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

@anteconfig5391

4 жыл бұрын

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

4 жыл бұрын

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

@HighestRank

4 жыл бұрын

Chef 11:49

@5Djoe

4 жыл бұрын

Practical Vs Theoretical strengths of math pls??

@ignaciosavi7739

4 жыл бұрын

Lol tru

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

@datchnac4577

4 жыл бұрын

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

@randomdude9135

4 жыл бұрын

@@datchnac4577 I like how Evolution and Mathematics are connected

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

@poopfart65

3 жыл бұрын

😳

@juliaanimates9765

3 жыл бұрын

actually, this is about math. not certain videos.

@avory7938

3 жыл бұрын

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

@cameo2277

2 жыл бұрын

@@juliaanimates9765 don’t be fooled Julia

@Callie_Cosmo

2 жыл бұрын

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

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

3 жыл бұрын

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

@yomumma7803

3 жыл бұрын

@@chaotickreg7024 wow, harsh bro lol

@chaotickreg7024

3 жыл бұрын

@@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

3 жыл бұрын

@@chaotickreg7024 At least engineers make useful stuff.

@aurelia8028

2 жыл бұрын

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.

“Not applicable but definitely artistic”

@ObjectsInMotion

4 жыл бұрын

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

@nonegiven3814

4 жыл бұрын

His non applicable curves are hunted for in stocks charts.

@genericusername4206

3 жыл бұрын

@@ObjectsInMotion sameeee

0:22 Hey Vsause, Zach here

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.

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

3 жыл бұрын

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

3 жыл бұрын

NortheastGamer lmao

@yatexasnycaflnvnigga

3 жыл бұрын

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

@yatexasnycaflnvnigga

3 жыл бұрын

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

@yatexasnycaflnvnigga

3 жыл бұрын

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

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

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

@techfreack8397

2 жыл бұрын

@liebe please what was the question please

@anshumanagrawal346

2 жыл бұрын

What question?

@NorseGraphic

2 жыл бұрын

@@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.

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.

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.

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.

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!

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

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

3 жыл бұрын

Actually it was the brachistochrone

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

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

@digitig

4 жыл бұрын

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

3 жыл бұрын

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

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

Жыл бұрын

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

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

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:)

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!

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

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

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)

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

Hmm yes, these curves do seem pretty cool

@tifahefendijagaming9606

4 жыл бұрын

I didnt know bigfoots were into math

@dondake3409

4 жыл бұрын

why are you everywhere?

@MAJORVBEZ

4 жыл бұрын

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

@Avinashkumar-ec9ov

4 жыл бұрын

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

@math_the_why_behind

3 жыл бұрын

Agreed!

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)

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.

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

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

@roygalaasen

4 жыл бұрын

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

@jaykaku7646

4 жыл бұрын

@@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.

This channel is increasing in quality

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

very cool video man, I love these

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

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

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!)

the infinite complexity of curvature is astounding - good video

Your channel is awesome. keep it up!

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

This is awesome! Thanks for sharing

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.

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

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

10:35 Isn't that the Lagrangian?

@michaeldamolsen

4 жыл бұрын

Yes it is.

@1495978707

4 жыл бұрын

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

@maxonmendel5757

4 жыл бұрын

Which one?

@michaeldamolsen

4 жыл бұрын

@@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

4 жыл бұрын

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

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

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.

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

@waterpidez6732

4 жыл бұрын

Bruh;hhhgddh

Please do more of these

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

Awesome stuff!

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.

Continue the serie!!!! Awesome!!

Thanks! I just learned how the Bezier curve works!

Excellent presentation.

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.

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()

Really interesting video!!! Thanks :D

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

thank you for a beautiful video.

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

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.

Fyi the brachistochrone is not drawn correctly as the optimal rolling path, it needs to be exactly half a period of the cycloid, such that it never has to roll back up.

Just loved it

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.

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!

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

I liked where you started doing applications

great work

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!

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.

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..

"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.

That intro though perfect 😂😂👌

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.

"even less useful tho(...)" Now you have my attention

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

Great video

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

Could you make video Hilbert curve ?

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.

awesome video

Loxodrome: en.wikipedia.org/wiki/Rhumb_line

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

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 :)

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.

Good stuff. The weierstrass curve was wild.

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.

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

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."

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.

But that cable at 8:45+ is stiff. That messes with the mechanics. So is it really a catenary?

@geromatezwanzig8789

4 жыл бұрын

Well, not really :-( The name catenary comes from "cadena" = chain (spanish or latin). This means, that each element of the hanging string, or whatever must have the degree of freedom of each (infinitesimal) element of a chain, which the cable definitively does NOT have. At least at that scale. A hanging bridge cable(s) definitively are at a scale where the stiffness is "negligent".

@geromatezwanzig8789

4 жыл бұрын

Well, not really :-( The name catenary comes from "cadena" = chain (spanish or latin). This means, that each element of the hanging string, or whatever must have the degree of freedom of each (infinitesimal) element of a chain, which the cable definitively does NOT have. At least at that scale. A hanging bridge cable(s) definitively are at a scale where the stiffness is "negligent".

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

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

Shedding tears from the beauty.

So this is how the pen tool works Damn it , Bezier Curves

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

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)

10:40 now I'm just imagining how funny it would be to just throw a basketball at a hoop and it just fly around randomly with cartoon sound effects in the background

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!_