So basically you've created the most complicated etch-a-sketch of all time
@TheoParis
4 жыл бұрын
XD
@Invalid571
5 жыл бұрын
You are the best coding teacher ever! Keep up the math related codes.
@onetzaid1
5 жыл бұрын
"Lets make an array, called "y" and this is signals" how about naming it "signals"?
@naxaes7889
4 жыл бұрын
Haha! While everyone can make such mistakes, it always trigger me when is see code like: var a = 20; // age var b = 40; // length instead of: var age = 20; var length = 40;
@spythere
4 жыл бұрын
@@naxaes7889 It triggers me when someone uses var instead of let without purpose
@doctorcardinal2322
3 жыл бұрын
@@spythere ES6 makes me cry.
@alexandermcclure6185
4 ай бұрын
@@spythere there's no difference though
@DirtySocrates
5 жыл бұрын
Jesus this is insane! I remember having to draw a image with piecewise functions in 10th grade and thought that was impressive.
@CalculusPhysics
5 жыл бұрын
i love how well you can explain topics, especially complex topics such as Fourier Transform! you’re definitely able to explain them in a really simple, but effective manner. great video as always!
@CodingWithUnity
5 жыл бұрын
My favorite part about the channel! Very inspiring
@michaelkotthaus7120
3 жыл бұрын
Yeah, this "Fourier machine" is really great! - And it's so inspiring how Dan explains the development of it.
@martinskodacek6567
5 жыл бұрын
I'm watching Daniel's channel since about 50k subs, every time he creates some amazing project, but this is maybe the most amazing thing I've ever seen here :D I heard about FT long time ago, but I've never been interested until now. Great job ;)
@TheCodingTrain
5 жыл бұрын
Thank you for this nice feedback!
@olegil2
Жыл бұрын
This is an awesome way to learn about the discrete fourier transform and fourier series. Even knowing a LOT about fourier transforms as applied to both sound and electronic signals, seeing the DFT used to draw a friggin train blew my mind.
@f.osborn1579
Жыл бұрын
This is the coolest explanation of Fourier transformation I’ve ever seen!
@pozzer3
4 жыл бұрын
Congratulations. The best video of all time. It encapsulates all necessary videos and sites on the internet related to fourier transform and fourier series so anyone with basic knowledge on math can understand and implement a fourier transform. It is simply amazing. I've spent many years of my life trying to understand it and now it it seams so easy.
@tfrangel
5 жыл бұрын
Hey Daniel! I'm a huge fan of your channel, and I always learn a lot from your videos. Although I do not do JavaScript or p5.js, by going through the algorithm step-by-step one can build and expand on the code in any modern language. More importantly, one can not only replicate the code, but also understand it! Combining math, code and graphics is so much fun! In fact, I've been interested in Fourier since ever, but the standard math has always daunting to me, a biologist without strong background in math. Your recent videos on FT are not only amusing, but also expanded the horizon of possibilities in my research. Keep up the GREAT work! Best!
@trumanknight873
3 жыл бұрын
I love that you shout out other channels and give credit to the inspiration of the video where it is due. Always love your videos keep up the great work
@ankitrai96
5 жыл бұрын
Although this video's sophistication only adds to its glory, I miss the time when your videos used to be based on p5.js and were so simple that I would start coding straight away!
@ajoealex1
5 жыл бұрын
isn't he using p5.js in this video?
@TheCodingTrain
5 жыл бұрын
@@ajoealex1 I am! But it does take me a while to get to it 😬
@ajoealex1
5 жыл бұрын
@@TheCodingTrain you are awesome.... keep going😊. love the way you explain.
@narveeviolet
5 жыл бұрын
FFT must means FULL of FRUSTRATING TRANSFORMATION. i had to watch more videos about fourier transform to catch up your 1 hour video. and more time to ponder. but only got glimpse of it. I only could start actual coding after i stop to try understand everything mathmatically. it was shame but when I finally reconstructed a straight line with x,y sin waves in my own code I am so satisfied. all my struggle was not in vain. thank you!
