The Continuity of Splines

why are splines? well my god I have good news for you, here's why splines!
if you like my work, please consider supporting me 💖
/ acegikmo
This project grew much larger in scope than I had originally intended, and burnout made it impossible for me to do more with it. It was already getting incredibly unwieldy, so I apologize in advance for not covering non-uniform splines, and the general jankiness of some parts. There are also quite a lot of places where I just talk about something without showing it, but since I couldn't animate it all, I opted to keep that information in rather than remove it altogether
• Lots of love to
💛 Jazz "queenjazz" Mickle for music & sounds ❱ queenjazz.bandcamp.com/
🐈 our children, Thor, Salad & Toast, for helping me out during recording
• How was this video made?
🔨 I made it in Unity, the 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
💡 Striped studio skybox by ProAssets: assetstore.unity.com/packages...
🎥 Final editing in DaVinci Resolve
• Links
🐦 Twitter ❱ / freyaholmer
📺 Twitch ❱ / acegikmo
💬 Discord ❱ / discord
🌸 Instagram ❱ / freya_holmer
Timestamps:
00:00:00 Intro
00:01:17 Examples of Paths
00:02:16 CHAPTER 1 - Bézier Curves
00:02:20 Lerp
00:03:16 Quadratic Bézier
00:03:54 Cubic Bézier
00:07:41 n-Degree Bézier
00:09:37 CHAPTER 2 - Bézier Splines
00:10:00 Cubic Bézier Spline
00:10:21 Spline Parameterization
00:11:01 The Anatomy of a Spline
00:11:32 Knot Values & Knot Intervals
00:12:21 Local Control
00:14:03 The flexibility of the Cubic Bézier
00:14:28 Tangent Points
00:15:07 CHAPTER 3 - Continuity
00:15:28 C⁰ (Positional Continuity)
00:16:12 Spline Derivatives
00:17:32 C¹ (Velocity Continuity)
00:20:25 C² (Acceleration Continuity)
00:21:48 Cascading Loss of Local Control
00:23:46 C³ (Jolt Continuity)
00:24:49 C∞ Continuity
00:25:19 Parametric Continuity Summary
00:26:20 CHAPTER 4 - Geometric Continuity
00:26:50 G¹ (Tangent Continuity)
00:27:46 G² (Curvature Continuity)
00:28:05 The Reflection Test
00:28:56 Seams in Circular Arcs
00:29:43 Angle Analysis
00:30:47 Curvature & The Osculating Circle
00:32:26 Curvature Combs
00:32:48 G¹ (Algebraically)
00:33:27 G² (Algebraically)
00:34:52 Geometric Continuity Summary
00:36:30 Continuity Summarized
00:38:06 Cusps
00:39:25 CHAPTER 5: The Extended Universe
00:39:40 Hermite Spline
00:43:35 Hermite ⇔ Bézier Conversion
00:44:09 Linear Spline
00:45:17 Cardinal Spline
00:48:20 Catmull-Rom Spline
00:49:33 Investigating Basis Functions
00:51:02 Basis Function Continuity Analysis
00:51:31 In Search of a C² Spline
00:53:15 B-Spline
00:56:35 Quiz Time!
00:57:12 Cubic Trajectory
00:57:37 Uniform Cubic Splines
00:59:09 Splines & Their Use Cases
01:01:37 Outro
01:02:16 NURBS
01:02:56 Color Spline
01:03:52 Credits & Freya Rambles
01:13:26 Cats helping me w. recording

[@temporaltoast9692]

“As you can see, it yeets off to fucking wherever” caught me off guard. But like in a good way

[@djangaver]

boy, it was a good way

[@MadplowDisease]

Came to the comments immediately after hearing it 😂

[@skiwarz]

Same. That was awesome.

[@djangaver]

21:45 just came to leave a shortcut

[@filipesrubio4015]

Of course you did. No surprise at all

[@carykh]

The animations of your videos are so beautiful! I'm impressed you pretty much made a full movie about how splines work, that's really one-of-a-kind on KZbin

[@spidunno]

I really like the use of colors to make things easier to recognize and understand.

[@rogogo1244]

cary, what are you doing here?

[@HomieSeal]

It really is, it’s super cool :D There’s a lot of different animation styles that I enjoy in educational videos like 3B1B’s/Manim, yours, the one from this video, and more I can’t think of off of the top of my head. Out of all of the ones I’ve seen, this is probably the most unique but still very clean ones I’ve seen, which is mostly through how it’s similar to Manim but also not, and the other fine details.

