The Beauty of Bézier Curves
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- čas přidán 7. 05. 2024
- Bézier curves - how do they do?
They're used for animation, text rendering, and all sorts of curved shapes! But how do they actually work? well, like, that's what the video is about, so, watch it to find out etc!!
• Lots of love to
💛 Jazz "queenjazz" Mickle for making the music ❱ queenjazz.bandcamp.com/
💙 Grant "3Blue1Brown" Sanderson for pushing me to finally do this ❱ / 3blue1brown
🐦 Pomax for their wonderful writeup on Bézier curves ❱ pomax.github.io/bezierinfo
🐈 Thor the cat for.. hanging out.. when I was trying to record audio with no interruptions
🇫🇷 French subtitles by @damqui on Twitter
• How was this video made?
🔨 Created in the Unity game engine
⭕ Using Shapes, my vector graphics plugin ❱ u3d.as/1W37
📈 Using Mathfs, my math library ❱ github.com/FreyaHolmer/Mathfs
...along with a bunch of hacky, partially broken, procedural animation tools I made specifically for this video
• Did you know that "osculating" isn't pronounced that way?
✅ yes I do now in fact please stop commenting on it ;-; ❱ / 1429472399389507591
• Links
💖 Patreon ❱ / acegikmo
🐦 Twitter ❱ / freyaholmer
📺 Twitch ❱ / acegikmo
💬 Discord ❱ / discord
🌸 Instagram ❱ / freya_holmer
00:00 Intro
01:06 Use Cases of Bézier Curves
01:56 Lerp
02:34 Quadratic Béziers
03:07 Cubic Béziers
03:57 De Casteljau's Algorithm
04:25 Bernstein Polynomial Form
06:18 1st Derivative (Velocity)
07:31 Tangents & Normals
08:03 2nd Derivative (Acceleration)
08:31 3rd Derivative (Jerk/Jolt)
09:07 Curvature
11:05 Bounding Box
14:14 Arc Length
15:07 Arc Length Parameterization & Uniform Animation
19:13 Outro
19:54 Credits
23:57 Thor helping me record this at 2am - Věda a technologie
🐱💖✨ Frequently Asked Questions ✨💖🐱
🤔: How did you make this video?
• Unity, the game engine, coding in C#
• using my vector graphics library Shapes
• using my own frame recorder tool
• using lots of hacky procedural animation tools made specifically for this video.
• exported as a .png sequence into DaVinci Resolve, for the final video editing
• audio was recorded in FL Studio because apparently the voiceover feature in Resolve is just bad and broken?
🤔: How long did it take?
33 days from start to finish, somehow. I spent the last two days or so on the video editing, while the rest was work in Unity itself. In parallel, Jazz Mickle made the music, which I believe she worked on for a few days!
🤔: Did you know that "Osculati-
yes after like 25 comments I finally know that that is, in fact, not how you pronounce "osculating" now please for the love of god stop commenting on it I've already added a note about it in the description but none of yall read that so please save me from more of this ;-;
🤔: I don't like your voice
..? okay? not sure how to answer that, it's a pretty weird thing to say. I don't like your comment! how about that?
Sick burn for the last one! XD
I dont like this coment
"i dont like your voice" who would ever say that?
i wish i had a voice any close to that. have been trying voice training for a year now, no progress, only tears every day. considering becoming mute
I like Your voice.
I absolutely love your voice!
And, I’m so proud at the sheer work and effort put into this month long project. Stunning work. 💜
This video was a joy to watch! Really well explained, and such beautiful visualizations.
This actually helped me to understand what Lerp() concretely was when watching your gamedev tutorials xD You guys are awesome!
18:23 did you help with this screenshot? Pasting in hard-to-decipher code seems up your alley
I couldn't agree more. What a great work with the visuals!! :)
@@katowo6521 😂😂😂😂😂😂
@@katowo6521 wait does that mean my video is a grocery store
Super cool. I'm really going to have to start using Shapes for my graphs.
You're that evolution guy! Love your vids
ayo blob wit da mango and wood? 🥭🪵
Hey its this guy!
Blob!
Was not expecting you here. Hello!
This is brilliant, it felt almost illegal for knowledge of this quality to be free!
cringe.
@@goodgoing4615lmao what
@@goodgoing4615 get help
@@goodgoing4615 That's not very nice.
