How to program a Drawing Machine

I will publish the code later.
Music used (in order of appearance):
Poison Message by Man with Roses
Winterbeams by Diffie Bosman
Empyrean by Dear Gravity (4000 pendulums part)

[@hexelldessin5080]

this is so cool the paint dispenser was a really nice touch

[@PezzzasWork]

Thank you! It took me way too much time :D

[@EscapedShadows]

@@PezzzasWork But really how did you do the Paint Dispenser it looked so good?!

[@itermercator114]

First time I've seen someone manage to explain how it works in a way that isn't overwhelming mathsy, cheers!

[@PezzzasWork]

Thank you! My background isn’t about mathematics, that might be why :)

[@GeneralSpazesSpy]

Wow great work! Hey, 3blue1brown has these informational maths video competitions once per year (I believe), maybe you can enter this video into the next one? Feels like the perfect fit! The judges value intuitive teaching and simplistic visualisations highly, which to me fits this video!

[@PezzzasWork]

Thank you for your comment, I didn’t know about this, I will have a look!

[@CliffHanger-fg6uy]

3b1b actually has a couple of very similar videos (including pretty similar animations), so I don’t think this would be a good entry for SoME (as I think being a somewhat novel explainer is part of the judgement criteria). I know of at least 4 major STEM KZbin channels who have done this topic (focusing on using the FFT/DFT to draw). However, I totally think this channel should consider coming up with an entry. The videos are very polished and the explanations are clear.

[@axelanderson2030]

I've seen these Fourier drawings on 3b1b before

[@Tryh4rd3rr]

It would be good for SoME, but he already has that type of video

[@박태수-t7o]

The little detail, such as "sensitive electric device" makes me little more pleasant. Always thanks for great content!

[@PezzzasWork]

Thank you! Glad you appreciate the details, it is what takes most of the time :)

[@karthikkumarrao6502]

You're not just a developer, but an artist as well - absolutely beautiful work

[@slmnchk]

OH MY GOD this animation at 1:10 explaining how parametric definition of a circle really works is just magnificent I wish I'd seen it in uni

[@Jay-Dub-Ay]

The continuity and grace of this is so beautiful ❤

[@PezzzasWork]

Thank you!

[@Sammysapphira]

This is one of the best videos on applied math in programming I've ever watched. I love that you show the coding part, too. Sometimes it takes a little bit of syntax for my brain to connect the dots between formulas and code, since my brain thinks in code more than it does formulas.

[@PezzzasWork]

Thank you :)

[@vnever9078]

12:58 bro didn't have to flex with that paint dispenser 💀

[@ahmedx3000]

I remember you've done that before but this explanation video is truly amazing 🔥🔥🔥

[@PezzzasWork]

Thank you! Yes indeed! I wanted to redo it for a long time since I truly find this fascinating :)

[@screwuk]

Wow, really beautiful work mate. Everything inc. vid production, narration etc. is really superb. Lovely touch with the paint dispenser.

[@sirhoog8321]

Your videos are always so clean and well edited. It's beautiful and informative. Thank you for them :) (and thank you for showing the code in the video :) )

[@rebeccarivers4797]

I have wanted a video like this from someone for years. Thank you!

[@robelbelay4065]

This is by far the best and most intuitive explanation I've ever seen for a DFT!!! I've been trying to grasp it for years, thank you so much!

[@PezzzasWork]

Thank you for your kind comment :)

[@caseyedson12]

The paint dispenser was fantastic! Your previous pendulum model with the pull of the Drawing Machine unbalancing it. So seamless I couldn't have noticed without those wheels haha. Great video, thank you.

[@tiagogarcia4900]

That part where you highlight the part of the function you are coding was pure Genius!

[@Pramerios]

This is some GORGEOUS math, my friend. Phenomenal work.

[@VivienLEGER]

it s trully amazing... as an embbeded software architect that have already use this methods, i never noticed how beautiful it may be ... thank you so much. I love your work on this channel..

