🐱💖✨ 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?
"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
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.
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!
"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.
@@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.
@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.
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!
@@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?
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.
@@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
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 ;-)
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.
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
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.
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.
not to take anything away from her but fwiw, we felt the same when it was a campfire not the glare of our screens that lit our faces with Oog (the feminine version of Ogg 🤭) teaching us how to knap flint into spear points or when we were apprentice shamans or metalsmiths (also seen as 'magical'). learning any and all science from a gifted teacher has always felt like a blessing 🙏 ✌️ 💜 👋
@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!
@@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 !
The production quality of this video is just out of the world. The animations are probably one of the best I ever saw on TH-cam 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 TH-cam like Veritasium, 3B1B, Kurzgesagt etc. Keep going, your work is flawless.
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 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 🙏🏼
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
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.
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.
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!
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.
Oh god, I start to feel my heart palpitate when I go back to these spline and curve videos. I've been in this rabbit hole for so much time I've got PTSD from it
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!
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!
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!
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!
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!!
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!
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 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
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
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!
@@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!
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
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!
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.
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.
15:54 The varying speed can be beneficial in some cases because when you use a Bezier curve to define the path of a robot arm you have to slow down in the sharper parts of the curve anyway.
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!
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.
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 swear I‘m so glad I came across this video. Even before starting this video I immediately knew: This has to be good… the video title already said it all. Nobody would talk about „the beauty“ of bézier curves if they didn’t LOVE what they’re talking about. I instantly fell in love with the video and could have cried (I‘m a student that was desperately on the lookout for a good explanation), because this is the best video for the explanation of bézier curves I‘ve ever seen. Thank you so much. And also it looks GREAT! You nailed it ❤
First things first, thank you for sharing Thor's voiceover. An important contribution. Second, you said you're not sure who your audience is. I am a solo game dev who enjoyed math in college and haven't looked at it since, and I am finding your videos incredibly helpful to approach intimidating topics like vector math and Bezier curves. Thanks for providing these incredible resources!
I've been using the "pen tool" and its equivalents for several years now, though I feel this is the first time I appreciated it for how elegantly it works. Really nice video and explanations!
What an amazing master piece. Love the phrase at 19:51 .. "There's a whole cinematic universe out there" This such a wonderful and liberating invitation to explore further. I hope you do more exploration. :)
This was incredibly well explained and so nice to watch. All of the animations were beautiful and helped illustrate the point super nicely. Looking forward to more in this format! Would certainly watch a follow up on splines.
Wow! What a beautiful video. I was immersed in the animations and the math. Your conceptual understanding of derivatives and higher order polynomials is simply superb. I'm a math major but unfortunately lack the imagination to visualise what I'm learning. I was slowly losing passion for the subject but your video has reignited my love for math and I'm motivated to approach my learning in a new way. Thank you so much for this beautiful video. I am very eager to now apply for a PhD and study a lot more math to discover the beauty of this world. There is something very special about your video which shows the a figment of the beauty of mathematics. It is something like a painting, a true work of art! Thank you for this. It is simply superb and amazing. Wish you the very best of luck and 100x the subscribers you have now..
I was really floored by the fact that the easiest way to depict the polynomials with a functional visual (i.e. one of the many that made this video satisfyingly understandable to us amateur, arm-chair mathematicians), was to use the very thing you were explaining, slightly upscaled, in RGB space. I relate that kind of joy discovery in a subject to a musician unlocking the next level of understanding with their instrument when they learn the physics of music. It adds a new wonder and awe to something that the typical linear learning path presented to the student might obscure. You definitely conveyed your passion for, joy and love of Bezier curves to us in a beautiful way that made me all about them as well. Whatever topic you tackle next, bring that same passion to the presentation and we'll be back in a heartbeat!
You are the best teacher of these topics I've ever seen. Thank you so much for your awesome educational content. I would recommend your videos to everyone interested in game related mathematics. Thank you thank you thank you!!
Absolutely amazing video!! Love everything about and want to see many many more in the future! It definitely has some 3blue1brown flair but also a lot of your personality and style! Really impressive
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..
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! 😁
I really like the animations. I have a strong affinity for these kind of seamless transformation animations. I loved to see it and I wish I could do the same...
Wow, this is an amazing video! Although I've studied math for my entire life and I have a PhD in algebraic number theory, I never really understood Bezier curves all that well and I must confess I still don't, but the few times I've used them, I've been quite intrigued by them! Perhaps this is an area of math in which I'm rather weak, but I can still appreciate their beauty and utility. Great job with the graphics as well!
That was beautiful! Been working with all kinds of splines for years, they're everywhere when you're doing animation stuff. But I'm pretty meh at math and never knew what exactlly goes on under the hood. Your video just made stuff click instantly, that was so insightful! Please make more videos like this!
This is an awesome introduction to Bezier curves and exceptionally well made. Even more so for a first production! I explored these things for animating a camera viewpoint through a 3D model I made a couple of years back, and came to admire the elegance and usefulness of Bezier curves, in spite of the math involved. You make it look so evident and elegant. You do an excellent job and I would enjoy watching more. :) Thanks!