@goodgoing4615 he's paying for you to watch extremely good education for free, what's cringe about that?
It’s hard to say which value is higher in this video - the pedagogic or the production one!
This is _the_ clearest and most gorgeous explanation of Bézier curves I’ve ever seen!
pedagogy is for children. Something OP can't have.
@@TheBcoolGuy Um... I think you may need to look that up again? Pedagogy is the name given to the practice and methodology of teaching, if something has good pedagogy that means it's good at teaching a subject. Pedagogy is basically the study of how to teach, the knowledge and understanding of why certain things work better than others at imparting knowledge and under what conditions, the craft and experience involved in doing so.
It's true that Pedagogy (etymology based on Greek for "child" and "leader") started out as referring to the teaching of children, and Andragogy was more meant for the teaching of adults, but this is basically a misnomer. Pedagogy actually means, especially in modern times, the craft or study of teaching from the perspective of a teacher-student approach to learning, whereas Andragogy is more learner-centric. In a pedagogic environment the learning is directed by an instructor who provides guidance, instruction and assistance to the learner who is learning a particular subject. In an andragogy environment the focus is more put on the learner's self-direction, where the learner takes responsibility for their learning, it's teaching the learner how to more efficiently learn on their own, how to find the information on their own, give the learner motivation to learn, etc, rather than on a particular guided course. Pedagogy can be used to teach adults, and Andragogy can be used to teach children, despite the origins of their names and both have advantages and disadvantages.
Regardless @crkvend was merely stating how much educational value the video has, not sure what the response contributes?
@@TheBcoolGuyyour power level is showing lmao
"3b1b but it's about gamedev" is a really good concept that I'd love to see you keep pushing!!!
If that's what you're looking for, I'd highly recommend Sebastian Lague!
why does everyone compare 3b1b to anything close to math lol
@@adammuse3541 because of the very clean, very clearly code-driven animation style.
@@adammuse3541 👆what they said. Plus he makes it easy to understand.
@@adammuse3541 what?
The way the animations assist the narrative - resetting when needed, blending smoothly from one concept to another - is a joy in itself and makes this so much easier to take in than it otherwise would have been. This is so well done.
Yeah nice!
@Freya there are no words to describe how valuable your content is to me. in 20 minutes, you have cleared up 2 years of curiosity and misunderstanding. Your explanations not only cater to the exceptionally educated but to mere mortals as well. Thank you. I never knew how interesting these subjects were until i began my journey into 3D, VFX and Animation but these videos are perhaps some of the highest quality I've seen to date. Also that includes being up there with the same quality I receive at my uni doing a degree in the field. Simply outstanding videos and work you do. Thank you for this content you build!
I'm so happy you like it maize!
@@acegikmo i have been a follower for some time and ill continue to be, i dont often comment but i really feel like you deserve the praise. loots of work goes into this and i can appreciate what you have built thus far is a mountain of exceptionally high quality, most dont realise the level that you are actually on, and im betting there is a whole lot of information that you know that simply we just will never learn due to its complexity. and im some one that loooves complexity, so much so ive built my own distro's in linux and i have my own git service on linux servers, i build software etc so i know a degree of complexity. but your on a whole other tier and I think you are right up there with the best the industry has to offer. Thank you for all you do !
@@acegikmo
Fantastic
👍😎👍
"Sometimes the value under the root is negative. These roots aren't real, ignore them"
Such a literal explanation gave me a chuckle
Hooooly cow, that intro animation was SO gorgeous.
Scratch that, ALL the animations.
right?! it's 3b1b level at this point
Especially the cat at the end
@a person
With all due respect, it's *way* better than 3b1b. Those animations are on a whole other level than anything out there. Procedural perfection, not to mention beautiful style.
@@ittixen not better, just a matter of taste
@@simohayha6031 Of course. Both "3b1b level" and "better than 3b1b" are opinions. Nobody claimed to be the arbiter of objective truth here.
Imagine a blind person suddenly receives their sight.
That is what happened to me during this video.
Now the course of my life is forever changed.
I humbly request the full breadth of all things splines. Thank you Freya!
...Now the course of my life is forever changed...
Hopefully along a spline. :)
I watched this not knowing anything
And still don't know anything
@@mikemack7933 brush up to pre-calculus and you should have the tools to then intuitively understand the process. That and geometry.