@majestechtn
5 жыл бұрын
if you don't mind, could i know how long you have been coding, and what is your level of knowledge in mathematics? i just started coding 5 months ago and feel like im picking it up very quickly, and i was always very good at math in highschool, just wanting to get an idea of how long until i'm at the level that I can start figuring this stuff out on my own like you commented that you did, thanks the last video he did on fourier series i understood like 95% of it, but on this one just got totally lost, lol, thanks again
@musondasalimu2986
3 жыл бұрын
I took computer graphics course in my third year of cs study and all I can say is you've done some amazing
@tcholly
4 жыл бұрын
When I saw this you gave me an idea, I took your code, and I added something with python. I made something like paint in black and white, and what you draw (if you refresh the page) it get drawn eith your machine. it's all a mix of phyton and javascript. But then I saw your other video where you did that but better :-( I'm still happy of my result Thank you for giving me fantastic ideas every time I watch one of your video
@Thepantino
5 жыл бұрын
What a beautiful challenge...it's fanstastic...unbelivable to see!
@Shipslaw
5 жыл бұрын
Beautiful - interaction between x- and y-axes. Put a similar wiper setup on the z-axis and you can make a 3D drawing.
@kevnar
3 жыл бұрын
Shiffman was a mad scientist in a past life. Lucky for us, he's on the side of good this time around.
@bapolino733
5 жыл бұрын
Im so glad that you uploud new videos! There was a little break that made me worry
@atrumluminarium
5 жыл бұрын
FFT if I remember correctly can be used to simulate sea waves in videogame graphics very efficiently Also, as a way to teach complex numbers, have you ever considered making a "complex number library"? You can then refer to it when you do videos like this that build on top of it.
@TheCodingTrain
5 жыл бұрын
I made a small start to one in the most recent video (part 3 of this series)!
Tip: Don't use "const" when you know you don't want to change a variable - just always use "const" except when it's _really_ not possible. You'll write different code that way because automatically you try to avoid "let" by using functions like .map() and stuff
@noodleexpanding3407
5 жыл бұрын
@@kreuner11 what a madlad
@angelcaru
4 жыл бұрын
and .forEach()
@NatetheAceOfficial
5 жыл бұрын
Don't joke about imaginary numbers Daniel! This is VERY serious business.
@0xDEAD_Inside
5 жыл бұрын
Please explain me!
@canaDavid1
5 жыл бұрын
imagine how bad puns would come out of that! Hehehe... "Imagine"-"imaginary"? No? Only me laughing? Anyway, bad pun intended
@danthiel8623
5 жыл бұрын
Daniel is my name to
@TheoParis
4 жыл бұрын
@@canaDavid1 lol
@Alistair
2 жыл бұрын
@Yo ming concur
@ashc5536
3 жыл бұрын
What can I say Dan..... You are a legend! (And you dance exactly like me!)
@sergten
4 жыл бұрын
I believe the performance issue for the logo path is due to the fact that in the dft() function the number of frequencies computed is equal to the number of the data points in the signal array. So if you have 1000 points in the data array - you will be computing 1000 frequencies. It's unlikely that anyone will need that many, most of them are probably close to 0.
@btwstr220
5 жыл бұрын
most helpful source for people who don't understand math functions but understand js :D
@EnriqueMartinezAgraAgra
5 жыл бұрын
Man i can tell you. I hate javascript and still love your channel. This is gold
@BlueyMcPhluey
3 жыл бұрын
for something so intimidating, I can't believe how quickly you reproduced that DFT algorithm
@mericinhikayesi8474
Жыл бұрын
I've been looking for this content for years
@cosif3001
5 жыл бұрын
greatest and most complicated programe ever created by a KZitemr
@Hoobizone
4 жыл бұрын
Daniel thank you!! so crazyy
@InspireToMake
5 жыл бұрын
So... he basically reinvented the Gcode?
@chri-k
3 жыл бұрын
Basically yes. But In a wierd way
@woulg
3 жыл бұрын
Isn't the path file already sorta like gcode? The Fourier transform just gives you the size and starting angle of the little circle/rotating things
@yeorinim2sida
3 жыл бұрын
This practical implementation is 100 x better than 10 videos dealing with the theorem.
@darkfrei2
3 жыл бұрын
Thanks for your help! It was always interesting how the fourier transformation works.
@kosmic000
5 жыл бұрын
amazing!!!
@steveburrus9347
5 жыл бұрын
"KOSMIC DUST" where the hell did you blow in from anyway?
@FirstLast-cl7xz
5 жыл бұрын
Yay new video!