[@jeremywoolley]

It's you. You are everywhere. How do you do everything.

[@AllExistence]

I disagree. Sebastian League does pretty much the same.

[@zynapz]

absolutely amazing. incredibly good animations, sense of aesthetics, and teaching style. and "yeets off" at 22:50 caught me completely off guard 😂and cracked me up. thank you!

[@M0du5Pwn3n5]

22:50 killed me. I had to pause the video.

[@ricomariani]

@@M0du5Pwn3n5 it's the best part of the video :D

[@xXx-un3ie]

this was so unexpected hahaha all serious and knowledge stuff and then just this in the middle of the explanation haha good job

[@hongkonger885]

bro dropped a literal "f-bomb" ngl 😂

[@phoenixshade3]

I had to stop the video for this exact reason. Completely lost it.

[@fersahahmet9597]

all in one; a perfect artist, teacher, programer, mathematician, animator and many many more. That is what it takes to produce such a perfect art. I bet it takes over 2 or 3 decades to be that good

[@kummer45]

I have four years working in parametric design. I have an idea HOW HARD this is. Freya UNDERSTANDS differential geometry, convex geometry, hyperbolic geometry, Fourier Series, Bernstein Polynomials, Legendre polynomials and DISCRETE mathematics. This is an educator, a teacher. I have a bachelor in architecture, mathematics and physics. I can't hardly imagine the insane amount of work it took making this content. Institutions are NOWHERE near this type of quality. I have a name for these individuals. They happen in mathematics, physics, architecture, education and so on. I call them ORACLES. Freya obviously became an ORACLE. I even bite my tongue thinking how much time it took to BUILD this video entirely with the animation, script, C++, Python among many other things. This is NOW the new standard. Freya established it.

[@mirkoforastiere4676]

First thing i thought was "I'm not gonna watch 1 hour of video about splines", well 1h and 13 minutes later I can tell you that it's definitely worth it. The video is incredibly clear and guides you step by step, also the animations are simply perfect. I wish there were more videos like this one.

[@SpiritmanProductions]

Same here.

[@bigpopakap]

Hahaha same here, and now I'm doing the exact same thing again 😂

[@authenticallysuperficial9874]

Same!

[@stijnvandrongelen5625]

Not that I expect you to never curse, but it really caught me off-guard when it happens when we're already 22:46 into an otherwise very calm explanation XD Amazing content as ever, Freya.

[@JosiahPurtlebaugh]

Came looking for this comment. I almost did a spit take!

[@ash.mystic]

@@JosiahPurtlebaugh same here 😂

[@diesdas9400]

But perfectly placed, for sure!

[@justice83]

It was this point she rage quit and had a 7 month break. For sure.

[@electrostats]

The best part of the video ahah

[@RobFulwell]

Not only is this material fascinating, the visualisations are absolutely world class! Outstanding work! ⭐

[@marklandgraf7667]

1:16:00 "I feel like I kindof struggle to find my voice." Stop struggling. You found it. And it's fantastic. Even, clear, well-paced, and most importantly, non-creaky! Even the parts with high jolt value.

[@BiologyTube]

This is a real masterpiece. If your little throw-away line about bivectors ends up blossoming into a video on geometric algebra, I will watch the hell out of that.

[@acegikmo]

it might!! I'm still in the research phase, and whether or not it'll be a standalone video or as part of a video about quaternions

[@zemoxian]

@@acegikmo Well quaternions can be generated as a geometric algebra or sub algebra, depending on the signs of the squares of the unit basis…Yes, I geeked out at the mention of bivectors as well. Funny story is I heard a rumor that quaternions were better at rotations than matrices. So in the process of learning about using quaternions I fell down the geometric algebraic rabbit hole. Now I’m mad that I didn’t learn about them 30 years ago in high school or college. I feel like getting drunk at a physicist bar and make fun at how they multiply vectors. _”Stupid cross products are only useful for three dimensions.” [Falls off stool.]_ 😂 But I’d really like to know more about higher dimensions and non uniform splines.😊

[@zemoxian]

@@acegikmoOh, and SLERP would be cool too.

[@tinkeringtim7999]

@Freya Holmér I have a degree in theoretical physics, used to be a professional coder, lifetime maths geek. Love what you've done here, I'm going a similar way. I might be able to help you with pointing to the linkages at a high level for basically whatever maths stuff you come across. I have a very extensive library too. You've given so much to make these things clearer, drop me a PM if I can help in return. Burnout sucks, you can lift more weight if you do it together.