The queen of the (Nordic) Pantheon, opening our mind to heavenly sight!
i just wanna say that that intro was pure art. the way that you were able to simply explain why bézier curves and splines are useful and cool in just over a minute with amazing graphics was super. thank you so much for providing this video for free for anyone on the internet
Being a digital artist / motion graphics artist, my mind was totally blown! You made the perfect fusion of knowledge and aesthetics!!
"This integral is a elliptic integral. In other words - sadness and despair"
As a math undergrad I LAUGHED SO HARD
The only problem in this video is that it ends. IT'S MARVELLOUS.
I LOVED IT
IT'S AWESOME
YOU NAILED IT!
I've only had to solve one like 3 times in my 9 years as an engineer. It's cool.
but yes, there was despair when I couldn't just plug in a formula and get a clean answer.
@@howdidthisgethere119 Would Wolfram Alpha be able to do it?
@@BboyKeny Most of the time if I have to ask Wolfram Alpha something, the solution I need is complex enough to be behind their paywall so i don't bother asking.
Wow thats smooth
Now you can think of eggs as Beziér curves! That is life changing
hah.
smooth.
like the video.. and the curves-
i am sorry
Not as smooth as vegetable oil
This felt exactly like watching a 3Blue1Brown video, very calming, very informative, and I got lost 2/3 of the way through, but that's on me. Please do more!
except 3b1b has a nice voice and this just ain't it
Your demonstration from 14:15 and onward not only explained why so many of my animations that follow splines had acceleration that I had to fix manually, but gave me an elegant solution, and a deeper understanding of the mechanisms that drive these pathways so I can have a complete understanding and control over my animation and model rigging.
You have opened my mind so much and even helped deepen my understanding of derivatives in the process.
Truly, thank you.
This is an absoulte masterpiece, especially the intro. That must have taken A LOT od work!
As Grant Sanderson said, this video is almost TOO good! Seriously, well done, and congratulations for the well-earned SoME winner selection. Since I watched this video, I've been literally thinking about Bezier curves a lot more frequently, and how I can apply them. Your visualization/explanation - of how Bezier curves are sort of like a moving weighted sum of the control points - was super powerful, and I've found myself thinking about that part for a while now.
thank you so much, I'm glad you like it
@@acegikmo no no, thank _you_ for making it! truly a masterpiece and i cant wait for more! Also 22:06.. a lot of us love dnd too lol. We would not mind dnd videos made in such a beautiful style
"The Beauty of Bézier Curves"? more like The Beauty of This Video!! Wow
Grant Sanderson has without doubt made _the_ most beautiful maths videos on CZcams ... until now. This was insanely good. Grant's got competition!
@@egilsandnes9637 even _grant_ said this video was beautiful. this videos got the seal of approval haha
i literally made my own javascript bezier curve visualizer because of this
I was captivated this whole time. My hat is off to you, Freya. Masterfully done. PLEASE make more videos like this! As a math nerd myself (trig and calc are my faves) seeing stuff like this is really satisfying to me, the chance to learn about this stuff in such a well presented way is simple amazing.
I can't get over how well-made this video is. The explanations are amazing, but the visualizations of everything are just next level!
I love that 3b1b inspired this, honestly you're giving him a run for his money! Incredible animation!
This video has everything. Math instructions that are both descriptive and clear. Beautiful animated illustrations. A gorgeous voice narrating. A few well-timed asides. Cat.
Please please make more videos like these. These are the best videos about geometry that I have seen. Your visual style and narrative are so well done. Congrats.
I have never seen an explanation on this level of quality. I absolutely adore Sebastian Lague, Veritasium and 3B1B, but this is such a beautiful combination of concepts, math, coding, smooth transitions and elegant visuals, I've only watched one video so far and it already feels like a drug.
many of 3b1b's videos are on this level.
look up scienceclic. he's also in this league of explainers
Goes into my "Responses to students asking 'what do I ever need this for?' in math class"-folder :-)
Very well done!
The answer is always video games. I have been playing a game called starbase recently that has required matrix math, unit vectors, converting between formal logic and equivalent math, and multilateration to understand some of the every day use things the community has made. Truly a beautiful time in history that play can inspire this kind of learning.