@username42
5 жыл бұрын
it was more fun to watch the live series :D
@loic.bertrand
5 жыл бұрын
"I made a terrible array" 😂😂
@shaymalkin6846
5 жыл бұрын
Brilliant work! One of your best ones.. Thank you :)
@gregcostanzo4724
5 жыл бұрын
3:06 I’m just glad you don’t put bananas in your smoothies.
@mattmar96
3 жыл бұрын
Thank you
@georgie3
3 жыл бұрын
Here's a code improvement that lets you use any time step size and see what happens between sample points. You can learn a lot about how DFT works this way, and it shows what a real "Fourier machine" would draw if you actually built it. The original code relies on aliasing for negative frequencies and plots an idealized path by only "hitting" the original sample points. To fix the negative frequencies issue change lines 31 & 32 to the following: let ang = 0; if (freq < fourierY.length/2) ang = freq * time + phase + HALF_PI; else ang = -(fourierY.length - freq) * time + phase + HALF_PI; x += radius * cos(ang); y += radius * sin(ang); Then you can reduce the time step size -- change the original line 50 to a small number or something like this const dt = TWO_PI / fourierY.length / 10; If your signal is undersampled you'll see a lot more oscillations since the DFT is only able to approximate the original signal. However the plotted line always goes through the sample points and you can increase accuracy by increasing the number of samples (signal array size). Functions containing sine waves don't need very many samples, for example this function works reasonably well for arbitrarily small time step sizes with as few as 16 samples: y = sin(x)*2 + sin(x * 3 + PI/7) for x from 0 to 2*PI. Note there is an ambiguity at N/2 so for some signals like a single straight line odd array sizes might work better.
@TheCodingTrain
3 жыл бұрын
Thank you for this! If you like you can submit it as a community contribution on the coding train website! thecodingtrain.com/CodingChallenges/130.1-fourier-transform-drawing.html
@Mikus_Ficus
2 жыл бұрын
Thank you so much Daniel!
@Dr.AdnanKhan
2 жыл бұрын
Amazing, The way you explained it, its very easy.
@Hagledesperado
5 жыл бұрын
A crude attempt at reducing the number of points in the drawing (to 375 in this case) instead of skipping 7 out of 8 points which gives 625 points, while keeping most of the details of the original 5000 point drawing: function simplifyDrawing(drawing) { const maxHeadingDiff = 0.4; const maxStrokeLength = 16; let newDrawing = []; let headingDiff = 0; let strokeLength = 0; let prevVec = null; for (let i = 0; i < drawing.length -1; i++) { let vec = createVector(drawing[i+1].x - drawing[i].x, drawing[i+1].y - drawing[i].y); if (prevVec == null) { newDrawing.push(drawing[0]); } else { headingDiff += vec.heading() - prevVec.heading(); if (abs(headingDiff) > maxHeadingDiff/8) { strokeLength += vec.mag(); } else { strokeLength += vec.mag()/128; } if (headingDiff > maxHeadingDiff || headingDiff < -maxHeadingDiff || strokeLength > maxStrokeLength) { newDrawing.push(drawing[i+1]); headingDiff = 0; strokeLength = 0; } } prevVec = vec; } if (strokeLength > 0) { newDrawing.push(drawing[drawing.length-1]); } print('Simplified drawing from ' + drawing.length + ' to ' + newDrawing.length + ' points.'); return newDrawing; }
@parvezmahmud6963
4 жыл бұрын
What is this actually?
@joaquinsifuentes7726
Жыл бұрын
do you know ho to create an array of points in js from an random image?
@kossboss
Жыл бұрын
This was Fwording insane!
@DaKnightsofawesome
5 жыл бұрын
yea but can you do it like 3blue1brown and make the picture from just one set of circles?
I loved it Dan, thank you very much for all this! Also i'm making it happens with dart/flutter!
@TheCodingTrain
3 жыл бұрын
Please share when you do!
@igotapochahontas
5 жыл бұрын
Thanks for making this!
@ElBellacko1
3 жыл бұрын
i have been deep in the rabbit hole after watching part one of this video
@SanuAntony
Жыл бұрын
Wonderful
@marcpaters0n
5 жыл бұрын
I would love to learn how the Coding Train logo path was generated.
@ahmetakil787
5 жыл бұрын
yeah me too
@TheCodingTrain
5 жыл бұрын
Ask twitter.com/tomfevrier, he made it!