Let me say that this was the first of your videos I've ever seen, and one of the two I have seen today, and my God, you clearly have an amazing talent for informative, educational and beautifully animated content! I've used Bézier curves in the past, but I never really understood how the control points affect them, although I developed a general "feel" for it. As a hobbyist game developer, I've also tried tackling them in different ways but never really even had the patience to start trying it since they just seemed like complicated beasts to me. But what do you know, I feel completely stupid now for not tackling them before, because your video has made me realize that they are fucking simple (pardon my French). Kudos for this video, I've learned a whole bunch just from watching it, and you can be certain you've just acquired another subscriber!
This video would have been very educational and entertaining even if it had been just a compilation of slides,but you went the extra mile and took the time to make these beautiful animations! Great video
Hot damn, these vector graphics are so clear I think my eyes will never need a wash ever again ^_^ Gorgeous educational vid, well worth the 33 days considering how many people you have helped with it
🐱💖✨ 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
I hate computer science. You stare at anything for long enough and you realize it's just vector-matrix multiplication
That's a lot of math. Group theory, graph theory, functional analysis, multivariate calculus... the tools are often linear algebra.
We can't get away from the spooky magic table
Moore's law?
the math ouija board is inescapable
😂😂
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?
"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
@@JewishThots because of the very clean, very clearly code-driven animation style.
@@JewishThots 👆what they said. Plus he makes it easy to understand.
@@JewishThots 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!
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!
"Sometimes the value under the root is negative. These roots aren't real, ignore them"
Such a literal explanation gave me a chuckle
Negative roots aren't real, it can't hurt you
*Negative roots:*
"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.
This is an absoulte masterpiece, especially the intro. That must have taken A LOT od work!
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.
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
I love that 3b1b inspired this, honestly you're giving him a run for his money! Incredible animation!
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 TH-cam ... 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
Being a digital artist / motion graphics artist, my mind was totally blown! You made the perfect fusion of knowledge and aesthetics!!
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
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 ;-)
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
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
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!
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.
not to take anything away from her but fwiw, we felt the same when it was a campfire not the glare of our screens that lit our faces with Oog (the feminine version of Ogg 🤭) teaching us how to knap flint into spear points or when we were apprentice shamans or metalsmiths (also seen as 'magical').
learning any and all science from a gifted teacher has always felt like a blessing 🙏
✌️ 💜 👋
@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
👍😎👍
The production quality of this video is just out of the world. The animations are probably one of the best I ever saw on TH-cam 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 TH-cam like Veritasium, 3B1B, Kurzgesagt etc. Keep going, your work is flawless.
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 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
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 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
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
My god, the editing here is amazing - great work! This described these things way better than Ive seen before
This video alone makes the call to action of 3Blue1Brown worthwhile. Incredible job!
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.
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
@@101Design_1 Hail Satan!
@@masteroogway2405 xD
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.
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.
Oh god, I start to feel my heart palpitate when I go back to these spline and curve videos. I've been in this rabbit hole for so much time I've got PTSD from it
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!
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
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!
Brilliant! Thanks! 😊🙏
This is an awe inspiring video, artistically, mathematically and technically.
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!
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
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!!
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!
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.
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.
This video is beautiful. As someone just moving in higher level maths like this, seeing the concepts we've been learning applied is fascinating
That was the nerdiest and most amazing way to visualize Bézier curves ever. This was amazing!
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
Could u explain the whole of maths like this?
yes! but only if you explain the whole of music
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
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!
Love that alternative interpretation of the lerp visualized as weighed vectors, I hadn't seen that demonstrated like that before.
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
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 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.
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
15:54 The varying speed can be beneficial in some cases because when you use a Bezier curve to define the path of a robot arm you have to slow down in the sharper parts of the curve anyway.
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!
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 immaculately edited, i'm so impressed with the effort in the presentation--thanks so much for making this!
I love your voice and your soothing and engaging style of teaching. Thank you!
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!
this video is a masterpiece, i wish every math lesson was this refined and clear, very good job!
Absolutely stunning. I definitely would like to see more videos like this.
I swear I‘m so glad I came across this video. Even before starting this video I immediately knew: This has to be good… the video title already said it all. Nobody would talk about „the beauty“ of bézier curves if they didn’t LOVE what they’re talking about. I instantly fell in love with the video and could have cried (I‘m a student that was desperately on the lookout for a good explanation), because this is the best video for the explanation of bézier curves I‘ve ever seen. Thank you so much. And also it looks GREAT! You nailed it ❤
First things first, thank you for sharing Thor's voiceover. An important contribution. Second, you said you're not sure who your audience is. I am a solo game dev who enjoyed math in college and haven't looked at it since, and I am finding your videos incredibly helpful to approach intimidating topics like vector math and Bezier curves. Thanks for providing these incredible resources!
I've been using the "pen tool" and its equivalents for several years now, though I feel this is the first time I appreciated it for how elegantly it works. Really nice video and explanations!
What an amazing master piece.
