Instead of rotating the curve, you can make the POV change during time. Parametrize the path of the POV, deduce the equation of the plane onto which the projection will be done in terms of the coordinate of the POV and make a change of basis to get rid of the z coordinate. Sounds more difficult but is actually not that hard.
I'm not sure, but I think moving pov away from the Z axis requires a lot more work. You need to use a perspective projection matrix, think about FOV. The camera is a vector and the direction the position looks is backwards, and a lot more complicated stuff. When I tried to code this using c# I messed up on so many levels that I gave up, I had a deadline for the project so I used an external library. I will definitely write proper poor 3d library only for sake of my linear algebra knowledge
@@slayvict I don’t really know what a perspective projection Matrix is but if you have a POV that always look at the center of the 3D space, you can deduce the equation of the plane you project onto, the equation of the projection of the original 3D curve, the change of basis matrix only with the coordinate of the POV. Which means that if you parametrize the POV, you can actually do exactly as much as with a fixed point. I did this a few weeks ago and it worked perfectly.
@@samylahlou Perspective projection matrix is simply division by z axis raised to some power like 0.4 to both x axis and y axis. it took me too long to realise this simple fact in making 4-cube.
4D: maybe, there's a map editor for 4D miner, and you could use something like marching [hyper-]cubes to determine which blocks would make part of the mesh.
Love it, before this video came out I decided to play around with 3D rotations because of the tease at the end of last video. I got it working using a rotation matrix and the formulae erre actually much simpler than yours, although I cannot as easily change the rotation axis. I'd probably have to rotate twice. Anyways earlier today I made an interactive program in desmos where you can move on the z axis to and from a unit sphere around the origin. You can change certain parameters like opacheness and auto rotation. Maybe I'll post it somewhere when I'm finished because I'm honestly a bit pround about it ^^' About the 4D case I think it would be quite similar if youuse the same formula to project it first from R⁴ to R³ and then to R². Maybe something like 30/30-w*[x,y,z,0] (w is the 4th coord) again and from there you have it in 3D again. Maybe I'll play around eith it tommorrow xD
Update: So I tried the 4D thing and in theory it seems to work. I've succesfully created a grid of the "surface" of a hypersphere. But I am not happy with my projection algorithm 1/((d-z)*(d-w))*(x,y). It is simply the one for 3D applied twice and because of that it treats the z and w axis equal which I do not like. Moving 1 across the z axis looks the same as moving 1 across the w axis. Maybe I'll find a better formula Update: I made a Hypercube and I got it spinning and I actually got an animation similar to the one we all know where it turns inside out. I prijected it from 4d to 3d by first creating a point like a lightsource and then calculated it's 3d shadow on the x,y,z space. The difference to before was to make that point independent from the observer
@@didodido883 It was still an interesting watch, so thank you. I think my formula actually works relatively fine. It would be p(x,y,z,w)=d2/(d2-w) * [x,y,z] to have it in 3D and then you can just use the method from the video to get in to 2D. At this step you can use the 2D rotation stuff from video one to rotate it around any plane like the x-w plane to get some cool effects. Edit: maybe I should add that d2 is just a constant referring to the 4D point from which to project to the x-y-z hyperplane. It has the coordinates (0,0,0,d2)
Why not? From the last video I got the idea that you could have a graph rotating round a point which is already rotating around another point and even that would be rotating around a third point. All at the same time. Or whatever craziness.
even easier easy way: rotate in geometric algebra, then the rotated function is just exp(-theta/2 B) v exp(theta/2 B) where B is a unit bivector representing the plane that the rotation is occurring in
Traditional linear algebra doesn't have vector multiplication beyond the dot and cross products, though those do respectively correspond to the cosine and sine functions. A true vector product, like the one from geometric algebra, would be akin to replacing cosine and sine with an exponential. In fact, in geometric algebra, rotating a vector v around an axis a by angle θ is done by exp(θa/2)*v*exp(-θa/2). (Why the /2? Because the coefficient is more related to the area of the circular sector than the arc length around it. Alternatively, it's quantum. Either way, quaternions work the exact same way.)