@defnotdeepakr
5 жыл бұрын
Probably something to do with SVG data
@swagagain8651
5 жыл бұрын
1) Load picture. 2) Create array. 3) Make function to fill array with coordinates when clicked. 4) Click only 5000x on picture. 5) Profit! Simple as that.
@alexandar1794
4 жыл бұрын
Actually, you can easily convert any image to array of vectors using any cnc machine software
@zer001
5 жыл бұрын
Awesome, like always!
@zhabiboss
Жыл бұрын
Bob Ross of Fourier drawing
@samueldelucas4412
5 жыл бұрын
This is basically Etch-a-sketch, but cooler!!
@CJBurkey
5 жыл бұрын
Sweet. I was only able to see a part of the stream :)
@lotarjedrasik4565
3 жыл бұрын
Why you need that Fourier calculations at all??? To be able to draw the logo you need to draw the points taken from the logo array input. The same result, but without taking Fourier calculations.... This is insane. You are using your input array to perform heavy calculations to get the output which in fact is your input!
@harikrishnametta8549
5 жыл бұрын
Dude I loved it! ❤❤
@tonyswain4174
2 жыл бұрын
Thinks; that is awesome :)
@anshul5909
3 жыл бұрын
Hi, i have learnt a lot from this channel. The content and references provided are great. Great job👍 I don't whether it's the right place to ask, but can you create a video on generating offset polygons. I think they are great, and challenging as well. Can you some techniques for working with them. I really appreciate
@braindeadgaming8430
4 жыл бұрын
Daniel: Always succeeds using Wikipedia Teachers: Nooooooooo!
@MrApploid
4 жыл бұрын
How to transform an image into a set of points like the train he made?
@keshavapiyushprasad5439
4 жыл бұрын
There is a p5 function called text to bounds for text, I dont know about images. You can check the p5 reference for other information
@iamr0b0tx
Жыл бұрын
Thanks
@TheCodingTrain
Жыл бұрын
Thank you for the support!
@iradnuriel9087
5 жыл бұрын
Cool one!!!!!!
@vinho3cocacola
5 жыл бұрын
i like how this relates to alternating current
@mafiozos2000
2 жыл бұрын
Was it only me or you stoped to find the "i will refactor it later" song as well?
@danielalejandrocordovadela657
2 жыл бұрын
I loved the video so much. I download your code, so I can play a bit with different pictures. However, I have problems to extract the x and y coordinates from a continuous line drawing picture. does someone know how to extract those coordinates other than manually?
@krakadoros8605
2 жыл бұрын
Hello. Congratulations for your video .I admire your talent to think, design and execute this. I have a big question. But let me explain with a simple example : Lets say that we have a very simple drawing that can described by this variables : (t , y) and these four points : (0 , 0) (0.1 , 1) (0.2 , 5) (0.3 , 11) (t on X axes and y on Y axes). I assumed that all the points between these 4 points are on a line between two successive points of the defined for points. I understood from your video that fft that applied on n samples (taken on n time points, lets say equal spaced), gives n complex numbers. From these n complex numbers, we can take the amplitudes and create an array of n bins. Now we can give labels to these bins, so that they correspond to frequencies, from 0 to fsampling, something like this : (0:n-1)*fresolution or (0:n-1)*(fsampling/n) or (0:n-1)/T were T is the total time of sampling. Now if we plot amplitudes of complex numbers (created by the transform), over fbins, we can see the frequency-ies from which the sampled signal have created. I can see that you separate real (a) and imaginary part (b) for each complex number, and calculate a x and a y like this : x=square root of (a*a + b*b) * cos (freq * time + atan2 (b,a) and y=square root of (a*a + b*b) * sin (freq * time + atan2 (b,a). I assume tha freq that is the "k" in fft equation, is each of (0:n-1)*fresolution. Am i correct ? And time is each of the time points on which each sample have been taken ? (Are these x an y, on a polar system, so if the signal that we have sampled was a sine wave, they will give a cycle as a graph ?). Now my main question is this : How, from the array of the 4 given points (in the beginning of my comment) we can define epicycles, to construct the graph-shape that these points define ? I am not interesting in to draw the graph, but only to define the epicycles, and i am not interesting in a system of 2 (epicycle "generators", like in your examples [one left-down and one right-up]), but for only one. What i am thinking to do is this : for the first point (0,0) 0=square root of (a*a + b*b) * cos (freq * time + atan2 (b,a) 0=square root of (a*a + b*b) * sin (freq * time + atan2 (b,a) for the second point (0.1 , 1) 0.1=square root of (a*a + b*b) * cos (freq * time + atan2 (b,a) 1 =square root of (a*a + b*b) * sin (freq * time + atan2 (b,a) for the third point (0.2 , 5) 0.2=square root of (a*a + b*b) * cos (freq * time + atan2 (b,a) 5 =square root of (a*a + b*b) * sin (freq * time + atan2 (b,a) and for the fourth point (0.3 , 11) 0.3=square root of (a*a + b*b) * cos (freq * time + atan2 (b,a) 11 =square root of (a*a + b*b) * sin (freq * time + atan2 (b,a) I do not know if i can calculate something from the above equations .... Confused...