@@Thaccus I think the main reason video games work so well for this from a coding and playing perspective, is that they still streamline a lot of stuff, which in 'reality' gets much too complicated for students still trying to grasp the basics. I've been working with a "realistic math questions"-collection for an advanced course and more often then not problems arise, when they have to debate significance of statistical data points and where to cut corners to make the math work.
@@maxpower2480 True. IRL GPS is filled with strange caveats and "We just add this constant because we found that it corrects errors for this part of this satellites path, but we don't know exactly why." and making one in a game with transmitters removes a lot of the more complex behaviors of physics. Working within virtual worlds does come with its own caveats, but they are way more understandable.
@@maxpower2480 I love math so much as a subject but I don't think this is a good answer to the question. I think most of the coolest things in math (the fibonacci sequence, bezier curves, fractals, phi, etc.) are never taught in school. And they are, in fact, right that they most likely will not uses most things they learn in math class.
Plus, using this (or other clearly real-world applications) as evidence of math's importance sends the message that math (or any other subject) is not valuable if it isn't actually useful in the "real world".
I think the best answer to "why do I need this" is admitting that, yes, they most likely will not need to know calculus or how to solve a quadratic in their every day life, but that that doesn't mean the skills they learn aren't valuable. I here's a quote from the book "How Not to Be Wrong: The Power of Mathematical Thinking" that explains it better than I can:
_"You may not be aiming for a mathematically oriented career. That's fine-most people aren't. But you can still do math. You probably already _*_are_*_ doing math, even if you don't call it that. Math is woven into the way we reason. And math makes you better at things. Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way."_
@@Carbon_Crow Very well put and let me just say that I agree. What I didn't say before is, that my students usually only get to hear my list of 'real world applications' right after they had to listen to a eyeroll inducing passionate speech about the intrinsic value of academic learning and the way math can entirely restructure the way your brain operates. That said: People are different and many students, if not most, will never see the world the way we do. To them fibonacci's sequence will never look like the cool mathematical guiding force of nature's complexities it so obviously is. And that is fine. Most of math lessons is to actual application of that knowledge what driving nails into a stump with a hammer is to building a beautiful wardrobe. And real world example exercises have to be kept simple most of the time to avoid uneccesary complexities, that even mathematically illiterate students recognize them as oversimplified. So providing them with an answer to "What can this be used for?" (Even if they likely never will...) with such great visualization can be invaluable... Maybe for that one student who, after getting a glimpse behind the curtain, gets caught up in my excitement on the next topic. If I manage to get a few handful of students to feel the way I did, when my teacher explained the idea behind calculus to me, I've done my job... Or in this case, Freya has ;-)
This was so elegant and beautiful. That intro is something from the outer world. I can watch it over and over again. And I will do that. Absolutely amazing piece of work!
These visuals are mind blowing. From the explanations to the animations, this was phenomenal. amazing job with this
It's videos like this that make me realize despite how awful the world seems sometimes that we live in the greatest point in human history. We are so lucky to have beautiful people like Freya to teach us things in such a simple and elegant manner.
This promises to be a very well-rounded video.
Although there is a slight learning curve
XD
I don't know. It seems pretty derivative.
We'll have to stay around and see
I would try to come up with a smart pun but, LERP
I was just browsing about Bézier curves and stumbled upon this. Please do more videos like this on applied math topics. The visualization are just perfect and after each segment i paused the video to think about it for a while. You definitely earned a sub..
The colours, the gradients, the smooth af animation! this video was soooo satisfying to watch as well as being informative
Bruh freyas artistic skills really came into play with this one, this type teaching would help a lot of people that don't find math's appealing actually care about this
Hi GIR
As an animator and game dev myself, these are levels of perfection I genuinely thought were practically impossible to achieve for mere humans.
You are definitely a genius.
Those procedural graphics and movements, the underlying mastery of modular design which must have gone into making those tools, the beautiful style all across motion, color, logic, shape, and the way everything clicks together... This is on its own level far, far above everything I've seen.
Mi pali e toki tawa sina pi nanpa wan!
I really appreciate how the video has a what seems like a single curve that flows from start to finish (some bits the line disappears but mostly the curve controls are still visible so I still count it as non-interrupted).