I'm assuming projection of 4D graphs would require the projection algorithm twice, right? The first projection would project the 4D graph onto a 3D surface, and the second projection would just be the same as shown in the video, to project the 3D projection onto a 2D surface. Sure, the first projection would be more complicated, and the entire graph function would look like a nightmare, but I guess that's doable? I'm not sure if Rodrigues' Rotation Formula works though, I still don't understand rotations in 4D... I haven't really thought about all this much, so what I just said may or may not be wrong, so if there's any smarter people here in the comments, it would be nice to share your insight.
When I try to copy these equations into Desmos, I'm getting very choppy animation, instead of the nice smooth animation in the video. My computer's CPU is an Intel Core i7-6700, which isn't blazing, but seems fine for most things. Is that the problem, or am I doing something wrong?
Hey! You probably aren’t gonna read this, but I’m a fellow abuser of desmos. I would really love to show you some of the wacky stuff I’ve made in desmos, if you’re interested. I have a graph about a rhombicosidodecahedron, all 3-d rendered with multiple variable side lengths, and it’s kinda the coolest thing I’ve made in desmos. I’d give you a link, but I tried that last video and I think the algorithm thought I was a bot and deleted the comment. Love the videos, I was one of your first thousand subs (although I just subbed like 2 days ago). If you wanna see it, just give this a reply and I’ll give you the link ASAP.
Could you link the working Desmos graph in the description, I can't figure out what's wrong with mine and the working one would help me find my problem.
Can we share links on youtube? I’ve tried before and it hasn’t let me before… I wouldn’t mind sharing a fully functional parametric 3D grapher I made on desmos.
@@codevector211So I just pasted the share link and posted the comment, but it doesn’t display as a comment when I reload the comments, so let me know if you can actually see the link.
WOOOOOOO PART 3 LETS GOOOOO
Yes part 3 lets goooooooooo
Alright. Let's go.
Lets goooooooooo
YESSIR
saying "LETSS GOO" is pointless
This is just like drawing a cube on a paper. We stretch one face of square linearly along the plane to make it a cube. This concept is interesting
Instead of rotating the curve, you can make the POV change during time. Parametrize the path of the POV, deduce the equation of the plane onto which the projection will be done in terms of the coordinate of the POV and make a change of basis to get rid of the z coordinate. Sounds more difficult but is actually not that hard.
I'm not sure, but I think moving pov away from the Z axis requires a lot more work. You need to use a perspective projection matrix, think about FOV. The camera is a vector and the direction the position looks is backwards, and a lot more complicated stuff. When I tried to code this using c# I messed up on so many levels that I gave up, I had a deadline for the project so I used an external library. I will definitely write proper poor 3d library only for sake of my linear algebra knowledge
@@slayvict I don’t really know what a perspective projection Matrix is but if you have a POV that always look at the center of the 3D space, you can deduce the equation of the plane you project onto, the equation of the projection of the original 3D curve, the change of basis matrix only with the coordinate of the POV. Which means that if you parametrize the POV, you can actually do exactly as much as with a fixed point. I did this a few weeks ago and it worked perfectly.
@@samylahlou Perspective projection matrix is simply division by z axis raised to some power like 0.4 to both x axis and y axis. it took me too long to realise this simple fact in making 4-cube.
NO WAY I WAS JUST DONE WATCHING PART 2. THIS MATERIAL IS PURELY BONKERS. Let's goooooo bean dude
To avoid the formulas becoming too unwieldy, I like to use shadertoy instead of desmos. Though that doesn't do as well with parametric equations
This is why math is beautiful
I saw the first two videos a few days ago and now I'm a desmos wizard.
Okay that was beautiful 3 part series you easily earned the sub
This is answering a lot of long time questions I just never bothered to go look for - but so glad I found
I love this series! And i love that it actually gives an insight into the concept of how 3D images on computers work!
Your videos are amazing! Thank you for making them.
"Spiral vortex"
So just an ordinairy spiral, but the radius is exponential?
That ought to be an interesting projection :)
Bro! Your channel is an absolute goldmine of youtube! Golden Content!!! Keep it up man:)
1 step closer to creating doom in desmos.
As one of the few people who actaully drew 3D stuff in desmos this is easy.
You are honestly just incredible
I love watching the number of subscribers grow!
God I love your content
even with b1 i can understand you, such a good job!
Wow, that is a monster of an equation!
Bro, your vids are amazing. BONS PRA KRL. Keep on the awesome work!
The fact that Desmos now has a 3d version makes this video very useful.