[@Kram1032]

​@@acegikmo I think this could actually be an entire series, perhaps starting with quaternions. I find quaternions far more illuminating as a subalgebra of Geometric Algebra, so it makes sense to mention them in that topic. But if you are willing to explore further (and given this video on splines, that's honestly kind of a given), there are so many other things that become clearer in this context. But it has the same sort of exploding-scope problem that splines as a topic have, so trying to fit it into a single video would definitely be too much.

[@thedebapriyakar]

Freya your video on Bezier curves was literally the one thing that kickstarted my journey into computer graphics research. I will forever be indebted to you for that 💖

[@chasechapman9302]

Dogg same

[@Capungo]

Bezier curves is still my favorite video too. I am not into anything like computer graphics or something like that. I just like watching these videos very much

[@yvrezzz]

Same for me it fully re-ignited the passion on the subject. Thanks a lot Freya !

[@whannabi]

This type of video is just amazing, we're being blessed each time there's a new video like that from her .

[@JesseLeeHumphry]

Same, she really helped me understand something I was having serious trouble understanding and it made me a better programmer.

[@charlessavoie2971]

I can only imagine the hard work that went into this. Thank you for teaching us.

[@FynnKliemann]

This might be the best video on youtube!

[@simon18769]

would ich too sagen👍

[@bennettgardiner8936]

OKAY. I already thought the animation was gorgeous, and then we entered 3D reflective surfaces. This is absolutely incredible and you deserve more views.

[@TheFinnish1]

"we're gonna have to enter the 3rd dimension, and turn on the lights." That felt like casually inventing a new dimension because the entire video up to that point, (besides the intro,) was 2D

[@cstiger4]

Are you kidding? Your presentation style is totally exciting and compelling. Please do not change it. It is so engaging. I could not stop watching.

[@fghsgh]

If she doesn't feel authentic like this... she is going to want to change things. Ideally she'd be as engaging or maybe even more, but you cannot enjoy a creative outlet if you don't get to be authentic.

[@fliptopjim]

Oh my goodness. 25 years of working in animation and illustration and I'm 4 minutes in to your video and it's a sheer delight. The intuitive learning I've made over those decades is falling into place seeing it fit together. Just wonderful. I can feel the fog lifting to reveal a familiar world put into context. Really looking forward to the next 70 minutes of this!

[@fliptopjim]

Just finished. Goodness me, this is so interesting. Really thoughtfully done and beautifully collated. Congratulations on making it through to finishing this (also really interesting analysis of it at the end). Thank you.

[@fliptopjim]

By the way your cadence, style and personality is just FINE! Your delivery is perfect in this context. I've been here to the end which is testament to your style..

[@fabkury]

This video production deserves some kind of Data Communication & Teaching award. What a fine piece of work. Thanks a lot. I learned so much from this.

[@damionmurray8244]

I spent years exploring these concepts. I had to pull from a variety of sources to fully grasp these ideas. Nothing I came across in all those years explained them as organically or as succinctly as you've done here. Truly beautiful work Freya. Well done.

[@Armadous]

Recently spent weeks in a university course treating most of these in an into to interpolation. It's funny/depressing how effective books and formal education are at being correct and abysmal at comprehension.

[@paulkanja]

Simply put Freya here is the 3b1b of graphics and technical aspects of game devevlopment

[@morkovija]

I've heard a really good saying somewhere that gives me hope: "If you dont understand something - come back in a few years, they might figure out a way to make it understandable for high schoolers"

[@janosadelsberger]

This is gold for the industrial design community as it's quite difficult for visual people the wrap their head around how CAD software works. Perfectly leads up to NURBS - which would be awesome as a future video in more detail! Thank you so much for going through all the research!

[@JohnDlugosz]

I don't understand. Isn't CAD all visual?

[@janosadelsberger]

​@@JohnDlugosz Yes, for simple parts it's not a problem but when you're working on more complicated surface transitions it really helps to know the mathematical concepts because otherwise tools just can't solve with the inputs provided. It gets really tricky when trying to blend three surfaces with curvature continuity for example.

[@RandyCroucher]

Such a great explanation of splines. I only wish I had seen this 30 years ago when creating the spline 3D modeling tools in Animation:Master. Thank you so much for all of the effort that went into this. It shows.

[@MetricFudge]

I've rewatched this video all the way through something like 4 times now. I love the intuitive explanations you give and the way you break down the concept of splines. Also the animations and the way you present them are visually appealing and relaxing to watch. They made a very intimidating topic feel much easier to understand.

