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

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

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

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

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

“Not applicable but definitely artistic”

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.

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

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.

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

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.

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

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!

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.

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

Hmm yes, these curves do seem pretty cool

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

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!

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.

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)

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.

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

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)

This channel is increasing in quality

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:)

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.

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

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

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

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

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.

10:35 Isn't that the Lagrangian?

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

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

the infinite complexity of curvature is astounding - good video

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

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

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

Loxodrome: en.wikipedia.org/wiki/Rhumb_line

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!

very cool video man, I love these

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

This is awesome! Thanks for sharing

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

Thanks! I just learned how the Bezier curve works!

Please do more of these

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.

That intro though perfect 😂😂👌

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

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!)

Awesome stuff!

Excellent presentation.

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.

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

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.

Continue the serie!!!! Awesome!!

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

"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.

thank you for a beautiful video.

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.

With the same amount of money I pay for Netflix I can pay Curiosity Stream, Brilliant. and still, have a leftover. Goodbye Netflix

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

I liked where you started doing applications

Really interesting video!!! Thanks :D

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.

Just loved it

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

"I'm sure many of you know where this is going..." oh yeah for sure.

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.

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.

great work

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!

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()

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.

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

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.

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.

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!

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.

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

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 :)

Great video

Shedding tears from the beauty.

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

Could you make video Hilbert curve ?

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..

Good stuff. The weierstrass curve was wild.

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

Wait, how can those create curves when sin(x) = x and cos(x) = 1?

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

awesome video

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

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

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

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

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.

Hahahaha beginning had me dead 💀