[@PezzzasWork]

Thank you very much for the nice comment :)

[@Radu]

Amazing work! And really nice touch with the paint dispenser at the end :-) I feel like a lot of work went into that.

[@kirankumark7679]

0:15 if bro's bad at maths we are all cooked.💀

[@minecraftprovie5076]

fr

[@xernas7880]

Finally, all your videos are delighful, Nice to see another french person coding projects like this

[@13-Diamino-246-trinitrobenzene]

Xerne5 clone? Lol

[@PezzzasWork]

Merci beaucoup ! :)

[@yansakovich]

Who is the other one you are referring to? Sebastian Lague?

[@EpicVideoGamer7771]

You killed it. Blew my mind in more ways than one. Keep up the amazing work!!

[@Kingdumkey]

Im in SHOCK at how ingenious this is!!!

[@pasmoluiso]

Great video, I love the quality of the animation and how everything looks so clean. The explanations are also very good, but my favorite part is the visuals.

[@DIMENSIONCODE]

Great video! I just have one question, some of the animations (like the ink pipe) seem really overkill for a project like this, especially in C++. Do you use a library or is it added during editing?

[@PezzzasWork]

Thank you! It’s hand made and runs in real time during the signal reconstruction. It is based on the XPBD method, the scientific paper is amazing and makes it very easy to implement.

[@-Yousof-]

Amazing work and explanation! You impress me more with each of your videos🤯 Keep up the good work 👍👍

[@PezzzasWork]

Thank you for your support!

[@Alayric]

Beautiful as usual, I like how you reused the self balancing machine graphics for the paint dispenser. I was wondering if that was how old writing automata like the Jaquet Droz ones work, but they are actually a bit different.

[@sanderbos4243]

Hands down the coolest use of FFTs I've seen

[@ajejebrazor4936]

Pure art! Thanks for this contribution

[@Theo_El_Gato]

This is funny because I did the same thing for a school project (math expertes) and I discovered that my teacher was a literal pro in this domain and show me every single problem

[@Astra_Dystopium]

Bro thinks he's bad at math. You have no idea what bad at math is lol. You are brilliant my man.

[@punk3900]

i ve seen it many time but each time it makes me wonder how simple actions can lead to unbelivable complexity. the world is just a combination of waves, isn't it?

[@lucyhalut4028]

I actualy understand the Fourier Transform now. Thanks ☺

[@PaulMetalhero]

Man, your videos are pure magic!

[@PezzzasWork]

Thank you!

[@rayenwiller]

Fascinating Work! really wanna know how you visualize all these disc animation. Definitely deserve a lot more views.

[@TheDailyMemesShow]

I've been daydreaming about something like this, but with 3D mathematical expressions 😊❤🎉

[@skromny150]

That is really good job! I am so happy that yt recommend me this video! Sub+like for your and I hope you keep doing your amazing work. Take care man! The movement of the circles according to drawn lines is magical and is so calming to watch

[@PezzzasWork]

Thank you very much for the nice comment!

[@raducristiandimitrie]

Once again, delivering an amazing video! Thank you!

[@gelerson1642]

God I hope you get 10x more subscribers. Your videos are incredible. Keep it up, please!!

[@1.4142]

Now I know why those drawing/ note ipad apps have those continuity glitches

[@roemischer]

Thanks for the coll video. I really like your style of explaining things.

[@PatrickHoodDaniel]

Such a great way to explain DFTs. I wonder if this could be used to forecast trends similarly to fitting polynomials to data.

[@CliffHanger-fg6uy]

Data fitting was why the DFT was originally invented. The FFT, a fast algorithm for performing the DFT, was originally discovered by Gauss while trying to predict the orbits of asteroids.

[@PatrickHoodDaniel]

@@CliffHanger-fg6uy Pretty cool!

[@nofacee94]

Very cool visuals. I'd love to see behind the scenes as in how you create the video with code.

[@PezzzasWork]

Thank you! I will do some videos about my way of doing animations as it seems there is a demand for it.