Not sure if this is intentional but it really gives a 'continuous flow' feel, much like how the topics seemlessly blend and flow into each other. Absolutely brilliant learning resource I can't believe I've put off watching this for months!
The production and educational value is insane, definitely linking this video to anyone who wants to learn about Bézier Curves
this is a presentation masterpiece, i really like how you started with the notion, built up to the relatability then you dive deeper and deeper into the math allowing people at different mathematical maturity to still appreciate it up to whatever ability they are at.
I have been using bezier (and other) splines for decades in my work. Seeing them come alive like this and described and deconstructed was honestly inspiring. Thank you so much!
This takes me back to when me and the homies were scrambling to figure out differential geometry the night before our first exam. Absolutely excellent content. Between you, 3b1b, and Seb Lague, I’m tempted to start working on graphics lmao
10:10 Note: Osculating is different from oscillating! When a circle “fits” nicely to a curve like that, it is called osculation. Oscillation is not the same thing. This video is still gold, though. Amazing work!
I knooooow~
twitter.com/FreyaHolmer/status/1429472399389507591
@@acegikmo My apologies if I sounded rude. I didn’t see any top comments on it so I wanted to make sure that was out there. Really though, great work on the video. I watched the whole thing even though I already knew a bit about Bézier curves!
@@JordanMetroidManiac you're good, I'm just sad I'll never escape this comment over and over again, haha
but I'm glad you liked the video!
This project is such an eye candy. Can't imagine how fun it would've been to make these beautifully crafted animations. Love your work! Lot's of blessings 🙏🏼
It takes brains to make it real
I’ve been coming back to this video so many times, I just love the clarity, the calm pace and the general beauty of the explanations. Thank you so much, this video made me nerd into math again
Wow, this is maybe the best video I've ever seen that explains a complex topic. Didactic, timing, graphics, everything on the point.
The production quality of this video is just out of the world. The animations are probably one of the best I ever saw on CZcams and the concepts illustrated along with the perfect narration makes it one of the best videos. I would love to see you make more of Science and Mathematics videos, the quality of the video is so high that it can give a tough fight to the giants of CZcams like Veritasium, 3B1B, Kurzgesagt etc. Keep going, your work is flawless.
This video alone makes the call to action of 3Blue1Brown worthwhile. Incredible job!
As the title tells, this might the be most beautiful visualisation of the subject. I have to appreciate the effort you’ve put into this.
This is a super high-quality content. I usually keep away from commenting on any youtube videos. But could not keep me from appreciating this superb work. I am presently studying at IIT but kind of a person that rely on visualizations for building conceptual understanding. This video helped me to have a solid understanding of Bezier Curves. I'm gonna watch all of your videos. Please keep making these videos.
This is a piece of art.
Its the first video of yours that i watch and i still cant believe how great it is. Im an undergrad in cs and i often question myself "WHY im studying this?", but you showing how beautiful math can be and how it can be applied in really cool things (like these games) actually inspired me again to keep learning.
Thank you again for making this video!
*english isnt my first language so i hope you all understand my message*
My god, the editing here is amazing - great work! This described these things way better than Ive seen before
19:37 I was wondering how you chose the colouring of the curve. Of course it was with the help of another bézier curve! Really showed the versatility and practicality of this tool.
Jesus Loves You Isak
@@101design5 Hail Satan!
@@Kenjuudo 😂😂
@@masteroogway2405 xD
This video is masterpiece! The visualizations, the explanations, the music - just everything
This is the most beautiful animation I've ever seen ❤ Great work!
The phrase is now, "Lerps all the way down"
Lerpception! 🌈🚂
@@gabrz75 I got that reference
Unless you're in Australia and a koala accidentally eats some lerp.
Wow, I just clicked on a random video expecting nothing, and got probably the most well-crafted video I've seen this month. Great visuals, great explanations, and great background music choice and volume. I'm only 6 minutes in, and can already tell this is a masterpiece.
Edit: I'm just gonna subscribe, this video was great. Also ooo I'd like the video on splines, especially from you!
That's always how it is when you have no expectations. The best thing ever hits you.
Freya, after watching this video, I'm really impressed, this is so great video. The idea of adding explanatory animated visuals is excellent.
I saw many videos and II haven't seen anything like it, I really hope you continue your work.