Damn the subs be growing at 2k a day
You deserve even more
Thank you a lot of knowledge, could you make a lot of math like this, I love you.
my favourite series :)
daily portion of desmos madness :D
you should make a series on transforming an axis into any curve, i did it for a parabola and it gave some pretty interesting results
Keep up the good work, Beanie!
This is beautiful.
Great video❤, could you make a video about how to mirror any graph over any point or any line?
Good job, and amusing persona
Weeeee, spining graphs!!
you are an artist
Beautiful.
4D: maybe, there's a map editor for 4D miner, and you could use something like marching [hyper-]cubes to determine which blocks would make part of the mesh.
Can't wait for Part 4
my brain exploded watching this video
This is gold!!
Love it, before this video came out I decided to play around with 3D rotations because of the tease at the end of last video. I got it working using a rotation matrix and the formulae erre actually much simpler than yours, although I cannot as easily change the rotation axis. I'd probably have to rotate twice.
Anyways earlier today I made an interactive program in desmos where you can move on the z axis to and from a unit sphere around the origin. You can change certain parameters like opacheness and auto rotation. Maybe I'll post it somewhere when I'm finished because I'm honestly a bit pround about it ^^'
About the 4D case I think it would be quite similar if youuse the same formula to project it first from R⁴ to R³ and then to R². Maybe something like 30/30-w*[x,y,z,0] (w is the 4th coord) again and from there you have it in 3D again. Maybe I'll play around eith it tommorrow xD
Update: So I tried the 4D thing and in theory it seems to work. I've succesfully created a grid of the "surface" of a hypersphere. But I am not happy with my projection algorithm 1/((d-z)*(d-w))*(x,y). It is simply the one for 3D applied twice and because of that it treats the z and w axis equal which I do not like. Moving 1 across the z axis looks the same as moving 1 across the w axis.
Maybe I'll find a better formula
Update: I made a Hypercube and I got it spinning and I actually got an animation similar to the one we all know where it turns inside out.
I prijected it from 4d to 3d by first creating a point like a lightsource and then calculated it's 3d shadow on the x,y,z space. The difference to before was to make that point independent from the observer
@@WhyneedanAlias steal the code from here.
ua-cam.com/video/4URVJ3D8e8k/v-deo.html
or i think source code is not being provided in that video.
@@didodido883 It was still an interesting watch, so thank you.
I think my formula actually works relatively fine. It would be p(x,y,z,w)=d2/(d2-w) * [x,y,z] to have it in 3D and then you can just use the method from the video to get in to 2D. At this step you can use the 2D rotation stuff from video one to rotate it around any plane like the x-w plane to get some cool effects.
Edit: maybe I should add that d2 is just a constant referring to the 4D point from which to project to the x-y-z hyperplane. It has the coordinates (0,0,0,d2)
Sense you proj from 4 space, can you proj double rotation to 2 space with your formula?
Very educational.
im waiting for part 4
An idea to spend a video on geometric algebra? It seems to do miracles on computer graphics.
Matbatwings did method two in minecraft with redstone, such a cool creation
this is gonna be fun
Amazing!!!
It would be nice to have a complete desmos tutorial, i dont even know how to do parametrics equations
Nice it's finally here
That man has a brain bigger than my mom
insane big pog content
Liked before watching
You explained to Principe of Ray Tracing by the way!
More please 🙇♂️
Ur awesome :)
Lets go🎉 episode 3
Your videos remind me a lot of a guy I used to watch called Sen Zen
yoooooooooo this was soo cool
gg my man jumped to 6k subs from 1.5k in 2 days
yeah everyone only subs to him for his content, not for anything else. They probably wanted to see this so they could copy him and brag about it.
At 20:00, would it be possible to have it spinning like it is now and simultaneously rotating along the line through itself like it was just before?
Yes but you will have to use the rotating equation multiple times
Why not? From the last video I got the idea that you could have a graph rotating round a point which is already rotating around another point and even that would be rotating around a third point. All at the same time. Or whatever craziness.
It's very hard to follow. Anyway, you are very clever. It was a MBA cause?
"We're gonna go for the general case" after setting the viewing point (0,0,30) rather than generalizing it.
lol
Woo hooo thank you!!!!