[@yuvalorp]

You can do it with just one set of wheels by treating the drawing plane as the complex plane

[@dormindurst349]

Never seen a Fourier transform do graffiti before. That's magic.

[@alejandromartinez-vp4sx]

Amazing! I will appreciate you sharing the code 😊.

[@Hycord]

I love that I could recognize the forier transform in the intro sequence within seconds!

[@Djellowman]

Ok

[@Firestorm-tq7fy]

Yep sme

[@Diabl0pl]

Just incredible!

[@brumomento-so2nd]

love your visualizations and contents!

[@errorhostnotfound1165]

mhm I love watching large clouds/larva draw lines :D

[@ddBenny]

adding int and bool like a madlad!

[@PezzzasWork]

so fearless

[@metactal]

I was certain that this would be about bezier curves and am pleasantly surprised it isnt! This feels like something i would come up with back in primary school (a bunch of circles rotating at different speeds can draw anything), of course, i wouldnt have had the mathematical knowledge to figure out to do it.

[@liftbridgetown]

this is beautiful and awe inspiring!

[@PezzzasWork]

Thank you!

[@dotero666]

I was hoping to see you use it to draw the chicken from your profile picture like you did in older videos, keep up the good work, I like your videos even thought I don't know much about coding

[@robelbelay4065]

If you haven't considered it already this code would be super useful to be implemented as a plug-in to augment CAD software(s) as including high-fidelity texts in 3D models isn't as intuitive and simple (for 3D printing in particular)

[@dottedboxguy]

man the animations are sooo nice, you could make a whole video about doing animations for your videos. on the initial premice of tranforming a drawing into a math function, i'd personally have done it as a long serie of bezier curves, which is definitely not as interesting, but still a subject you can get lost into for a while

[@Vaaaaadim]

Is there an algorithm for turning a curve into a bunch of stitched together Bezier curves? I am under the impression that Bezier curves are typically made by hand.

[@PezzzasWork]

Thank you!

[@totoooenf5]

The visuals are super nice!

[@Acuzzio]

Super good. Extremely well done. Thanks.

[@PezzzasWork]

Thank you :)

[@kyleeames8229]

‘Ey! Collin Furze, I think I found a machine that needs building!

[@timuren6422]

Totally looks like an orthographic projection of an alien tentacle drawing

[@programagor]

Great video! I think I need more detail on going from two sets of rotating disks to one set, but Fourier Transform/Series is such an important concept!

[@programagor]

Oh never mind, just figured it out, you just feed it as x+y*i complex number

[@PezzzasWork]

Yes exactly!

[@chris.hinsley]

Beautiful work :)

[@spinteamok6877]

who else noticed how his english seems to get better with every new video?

[@lucasxia2232]

Your animations are really cool and make it easy to understand. What do you use to make them ?

[@PezzzasWork]

Thank you! I am using my own animation framework I wrote in C++

[@caledonianamerican41]

Very nice work, great job!

[@skilz8098]

FFTs are one of my favorite algorithms. Perhaps take this application one step further and have it draw or plot the Mandelbrot Set. Now that would be infinitely amazing to watch!

[@ronaldmullins8221]

I want this as a PC screensaver

[@sdjhgfkshfswdfhskljh3360]

Exploration of image encoding methods not only funny, but also useful. Next logical step is to think about how coefficients can be effectively packed into bits. Bonus level: think about how to effectively pack decompression code into bits.

[@mizoik9893]

I've had a very similar idea just a few days ago using sound waves as a way to cypher information for my arg using separate sound waves

[@CliffHanger-fg6uy]

If you did watch other KZbin videos on this subject, I think it would be good to reference them or link them in the bio. Your video is still a cool contribution (due to not getting to into the weeds, posting the code and talking about the code more, more nice animations, etc.). It just rubs me the wrong way that it seems *very* likely that you checked out some comparable videos for inspiration (this topic has already been hit by several large math/programming channels) and make no mention of them.