@krakadoros8605
2 жыл бұрын
Anybody can help ?
@user-rq4zw8qv9k
7 ай бұрын
Nice work! Here you used two signals x and y as pairs of the coordinates to draw it. You can also draw it with one signals e.g., y only, right?
@indianhuman356
4 жыл бұрын
Can you drow projection of 3d train into n dimensions greater than 3. Using this technique?
@yurinagel2270
5 жыл бұрын
Hello Daniel, I'm now following your tutorial videos about Java, after i saw you making cool things i wanted to make my own sort of minigame. The thing is i want the bal to bounce back after that the ball hits the pedal, but the problem is i don't know how. Here is the thing i tried following the whole script. ellipse(x,y,24,24); if ((x==mouseX) && (y==300));{ x+=9; } rect(mouseX, 300, 94, 24); float x=width/2; float y=height/2; float kleur1; float kleur2; float kleur3; int xspeed = 2; int yspeed = 2; boolean xgaan = false; boolean ygaan = false; boolean xwaar = false; boolean ywaar = false; void setup() { size (640,360,P2D); background(kleur1,kleur2,kleur3); } void draw() { background(kleur1,kleur2,kleur3); fill(0,255,0); ellipse(x,y,24,24); // if ((x==mouseX) && (y==300));{ //x+=9; //} rect(mouseX, 300, 94, 24); kleur1 += 0.8; kleur2 += -0.8; kleur3 += 3; x += xspeed; if (x > width){ xspeed *= -1; xwaar= true; } if (x < 0){ xspeed *= -1; xwaar =false; } if(mousePressed == true) { xspeed = 0; } if(mousePressed==false){ if(xwaar == false){ xspeed=2; } } if(xwaar == true){ xspeed=-2; } y += yspeed; if (y > height){ yspeed *= -1; ywaar= true; } if (y < 0){ yspeed *= -1; ywaar =false; } if(mousePressed == true) { yspeed = 0; } if(mousePressed==false){ if(ywaar == false){ yspeed=2; } if(ywaar == true){ yspeed=-2; } } }
@TheCodingTrain
5 жыл бұрын
Would you mind asking at discourse.processing.org/! It's a better platform for Processing and p5.js related code questions. You can share code there easily! Feel free to link from here to your post.
@diophantine1598
4 жыл бұрын
Now you just have to do it with a single bunch of circles. (Nothing different between x and y)
@zxuiji
3 жыл бұрын
So basically draw in a convoluded way when it is simpler to just normalise those values and multiply them against size and add them to the xy positions wanted
@adilkhatri7475
5 жыл бұрын
really really amazing!!
@neopalm2050
5 жыл бұрын
This isn't epicycles. Using the Fourier transform that works with complex numbers, this would have used actual epicycles.
@gloubiboulgazeblob
5 жыл бұрын
Awesome ! Really awesome ! But how did the guy to generate the logo ???? I think he has "scanned" the logo with a vertical line, then, for all points intersecting that line, he has summed up their y coordinates returning him a value, a kind of signal value. Then he has processing that signal as any signal with an FFT giving him all the values needed for the drawing. Isn't there a more elegant way of writing the SUM function by taking advantage of Javascript ? I tried the something like the following but there's a little problem with passing parameters to the callback function for the real example of your video, I need help, but I tried this and it works fine : //this one is valid for ANY sum you want to implement const Sigma = (start, end, callback) => { sum=0; for (let n=start; n < end; n++) sum += callback(n); return sum; }; //Then, write the callback function you want, like: const f = (n, obj = {"x" : -5}) => { return Math.cos(2*Math.PI*n) * obj.x; }; //And call Sigma from the main code, passing it the f function.Well, roughly speaking of course... X = Sigma(0, N-1, f);
@thecodingcaravan9013
5 жыл бұрын
kzitem.info/news/bejne/1KZovmerfH6Cpn4
@JohnPaulBuce
2 жыл бұрын
im lost halfway in the video but i still love watching your videos, i wish i could program like you lol im so bad at this
@thecalculatedcreativecoder1428
10 ай бұрын
Its an accomplishment that you got the first half though!!! Keep working at it and over time you will improve.