[@LasseVagstherKarlsen]

You have a language discontinuity at 22:44 and it made me spit out my coffee in laughter! These videos you've made about splines and curves are, in my opinion, some of the very best videos on KZbin. They're extremely well made, in terms of teaching videos they rival the quality that top movie studios create movies at, I can't even begin to imagine the amount of work that went into just the animations and illustrations you made for these videos. Your obvious deep knowledge saturates these videos, but deep knowledge is not enough, there are plenty of people that have that. What you have is the ability to communicate that knowledge in a clear and understandable manner. You're a teacher, pure and simple, and an extremely good one at that. I hope you know how good you are at this. Thank you for these videos!

[@AlexStormDrake]

hahaha, language discontinuity yes. That caught me off guard as well.

[@Chronozia]

I was *not* expecting that in a 3b1b-esque video. I burst out laughing since it caught me so off guard.

[@DavidJGall]

It's not a 'discontinuity' so much as it is a... yeet!

[@DavidLindes]

@@DavidJGall whatever it is, it's delightful. :)

[@trevorasargent]

i came here to say this. so goooooood so stoked to watch the rest of this :D

[@JohannesSchmitz]

Unbelievable level of animation quality. Again one of the best produced KZbin videos of the year. Hopefully this will win an award.

[@VeganSemihCyprus33]

Truth should be shared 👉 The Connections (2021) [short documentary] 👀

[@deiphosant]

Why don't you give it an award?

[@infinitesimalphilip1470]

@@deiphosant Because they aren’t an official organization capable of doing that.

[@DavidLindes]

@@infinitesimalphilip1470 being "official" isn't required for doing a thing, including giving an award. It just may (or may not!) make the award more meaningful -- to the recipient and/or to others. Freya, I hereby award you the status of "creator of the video most demonstrating something I want to be doing more of" for 2022. FWIW. ;)

[@DavidDelSolOMG]

totally

[@seven_keys]

Your skills in coding, math, animation, knowledge sharing and keeping viewers focused are amazing! Thanks for inspiration :)

[@l0gansw0rld]

i started to watch this last night and the smooth animations and soothing narration knocked me out. then, realizing the quality of what i had just missed, i watched the whole thing this morning. it was very enjoyable both ways. keep up the good work!

[@TheChillosophizer]

I didn't expect to watch this in it's entirety. I can say without question this was the most useful lecture I have ever received and the phenomenal visuals were the key. I've used splines for years, but I had a lot of difficulty implementing them in code because I lacked the fundamental understanding. Thank you so much for this deep dive on math noodles :P.

[@eroraf8637]

Okay, KZbin, you’ve convinced me. I’ve seen this video in my recommendations for a while now, and WOW have I been missing out! This is easily of the same level of quality as 3B1B, and I’m definitely subscribing. Hope you’re doing well!

[@acegikmo]

thank you ❤

[@eshafto]

I played hard-to-get for a while, but it was just a game and KZbin knew it. It put "The continuity of splines" in my recommendations. I pretended not to notice. It kept putting it there. I opened the link into a new tab. I closed the tab a few days later. But the algorithm knew it had me. Our little dance continues. I did not expect what I got, though. This is an utterly amazing video. It took me so far into mathematics that would otherwise have been impenetrable, and tied it into real-world experience and intuition. On a meta-level, it's a master course on how to create a master course. I think anyone involved in education--from KZbinrs to curriculum and textbook creators--could learn a lot from this video.

[@SmokeyChipOatley]

@@eshafto Yes. What you said, 100%.

[@geekmuffin]

I would also like to echo every single thing that Eric said, and then add a couple of my own observations: IMHO, your voice and cadence are SUBLIME for this! They're the first and biggest factors that kept me from switching. I was going to check a few seconds and then add it to my Watch Later if it seemed halfway decent. NO WAY was I going to watch a 1+ hour video tonight! But I was hooked more and more at every transition. And after almost 74 minutes, I'm actually wanting to watch another video of yours instead of going to bed like I should. Oh, and regarding flatness of the delivery or your personality not shining through, I think it was extremely entertaining. I sensed your personality, to some extent, throughout. Having said that, I would certainly welcome even more of it! I just don't want you thinking it was dry. It certainly wasn't to this casual math nerd! BTW, I have to feel very strongly about a video to leave a comment. The best educators show you that you're smarter than you thought and leave you floating on a cloud of "I can't believe I understood that!" That's what you just did for me.

[@acegikmo]

@@geekmuffin I'm happy to hear that, thank you Walt!