[@greatoak7661]

This is a beautiful video. Thank you.

[@JavierAndres-o6w]

Man, this is soooo cool, I have just discovered your channel and I am loving it. What engine/workspace are you using to make this kind of videos and simulations?

[@PezzzasWork]

Thank you for your kind comment! I use raw C++ to program my own simulations, I am currently creating a series of video explaining how to do rendering using C++.

[@belkacemF]

interesting how much little videos about this topic,,u made a good video as of others

[@zx3215]

Eureka! The Antikythera mechanism was just an ancient drawing machine!

[@DefinitlyAPerson]

The white click state segments can be a straight line just like the padding samples to decrease the amount of information within the Fourier-transformed version. (P.S. I'm not a mathematician as well, but a straight line is not efficient in FT.)

[@Vognar6]

Un commentaire pour l'algorithme youtube, très instructif, bravo.

[@mutiev]

Masterpiece 💪🏼

[@ENDESGA]

this was quite hard to follow on my phone, the code was really small so it's hard to read. other than that I loved this a lot

[@MysteryPancake]

Cool video! You might be interested in "An Interactive Introduction to Fourier Transforms" by Jez Swanson, they made an interactive demo of this as well

[@bytesandbikes]

You can also do this so you don't need to track the on/off states outside of the signal by extending into 3 dimensions and drawing only a slice inside the 'on' depth. Could treat the 3rd dimension as 'paint dispenser flow'.

[@CliffHanger-fg6uy]

He never actually used a 2D DFT. The DFT by default allows for a complex-valued signal. You can just set the real and imaginary parts of each entry to the x and y coordinates of each sample. Typical generalizations to higher dimensions rely on adding up plane waves, which certainly wouldn’t create a comparable graphic. For that reason, generalizing this sort of animation to 3D would be significantly tricky. Apparently there’s such a thing as a quaternion Fourier transform. Maybe that would do it? I’ve never studied them, though, so I don’t know.

[@bytesandbikes]

@@CliffHanger-fg6uy Yes, you can sum plane waves, but in every actual use I've ever seen it's done by decomposition... just like in this video.

[@NatCo-Supremacist]

I wonder, what will the traced patterns look like if every circle had a tracer where the next circle was fixed? So for the first circle, it's just a circle, the second is slightly more complicated, and so on until the last circle traces the final pattern. It would be like a big sheet of lines that are really close to each other.

[@gustavosena7876]

Mind blowing channel

[@qfurgie]

could you do the other way around? Define the circles and see what designs they create?

[@yijhebsldiv3gyxi88]

your videos are so good! they look really clean

[@Lucas-pj9ns]

epic video, im somewhat lost on combining the x and y signals part, how did you give the dft both signals at once, did you simply have the arm be one x circle than one y circle, repeating or something more complicated?

[@ccost]

if you put the circles that made the y signal on top of the circles that made the x signal or vice versa they simply combine, that's all, if this were real life u literally just stick them on top of each other (obviously aligned correctly) yes he could've done it by altering the circles and likely did as this would make it look more satisfying but it is not necessary, the order of the circles don't matter it just looks nicer when it decreases in size

[@chartroniumdude5870]

i believe it works by feeding the positions as complex numbers (x+iy) into the formula.

[@Vaaaaadim]

In the code the circles are really represented as complex numbers, But I'll still frame my comment here in terms of those circles, and sine/cosine. Also my notation here won't necessarily match the video. I don't think he computed x circles and y circles individually. For each frequency component, there is a single circle that handles both x and y. Still, if you do have a set of circles for x and y here is how you can combine them. Each circle will have some radius r, some speed ω, and some offset(phase) ϕ. So a point moving in a circle around the origin can be described as two functions x(t) = r⋅cos(ω(t-ϕ)) and y(t) = r⋅sin(ω(t-ϕ)) The x circles and y circles will have circles of possibly different sizes and phases, but same speeds. By which I mean, if the x circles will have one with a speed of 5, so too will the y circle. (Actually the speeds will be a multiple of 2*pi/N but whatever). Let's say the x circles include one with radius a, speed ω, and phase ϕ_1, and the y circles include one with radius b, speed ω, and phase ϕ_2. We can add these two circles to get a single circle. To do so, you can plug in t=0 to get the x,y values of one point from each circle, let's label them (x1,y1) and (x2,y2). Add these points component-wise to get a new point. So (x3,y3) = (x1+x2,y1+y2) . The radius of our resulting circle will be the distance of (x3,y3) from the origin, so sqrt(x3^2 + y3^2). The phase of our resulting circle will be the angle that (x3,y3) makes with the origin and the x axis, so atan2(y3,x3). And finally the speed will stay the same.