PART 3 LETS GOO
even easier easy way: rotate in geometric algebra, then the rotated function is just exp(-theta/2 B) v exp(theta/2 B) where B is a unit bivector representing the plane that the rotation is occurring in
i see the mysterious youtube algorithm has caught you ;)
you are awesome
bro is cracked at desmos
this is real cool
Brilliant
Part 4!!! Plz
Thanks
18:08 I tried in a different axis of rotation but forgot to normalise and crashed Desmos
🔥🔥🔥🔥🔥🗣️🗣️🗣️🗣️❗❗❗❗💯💯💯💯 Gatech quality
You just had to upload while I was recording, now I had to restart :(
I wonder if this could be simplified with vector multiplication instead of sin and cos.
Traditional linear algebra doesn't have vector multiplication beyond the dot and cross products, though those do respectively correspond to the cosine and sine functions. A true vector product, like the one from geometric algebra, would be akin to replacing cosine and sine with an exponential. In fact, in geometric algebra, rotating a vector v around an axis a by angle θ is done by exp(θa/2)*v*exp(-θa/2). (Why the /2? Because the coefficient is more related to the area of the circular sector than the arc length around it. Alternatively, it's quantum. Either way, quaternions work the exact same way.)
@@angeldude101 no shit! That's crazy. I just started learning about dot and cross products. The rabbit hole just keeps getting deeper
I'm assuming projection of 4D graphs would require the projection algorithm twice, right? The first projection would project the 4D graph onto a 3D surface, and the second projection would just be the same as shown in the video, to project the 3D projection onto a 2D surface. Sure, the first projection would be more complicated, and the entire graph function would look like a nightmare, but I guess that's doable?
I'm not sure if Rodrigues' Rotation Formula works though, I still don't understand rotations in 4D...
I haven't really thought about all this much, so what I just said may or may not be wrong, so if there's any smarter people here in the comments, it would be nice to share your insight.
no part 4? big sad 😭
Woooooot!
I don’t suppose there is a desmos link to this? Just type in some numbers and it does the rotation for me?
That would be great
When I try to copy these equations into Desmos, I'm getting very choppy animation, instead of the nice smooth animation in the video. My computer's CPU is an Intel Core i7-6700, which isn't blazing, but seems fine for most things. Is that the problem, or am I doing something wrong?
Por qué no subes más videos? 😢
Thanks to this mindblowing video I managed to make a square graph without infinite exponents.
Here it is btw:
|x-y|+|x+y|=1
Hey! You probably aren’t gonna read this, but I’m a fellow abuser of desmos. I would really love to show you some of the wacky stuff I’ve made in desmos, if you’re interested. I have a graph about a rhombicosidodecahedron, all 3-d rendered with multiple variable side lengths, and it’s kinda the coolest thing I’ve made in desmos. I’d give you a link, but I tried that last video and I think the algorithm thought I was a bot and deleted the comment. Love the videos, I was one of your first thousand subs (although I just subbed like 2 days ago). If you wanna see it, just give this a reply and I’ll give you the link ASAP.
UA-cam recomendations in 6am be like
Could you give me the matrix operations that produced graph of projection on 9:24 minutes of Part 3? Thanks
you should work for desmos
what if you wanted the rotation axis to rotate around a different axis
jesse from breaking bad explaining maths (meths) yo
Could you link the working Desmos graph in the description, I can't figure out what's wrong with mine and the working one would help me find my problem.
it doesn't work for me :(
yes this would be so cool! I hope they get around to adding the link to the description.
Can we share links on youtube? I’ve tried before and it hasn’t let me before… I wouldn’t mind sharing a fully functional parametric 3D grapher I made on desmos.
@@QP9237 Yes you can! Just copy and paste the link...
@@codevector211So I just pasted the share link and posted the comment, but it doesn’t display as a comment when I reload the comments, so let me know if you can actually see the link.
here we go...
WOW!
what about if i waht to rotate points in 3D, and i can input XYZ as i want in that point?
Game of Life.
What was the MOB function?
If we can project 3d onto 2d... maybe we can project 4d into 3d...?
YEEEEES
18:18 he sounds so fed up 💀
Ok, now what about rotate a 4D graph in a 3D space?
😢 y did blud stopped making videos
Yeeeeeeaaaaaaaaaaaaaaaaahhhh !!!! :D
Ayyyyyyyyyyy