Probably the best video I've ever seen on youtube! Just started looking at curves and paths with javascript( svg ) and this video was like "where have you been all my life" .. Thanks Freya!
Please more videos! This is amazing, and I wish I had this sort of instruction back in high school. I would have been interested in math as it relates to game development. So more videos along the line of where to use the curves, how, why? Arc of a grenade, etc.
Bézier curves are commonly used in animation and interpolation of various things! Trajectories are a little different, generally you simulate them in real time instead of planning the entire path, but for AI prediction etc. it's useful to know the entire arc, travel time, and so forth
now, trajectories with a simple gravity vector, are themselves quadratic bézier curves! though they're rarely represented as such or coded as such. But yeah! might be a neat topic for a future video :)
@@acegikmo I enjoyed your video very much, however I felt a bit misled. I was hoping you will explain what I saw in the intro. The point is (no pun intended) what you showed is the outer control point curve and the intro hinted on-curve control point workings. I would like to learn of that.
Again, thank you for creating this visualization. When years ago I learned about these in class was very dry and boring. This video showed everything in 20 minutes, while we spent at least 90 to sketch approximations and scratch formulas to a notebook we never ever used later...
@@Sekir80 the intro is using several bézier curves joined together, it's just that the control points aren't always visible! (but they are in some sections of the intro)
if you're looking for a spline that passes through all points, without explicit control points, that would be a catmull-rom spline usually :)
@@acegikmo Thanks for the answer! I might have mistaken the point and the perpendicular blue lines with little circles at the end as control points (visible from 0:19 - 0:23). What I meant is this:
en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#/media/File:Beziergon.svg
Where the curve passes through the points and tangential line is the control, which can be broken in a way as you see on the left points. So, this is just another representation of the same maths, I guess.
@@Sekir80 those blue lines in my video are the gray tangent lines you see in the image you linked - it's just a matter of putting multiple bézier curves in a row, usually called a spline
16:55 When you did the equally spaced t value animation it looked like a 3d animation. My brain was trying to telling me that the dots were traveling at a constant velocity and only appeared to be slowing down in the middle because they were traveling towards me.
yeah now that you say that I can see it too.
It looks like the speed of the dots is inversely proportional to the curvature or the speed of the dots is directly proportional to the radius of the curve.
i directly related it to the roller coaster she showed before lol
Yes, more of this! An incredible resource on this subject which will be useful for years to come.
This video is so rich, in the sense that, you'll hardly find any congruent material like this video covering such an underrated topic!
Freya, this was incredible! I do motion design in my video production work and I am SO blown away by what you created with Unity and Shapes! Incredible. One of these a month would be an absolute dream. Congrats on this wonderful video!
The intro is absolute art. Also, I love how you use FF14 as example both in the shader course and this, I started playing partly because of you :D
That first demonstration of the construction of the quadratic and cubic curves was so mind-blowingly intuitive, I felt like I immediately understood something I had only ever vaguely brushed up against up until now!
One of the most wonderfully animated video I have ever seen. What a pleasure to watch!!
This is an awe inspiring video, artistically, mathematically and technically.
Love that alternative interpretation of the lerp visualized as weighed vectors, I hadn't seen that demonstrated like that before.
Incredible video! Thank you so much for explaining each portion so thoroughly. It was a joy to watch and learn with you!!
The production quality of this video made it really enjoyable to go through. Big ups to you
As a CAD guy who occasionally teaches this stuff I would love to see a video on splines! Thank you. You explain this so much better than I do. I'll be sending this link to my engineers in the future.
I've been using bezier curves for years now, but had never really understood what they were all about. Awesome explanation and mind-blowing visualizations. Thanks for the insight!
These animations are really good, probably the best I've seen in a long time
I love learning more about things I knew/heard about, but never knew the gritty details about.
This was an incredible video, thank you!
Everything about this is amazing, from the mathematics, the animations, the music and vibe, and just the sheer elegance of how it's all interconnected. Very excited to see more!
This video has blown my mind. I have been using curves in parametric modeling for decades and have never seen such an elegant, clear, concise explanation of these concepts. Thank you for your efforts. The amount of thought and energy that went into this is astounding.
The animations were beautiful, amazing explanations too. Thank you for making this!
A lot of quality content on this channel that is either hard or impossible to find anywhere else, I'm amazed you don't have more subs.