[@valerykyungu]

That was amazing! Good job

[@PezzzasWork]

Thank you!

[@Haagimus]

Now replace all the different sized Wheels with one gangly arm so it looks like a weird Mr. Fantastic arm drawing all the characters

[@truiteteam3428]

Everything is normal on this channel

[@kpopalitfonzelitaclide2147]

Mri machine generate images using the fourier transform. Riemann used the mellin transform to anylitically calculate the prime number counting function.

[@VladimirDemidovIllusiveMan]

ive always wondered if any 3D or 2D object can have a function, and drawn like this.. and now i know. 3D would just need another axis. astonishing. BTW if ur math is bad, im afraid im a cave man in stone age, despite being a dev myself.

[@sdjhgfkshfswdfhskljh3360]

Signed distance function is more suitable for 3D I think.

[@jahbini]

In the regions of sharp angles, the higher components are needed. This seems like only needing to compute enough components to keep any error within bounds.

[@TechWithAbee]

i love it! ✨ thank you very much !

[@David_Raab]

Looks very intersting. But I think having low "circles" and don't matching the original as close as possible is a feature. It makes the lines more smooth instead of jagged. It's actually a feature many drawing programs have. Otherwise when it just reconstructs everything exactly then i could just save the points that are drwan with a timestamp to and replay it as often i want. Or does it have any other advantage of doing it like that?

[@lava_tiger]

you really should have more subscribers

[@chris.hinsley]

Great to see you using C++ and not Python ;)

[@TheSilentVictim]

So am I understanding that each circle is rotating at the same speed? Since it is a periodic signal? That is absolutely wild

[@shrootskyi815]

TL;DR - No, the circles are rotating at different speeds. The maths: to make things simpler to explain, I'll focus on a single "arm" of the drawing machine with two arms. The X or Y input signal is transformed into a sum of periodic (sinusoidal) functions. Each circle of the arm represents one of these functions. Each of the functions has a different period, so the rotation of each circle has a different period - in other words, the circles are all rotating at different speeds. I don't really understand how the machine with only one arm works, so I won't try to explain it.

[@CliffHanger-fg6uy]

There are two circles that rotate once per repetition of the animation (one CW, one CCW). Then, there are two that rotate twice as fast. Then, three times as fast. This keeps on going for as many circles as you choose to include (though for math reasons, the number of circles cannot exceed the number of data points sampled for the drawing).

[@CliffHanger-fg6uy]

@@shrootskyi815The DFT works on complex numbers. If you encode the x and y coordinates as the real and imaginary components of the signal you take the DFT of, the single arm machine is the result.

[@ignatikklokov]

this is really cool, would be interesting to try and replicate in javascript

[@wesleyjones1667]

This is an amazing video, I wanted to code this myself a while ago but got scared away because of the math. I want to try again now, what framework or engine do you use for your visuals?

[@PezzzasWork]

Thank you! I hope this will help you :) I am using my own C++ framework !

[@itsmodsiw]

as usual! great content. thanks for sharing.

[@PezzzasWork]

Thank you! And I think I recognize your profile picture from GitHub, so I take the opportunity to thank you very much for your generous support!

[@wack1305]

I would love your code to help with my research! Edit: halfway through now and I’m guessing you show most of it anyway :)