@MarkovChains223
5 жыл бұрын
Fun additional challenge: have the epicycles draw their final position when the drawing is finished.
@pierreandferb
5 жыл бұрын
ah i saw it on reddit hahah would be cool though
@eitanshirman9072
3 жыл бұрын
you just broke my mind. wait, it`s a paradox, right?
@hedibenfraj6158
5 жыл бұрын
Can u do the cristian ilies vasile visualization of pi it's so awesome !!
@dharmikmistry8781
4 жыл бұрын
Super 🔥🔥🔥
@Rousnay
5 жыл бұрын
I love this mad guy 😍
@stevew4213
4 жыл бұрын
Upon watching 'Coding Challenge #130.1: Drawing with Fourier Transform and Epicycles' I had a few questions that I am hoping the board can answer. 1) Given all the points from the picture being drawn, couldn't the picture have been drawn just connecting the points? 2) The equation included the imaginary part 'i' and i is defined a sqrt(-1), but sqrt(-1) was not used in the algorithm, that is a little confusing.
@eeerrrrzzz
Жыл бұрын
Teşekkürler.
@TheCodingTrain
Жыл бұрын
Thank you for the support!
@JwalinBhatt
Жыл бұрын
Could someone please explain why we add phase at 25:48?
@satanicmonkey666
4 жыл бұрын
I like this video, earned a sub. Thanks
@arka9504
4 жыл бұрын
I hope my FTIR (Fourier Transform Infrared, in chemistry) can drawing like this.
@shyambutani8618
2 жыл бұрын
Wow
@akshaypratap2010
5 жыл бұрын
Great video sir ji
@timk4904
5 жыл бұрын
Looks like a 3D printer 😂
@TheCodingTrain
5 жыл бұрын
I would love to see someone build a physical version of this!
@MasterMindmars
Жыл бұрын
Amazing work. Do you know how to get the x,y coordinates or the fft parameters of Catherine Zeta jones ? 😀
@umbertohe5723
5 жыл бұрын
😨 Amazing
@kumarprosenjit9714
5 жыл бұрын
Great. Thx .
@PeterVanAlem
5 жыл бұрын
23:57 didn't know Jim Carrey could code!
@sujals7108
5 жыл бұрын
4:03. infinitely better, that hit the mathematician in me so hard
@TheCodingTrain
5 жыл бұрын
😂
@zokalyx
5 жыл бұрын
This is epic! How can you make it like that, but with only one set of circles? (3:28) If you had a link or something I have to search, that's more than enough.
@TheCodingTrain
5 жыл бұрын
I'm hoping to tackle this tomorrow.
@AryK6227
Жыл бұрын
i'm not sure why but for me, only the 7th index of the transformed hard-coded square wave gives wrong values. all the other values are fine. Why does this happen?
@soumyodiptanath2917
4 ай бұрын
How do I generate the correct sequence of X and Y for an image? Is there any algorithm for that??
@emjizone
Жыл бұрын
@TheCodingTrain It sound silly to me to use two fourrier transform when *one fourrier transform already gives the two component vectosr* you need (radius and angle, or real part and imaginary part of the complex number). Obviously, *you don't need four component vectors to trace dots on a plane.* A two motor drawing machine would also be much simpler to build, operate and maintain, and more reliable, than a four-motor machine with gliding arms for intersection, so this two-FT method doesn't make much sense from an industrial point of view either. However, *you could use this two-FT idea to create 3D wire* instead of 2D drawings, that looks like a 2D image when seen from one point of view and looks like another 2D image when seen from another point of view. That would be insteresting and would completely justify the use of two FT. Feel free to ask me to collaborate with you on such project if you want.
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