[@Shadeslayer-is9cn]

One of the best video essays I've ever watched! Im currently going through some linear algebra and differential equations courses and it was amazing seeing the applications of all these concepts along with such beautiful animations. Keep up the wonderful work!

[@TheAgamemnon911]

It took me a while to find the time to watch this hour long essay, but since I am subscribed, YT recommended I watch it. Now I finally found the time and it was worth every second! I especially liked the part where you brought the reflections in 3D in. Such a WOW effect!

[@kdbin101-1]

This is going to be an incredible resource for so many people for many years to come. You've done something truly amazing here.

[@georgecorney9073]

Wow. Freya this video is absolutely fantastic, I've been on a similar spline journey recently and I've learnt more in the first few minutes than I have in weeks of reading around online It's a resource for the ages I can only thank you (and join your patreon!)

[@teacupanimates]

50 pounds, thats a signal for people to watch this channel, someone willingly donated 50 pounds as a thanks

[@aze4308]

money

[@sebbog]

you gave your money

[@whatsappvideos9665]

why does freya talk from the nose?

[@georgecorney9073]

@@_v_m_ Would never expect it! She put in a year of work to share this information which I wanted to thank her for Better she's able to focus on her day-to-day than read through thousands of comments (I know I'd find this hard too!)

[@Mezgrman]

This is absolutely incredibly well done. I wouldn't have expected to stay this hooked for this long in a video just about splines. Massive props to you!

[@arteks2001]

I take my hat off. This is one of the best videos I have ever watched on KZbin. Excellent explanations. As a mathematician and numerical analyst, I will certainly recommend this video to my students.

[@cineblazer]

1:11:12 Obviously I'm not you so I won't ever hear your voice the way you hear it, but I thought your narration was absolutely stellar! You sounded genuinely excited about everything you talked about without going over the top, your pronunciation is crystal-clear, your audio/recording setup is very professional, and never once did I feel like the video sounded "boring"! Honestly, I was more engaged watching this video than I am watching a 3Blue1Brown video, which is high praise because I love 3B1B!

[@Merthalophor]

Agreed, this video is a true gem! I wish it existed back in my numerical methods class... might have gotten more than 40%.

[@JohnDlugosz]

@@Merthalophor Yes, watching this I contemplated that animation is a teaching tool that really adds to the explanatory power of a video, compared to traditional media like textbooks and live teachers.

[@luisfonseca2086]

Agreed!

[@stulora3172]

same, your voice is perfect.

[@78Mathius]

Your voice is wonderful. You need not change anything.

[@molgeraxproductions6671]

Your very conclusive, intuitive explanations, your stellarly smooth animations and your calm voice have lulled me into such a meditative state of understanding along such a nice mathematical topic, that 22:47 just annihilated me and threw me laughing to the floor. I hope you recover from burnout and are able to take your time with that, because goodness, you have given us too much with this. Such a personal toll is never worth it, yet I can't not say that this video is such an amazing achievement, Freya. A new cornerstone of education that people will point back to for years to come❤

[@antanis]

oh my god, it sent me. super apt time to use it.

[@solhsa]

I was eating lunch. Luckily, I did not suffocate. =)

[@SpicyMelonYT]

I basically had a heart attack from that lol. I was chilling with the calm music, and I was making something in blender and that exploded my brain

[@philippschmidt78]

My thoughts exactly :D

[@Cappurniggas]

Seriously? He sounds like he's got throat cancer or something.

[@aaron_gomes]

I hope you can grasp the impact of this kind of content. Thank you for your amazing contribution in building something we can all be proud of.

[@jbejaran]

"It yeets off to fucking wherever" is my new favorite quote that I'll be incorporating in my own conversations.

[@Khytau]

Having implemented b-splines back then at university, I can't even fathom how well documented and directed this video is. This is an insane amount of work and you have earned my eternal and sincerest respect. Thanks! Definitely looking forward next topic!

[@kbusse]

Freya, this is reference stuff for the ages. This combination of clear visual presentation, deep understanding, good narrative and an astonishing grasp for speed and rythm is outstanding.

[@NLaQ2]

I don't comment often, but I wanted to express my appreciation for such a well assembled video! I thoroughly enjoyed it and learned a lot about things that I've always wanted to learn but assumed were beyond me. This video was stuck in my recommendations for a while but I kept procrastinating because of the length... but I'm very glad that I finally watched it! In retrospect the length wasn't an issue at all once I actually started watching.

[@olchobolcho]

Really great in-depth explanation, I honestly wanted more! You're animations are probably the best I've seen out there and frankly your delivery sounded quite natural and easy to follow. Def not boring! Thank you so much, I actually learned quite a lot from your video! Please keep it up!