Love the phrase at 19:51 .. "There's a whole cinematic universe out there"
This such a wonderful and liberating invitation to explore further. I hope you do more exploration. :)
This is an awesome video, like seriously I don't even understand how you made this concept so appealing and clear in 20 minutes
This was incredibly well explained and so nice to watch. All of the animations were beautiful and helped illustrate the point super nicely. Looking forward to more in this format! Would certainly watch a follow up on splines.
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.
Wow! What a beautiful video. I was immersed in the animations and the math. Your conceptual understanding of derivatives and higher order polynomials is simply superb. I'm a math major but unfortunately lack the imagination to visualise what I'm learning. I was slowly losing passion for the subject but your video has reignited my love for math and I'm motivated to approach my learning in a new way. Thank you so much for this beautiful video. I am very eager to now apply for a PhD and study a lot more math to discover the beauty of this world. There is something very special about your video which shows the a figment of the beauty of mathematics. It is something like a painting, a true work of art! Thank you for this. It is simply superb and amazing. Wish you the very best of luck and 100x the subscribers you have now..
This has to be the most GOATED intro to a math video of all time, that intro animation was SUBLIME
I was really floored by the fact that the easiest way to depict the polynomials with a functional visual (i.e. one of the many that made this video satisfyingly understandable to us amateur, arm-chair mathematicians), was to use the very thing you were explaining, slightly upscaled, in RGB space. I relate that kind of joy discovery in a subject to a musician unlocking the next level of understanding with their instrument when they learn the physics of music. It adds a new wonder and awe to something that the typical linear learning path presented to the student might obscure. You definitely conveyed your passion for, joy and love of Bezier curves to us in a beautiful way that made me all about them as well. Whatever topic you tackle next, bring that same passion to the presentation and we'll be back in a heartbeat!
I don't wanna imagine how hard it was to create something like this video, just insane
Freya, this is a masterpiece! Well done. Thank you for making it.
Unironically, the smoothest video in the math exposition competition. Stunningly gorgeous math communication.
You are the best teacher of these topics I've ever seen. Thank you so much for your awesome educational content. I would recommend your videos to everyone interested in game related mathematics. Thank you thank you thank you!!
Absolutely amazing video!! Love everything about and want to see many many more in the future!
It definitely has some 3blue1brown flair but also a lot of your personality and style!
Really impressive
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..
I can't get over how well-made this video is. The explanations are amazing, but the visualizations of everything are just next level!
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! 😁
I really like the animations. I have a strong affinity for these kind of seamless transformation animations. I loved to see it and I wish I could do the same...
I think you would be delighted to know that the smooth start/stop timing function is also cubic bezier!
Way to go! This is just beautiful, incredibly well done. I am sure it will be useful to many people, from entry level to professionals. Thanks!
Wow, this is an amazing video! Although I've studied math for my entire life and I have a PhD in algebraic number theory, I never really understood Bezier curves all that well and I must confess I still don't, but the few times I've used them, I've been quite intrigued by them! Perhaps this is an area of math in which I'm rather weak, but I can still appreciate their beauty and utility. Great job with the graphics as well!
These animations are really good, probably the best I've seen in a long time
This is really an incredible interpretation, absolutely love it. Thank you!
I've never heard of these beautiful incarnations until now. Thanks for sharing your talent in a way that really makes sense.
This is crazily good. I tried multiple times and I have never been able to understand it that well. Thanks for the intuitive explanation!
That was beautiful! Been working with all kinds of splines for years, they're everywhere when you're doing animation stuff. But I'm pretty meh at math and never knew what exactlly goes on under the hood. Your video just made stuff click instantly, that was so insightful! Please make more videos like this!
This really takes me back to the differential geometry course I took in undergrad! Well done
This is an awesome introduction to Bezier curves and exceptionally well made. Even more so for a first production! I explored these things for animating a camera viewpoint through a 3D model I made a couple of years back, and came to admire the elegance and usefulness of Bezier curves, in spite of the math involved. You make it look so evident and elegant. You do an excellent job and I would enjoy watching more. :) Thanks!
The production and educational value is insane, definitely linking this video to anyone who wants to learn about Bézier Curves
Let me say that this was the first of your videos I've ever seen, and one of the two I have seen today, and my God, you clearly have an amazing talent for informative, educational and beautifully animated content!
I've used Bézier curves in the past, but I never really understood how the control points affect them, although I developed a general "feel" for it. As a hobbyist game developer, I've also tried tackling them in different ways but never really even had the patience to start trying it since they just seemed like complicated beasts to me. But what do you know, I feel completely stupid now for not tackling them before, because your video has made me realize that they are fucking simple (pardon my French).
Kudos for this video, I've learned a whole bunch just from watching it, and you can be certain you've just acquired another subscriber!
This video would have been very educational and entertaining even if it had been just a compilation of slides,but you went the extra mile and took the time to make these beautiful animations! Great video
This is so beautiful, and coincidentally aligns with what I've been learning in calculus lately! Great job!
Hot damn, these vector graphics are so clear I think my eyes will never need a wash ever again ^_^
Gorgeous educational vid, well worth the 33 days considering how many people you have helped with it