Freya, this was stunningly beautiful! I teach Bézier curves every semester in my intro game programming class for nearly 20 years, and now I'm going to show your video as an intro. This is also the best advertisement I've yet seen for your Shapes library. I also really appreciated hearing the details of how you made the animations. Thank you. :)
and the narrator has a completely transsexual voice so will fit right in with the way you university professors like to brainwash people into being mindlessly woke. win win!
you have a really good voice/cadence for this! i've been hard pressed to find more channels with the same "friendly explanation" energy as sebastian lague videos for example. i look forward to more!
I like that the way many of the 2D curves you showed in this look like they go into and out of the screen and almost look 3D, a nice effect!
I saw your presentation at ESG and now I'm checking out your channel. This was really a great video! It is honestly rare to see this quality of content on youtube.
Ever since I got hooked on to 3b1b videos, all other sources of math kind of felt incomplete. And I assumed that it wouldn't be very likely that I come across another medium that was as unique, as informative, thought provoking and a million other adjectives. But your video was a breath of fresh air. It felt perfect in every regard. It felt less like an educational video and more like an excellent work of art (without compromising content!). I thoroughly enjoyed every single second of your video. And the cameo of Thor was the cherry on top! Absolutely loved it!
This video is beautiful. As someone just moving in higher level maths like this, seeing the concepts we've been learning applied is fascinating
I have been working with Bézier Curves for ages, but I have never seen such a nice explanation.
Many thanks for the great video 🙏
I'd like to take a moment to express my profound appreciation to the superb quality of this video. Excellently scripted, edited, visualized, narrated and paced. A long period of time have passed since I enjoy an educational presentation with this level of production value on this platform.
I'm keenly looking forward to your next brilliant contents. Subscribed.
This was amazing! You definitely have the charm of Grant, you’re voice is so smooth like his and you explain very intuitively. I’m looking forward to your future projects!!
This has got to be one of the best videos in the history of mathematics! You have done a cracking job and I hope you make more. Bravo.
This was incredible... I consider myself intelligent, but never really tried to figure out Bézier curves, because I assumed they were too complicated - especially for practical game development use. But the way you explain things made it nearly impossible for me to not get it. You have a very elegant and artistic teaching style that converts left brain data into right brain data, which is so much easier to absorb. Thanks so much for making these types of videos, and please make more!
I'm old, I'm bad at english, I was bad in math, especially the last years of school.
But I've understood more about math in this video than lots of lessons at school.
The visuals are simple but elegant, focusing on a clear explanation. I'm loving It.
The voice is clear, relatively slow, a little weird maybe, but I think it's wonderful for the ones who aren't english mothertongue and genuinely conveys passion for the subject.
The subs obviously gives additional help to the ones like me.
The music is simple and it allows you to focus without distractions, and reminds me of the heyday of hanging out on modarchive.
Thanks very much, I really enjoyed It.
before I read your writing on bezier curves and how to make it out of linear interpolations I had no idea it could be that simple and intuitive to understand,
this video just makes it all that much cooler and better, love it!
I know everyone else is saying this already, but this is truly one of the most spectacular math videos I've seen in a long time. Thank you for all your hard work
I fall in love with your content just watching the first video. And also your explanation is awesome.
Work of art thank you for explaining so clearly how works the tools like the feather in Photoshop. I’ve been using them for years and you really made me understand the beauty of it :)
Oh my gosh, the "Kappa" had me laughing so much 😂
This is great. I love the animations and your voice is so relaxing, while explaining complex subjects in such a simple manner. Keep the good work.
It's a beautiful video explaining Bezier Curve with visuals. Helped a lot. Thank you. Keep us enriching.
All of your videos, I find very entertaining due to the fact you make everything seem so simple, or the greatest videos, well; you're awesome!
There were so many points throughout this video where I wanted to like it again!
Beautiful animations (besting some of the biggest math channels), wonderfully explained, and fascinating ideas I've always wanted to learn but never taken the time!
Thanks so much for taking the time to create this! 😁
Wow, this was amazing! Can’t even imagine how much effort it takes to create something like this. I would certainly love to see more such videos. Just a pure mathematical delight.
this is so beautiful! Amazing work! Beyond words!
I'm not 100% sure why CZcams recommend me this video right now, but it was just beautiful to watch!