[@iout]

The video itself is fantastic, but everyone else has already mentioned that, so I just want to take the time out to appreciate that for all the effort that was already put into this hour+ long video, it also has subtitles to go with it. And really good, intentional subtitles to boot. They've got subscripts and the character β and proper timing. Just above and beyond, well done. That along with the stellar visuals, clear narration, and even the chapters make this a very educational and accessible video.

[@acegikmo]

thank you

[@nicolaesposito6490]

@@acegikmo it doesn't, especially for non-native English speakers. Congrats on the great work. The script, the voice, the animations, the visuals, the subtitles, everything is top-notch quality.

[@robharwood3538]

@@acegikmo I notice it, too! Even though I'm a native speaker, for certain kinds of videos, in this case ones that involve technical and/or math-related terminology and the usage of precise meanings, I often turn on subtitles just so I can both listen (aural) and read (textual) at the same time as watching the video (visual). It's like being able to read along in a textbook or the professor's notes during an audio-visual lecture/lesson. Helps with comprehension and retention, IMHO. Thanks for this video and all the glorious care and effort put into it! It is *very much* appreciated!!! I'm actually going to recommend it to a math professor I follow on KZbin who has done stuff in the past about splines, because your video actually goes deeper into the topic than he did, especially in terms of providing serious, yet intuitive motivation for why higher sophistication with C and G continuity are necessary to treat in a mathematical treatment of splines.

[@oliverwagner2453]

This is such a gem-how I wish all topics could be covered.

[@daanza5597]

Absolutely perfect, thank you so much for all of this! ✨

[@sharmsma]

An absolute joy to watch. THANK YOU. It summarizes many things I've learned during my last job, which was all about splines. I hope you'll make more videos about splines.

[@jefwesb]

I'm in awe. The facts, the animation, the storytelling

[@calloq1035]

I honestly think this is the single most impressive KZbin video I have ever seen. The research that went into this, the animations, the chapter continuity, everything, really incredible

[@Erik_g33k]

This video definitely deserves special recognition from KZbin/Google

[@doishid]

very relaxing, incredibly informative. As someone who had only heard the word spline once or twice, I understood 99% of what you were saying, and this is definitely helped by the amazing visuals (not in the least to say the script does an amazing job as well)

[@ywenp]

31:38 "It's the wedge product returning a bivector" That little geometric algebra tease didn't go unnoticed. Subscribing for what's to come ;)

[@torstenoakes]

YOUR ANIMATION AND SOUND DESIGN IS ABSOLUTELY AMAZING I just got to the intro for Bezier Splines and I had to rewatch it three times, my jaw dropped every time. Your attention to every tiny detail is astounding.

[@jaredcramsie182]

4:22 The most common font in use today, TrueType, uses only quadratic splines. There are some good reasons: 1. Fonts are only made once, but are displayed continuously, so flexibility is not as important as performance. 2. Glyph outlines are usually straight lines or rounded curves. This means that the extra point needed to define a cubic spline is normally redundant. Additionally sharp changes in direction aid the reader in parsing letters, so more points on the path is often beneficial. 3. The rendering algorithm is apparently much simpler, which was a concern for less powerful devices.

[@jaredcramsie182]

Avoiding cusps, mentioned at 39:15 , was another reason mentioned for using quadratic splines.

[@destroyoid]

Yeah. I work with fonts a lot. The comment about cubics being more usual surprised me. I think before truetype then in might have been the case. And graphics libraries tended to have a cubic function and not a special quadratic one because if you set the controld points of a cubic right it equals a certain quadratic. But even in graphics libraries not I think quadratic is more common.

[@jaredcramsie182]

@@destroyoid Before -TrueType- _outline fonts_ most of the fonts were bitmap based, but I think there was a period in the -1970's- _1980's_ where some -non-standard formats- _fonts uncommon in modern usage_ used cubic splines. _The font is "Type 1", see Alexis' coments below._ A lot of the new, modern font formats do allow for cubic splines - likely because performance is no longer an issue. _Thanks again to Alexis for the corrections, edits are in italics._

[@destroyoid]

@@jaredcramsie182 Type 1 fonts were introduced by Adobe in 1984 for use with its PostScript page description language, and became widely used with the spread of desktop publishing software and printers that could use PostScript. That was what I was thinking of. So before truetype it was postscript. I would not say most were bitmap fonts. Sure for 8 bit computers and games. But not professional workstations...

