TheTim466]

And then he continues to show the "action" or rather the principle of stationary action.

[@lukecabral5532]

Hey Zach I found your shows in treating I was wondering what schools you teach out cuz I would like to come visit that school

[@BrendanPJames]

I do motion graphics for a living and Bézier curves are a big part of my life haha. To the point where after many hours sitting in front of the computer, I close my eyes and my brain is just making them

[@TeboeCubes]

Hm, always used bezier curves in video editing but never knew the math and reasoning behind it. Cool!