[@destroyoid]

Truetype was 1992. So Between 1984 and 1992. Type 1 fonts were most common. And I was talking about spline fonts not bitmap fonts anyway,

[@AnarchistArtificer]

I feel like my experience watching this video was very authentic: I started watching a few months ago, but I stopped part way because I was burnt out by life stuff, but I've come back and finished it and I am so glad I did. I don't even know why this was recommended to me, but I am suddenly intensely interested in splines. Incredible, beautiful work, thank you so much this video, I feel privileged to be a part of the statistical anomalies at the end.

[@gubscruz]

Amazing work. The animations were flawless and the teaching was amazing! Please keep up with the work

[@cedrict.6617]

A lot of what I want to say has already been said but anyway : I saw ( and felt in love with) your first video on bezier curves and splines a year ago and this is an incredible and unexpected sequel. This is simply one of the best video I have ever seen : -You have made professional animations -You're passionate by the topics to a point where we hear it in your voice, we instantaneously want to be passionate with you -I believe you've used the topic of the video to bring us into a trip in a beautiful (and really smooth) world, it's been hard not to start dreaming and floating around in this world you've created -The way you explain you're creation process at the end helps further understand you're vision of video's creation, and what it means to you. It encourages us to appreciate even more the work you've done I'd love to see more videos (small or 1h long) and I understand why this might take a long time. I hope you're getting better, I wish you all the best In hope you make us voyage again soon Thank you from a cat lover 😻😽 PS: I'm a 20 years old French student please forgive me for any language errors I could have done (no hard feelings about Hermite I promise you) ❤️

[@DavidLindes]

I read and liked this comment after pausing earlier in the video. Then I got to 42:53, and tried to remember what you'd said. Glad it was OK, and... very much agree with all you've said (which came through fine to this 48 year old native English speaker -- while it's perhaps not "perfect", see 22:48 for my feelings on what perfection can do (😉), and it was certainly understandable, and frankly I wouldn't have known you weren't a native speaker until I read as much in your P.S. line.).

[@amrabozaid1942]

You made the world beautiful ❤️❤️😍

[@Reed81315]

this is better than my grad classes. it's easy to see how much heart and effort you made this and my attention was kept the whole time. i will never look at a shiny car surface again the same way now after seeing g2 surfaces

[@DanKaschel]

I think it is in the nature of art that pushing ourselves past our limits produces a voice that feels unfamiliar. It is like seeing a texture through a microscope; we see and know too much. Every part is the countless hours that created it instead of time spent experiencing it. In that mindset it can be disconcerting to hear that your art is perfect or complete. I want to encourage you that even the most frustrating parts of making this video--time wasted in fruitless rabbit holes, months spent in deep avoidance, painful compromises to end an endless project--bring no less value than your beautiful animations and charming, soothing voiceover. I want to encourage you that every great work sits atop a tower of blood and tears that few ever see. I want to encourage you that the world is littered with such piles that lacked the courage to take the last few steps. I've made many such. But you did it. It was a mistake to take on such a big project. But sometimes ignorance is better than bravery. May you continue to make art that challenges you and inspires us.

[@jorgevilchis1047]

Thanks Freya for this video, you have done an outstanding job covering in depth all the details about B-splines used for animation. It is a precise work in terms of the theoretical content, and extraordinary work in terms of didactic by generating those amazing animations. Thanks for sharing !

[@nelamazon8530]

As an architect and 3d modeler, I really appreciate this type of content. This is by far the best graphic explanation of splines and curves in general. This topic is very hard to explain, and very hard to understand when I was a student. This video does not only explain the terms very well but also visually represent the ideas behind boring definitions. Most people (in 3d and graphic industry) take these concepts and tools for granted, like it was just there, and the principles behind just don't matter as long as we know how to use it and its applications.

[@thallesaraujo7814]

People taking concepts and tools like these for granted is unfortunately very true in many fields, I believe. I'm not from the graphic industry (I'm an oceanographer) and feel the same.

[@bonovoxel7527]

@@thallesaraujo7814 I draw race tracks into sandbox videogames, learnt now this from scratch. I could even have had a badass sofware but before this I didn't know why some curves I drew are more interesting to drive thru than others. I wondered why since some are flat so it wasn't altitude change or banked turns. Well it's speed, but not just that and this video was mindblowing as she started to visualize the various steps at which two sections can be continuous. Ideally, C3 G3 steps are the kind of continuity you want in a rail track or highway. C1 is the dammm corkscrew at Laguna Seca. I loved this content !

[@Erik_g33k]

As mechanical design engineer and industrial design lover I couldn't agree more

[@mcny40]

This animation work is nothing less than an masterpiece. Astonishing!

[@rustedshader]

It's all coded

[@Kenionatus]

@@rustedshader She didn't just throwing the splines on the screen and calculating movement over them, she also animated all the little highlight circles, arrows and lines and probably a whole other host of things that I didn't consciously notice. Those were 100% creative animation.

[@rustedshader]

@@Kenionatus yeah i have seen her doing that in her streams truely a great work

[@shivmsit]

The explanation and in-depth video on splines is amazing! I had casually started reading about splines after studying mathematics and watching some confusing videos that lacked detailed explanations. However, I came across this video, which is absolutely amazing!

[@josipjosifovski3262]

Wonderful video, it helped me a lot in understanding the underlying concepts and math, and hearing that you made it in Unity motivated me even more as I am using it for my current project at work, which is (kind of) related to splines. Thank you very much!

[@Sewdn]

The world needs more of this kind of premium content(-creators). Thank you so much for your enormous efforts and the amount of risk you take when investing this much time, passion and expertise without any direct return.

[@acegikmo]

thank you

[@bigpopakap]

I can't agree more with the other comments. This is truly one of the best math videos I've seen on KZbin! You provided great motivation and context to make it easy to understand, and each chapter nicely builds off the previous. It's just enough information to be engaging and illuminating without getting too pedantic. And the cherry on top is the super slick animations, particularly the smoothest chapter transitions ever!! Amazing work. This is my favorite movie of 2022

[@acegikmo]

thank you so much kapil!

[@bigpopakap]

This is still going strong as my favorite video of 2023... Sorry, Barbie! This time what hit me was the way in which you structured the whole journey, slowly building up concepts one by one. Particularly, at the end I loved the trick quiz to drive home the point that splines are "curve generators - transformations from control points to curves that make certain promises about continuity". Well summarized, and easily forgotten over the course of the video There needs to be some award for didactic film, so that you can win it 😂

[@multiarray2320]

​@@bigpopakapwhat about oppenheimer? ;)

[@unils3891]

Couldn't have said it better myself!

[@jeffkal]

This was fascinating and made a complicated topic very accessible. Also, 22-something minutes into it, you made me laugh harder than I have in months. Thanks!

[@Oscar-vs5yw]

That was awesome! beautiful aminations, wonderfully packed with dense yet digestible information presented at a good pace, can't find any critism, one of the best videos I've seen on youtube! I am extremely impressed. Thank you for your work

[@geekmuffin]

Thank you for that experience, the knowledge I gained from it, and especially this feeling that it left me with! I hope you and your favorite peeps are having a wonderful winter solstice.

[@lekeshala3735]

How did you send 50$ over comments how can I do that

[@geekmuffin]

@@lekeshala3735 Look for a button with a heart shaped icon. It might say something like "Thanks" depending on the app and your settings.

[@zh9664]

simp

[@ArmanNobari]

Extremely practical, well-taught, and incredibly well produced. Thank you so much for putting this out into the world.

[@FreyaHolmer--]

Thanks for watching and leaving a comment😊. ᴛᴇʟᴇɢʀᴀᴍ the above username, got a package for you📦🎉..

[@KaiRosenkranz]

Your interactive visualizations are incredible. ❤ What a great way to illustrate a video.

[@p9fd3mt]

Brilliant!!! The effort put in is clearly visible! THANK YOU for one of the best YT videos I've come across. Great content, fantastic presentation, intuitive layout and crisp voice over. Can't stop myself from sharing. Hats off!

[@devonparsons687]

Thank you for making this video. I have been eagerly awaiting a thorough examination on this exact topic for years and you have delivered with a beautiful, mesmerizing, illustrative and approachable masterpiece.

[@BlameItOnGreg]

Feels like you condensed an entire graduate degree’s worth of content into an hour long video!

[@drobin9040]

Outstanding explanations and in-depth visualisations. Excellent content!

[@rafagamedev3606]

Thank you so much for this video Freya, this was so insightful and the visuals really helped me understand the topics at the intuition level. You rock!

[@HighWindd87]

This is such perfectly crafted, visually pleasing and comprehensive material. I am an automotive engineer specialized in interior design and I have to train fresh engineers in class A surfacing from time to time. This material will be immensely valuable for the particular interested in what's happening under the hood when drawing those unruly splines. Thank you! BTW: G3 in various CAD software is called "flow" continuity.

[@acegikmo]

oh neat! I never came across that term in my research somehow

[@CharonNg]

You know what? I didn't realise this video is hour long until you mentioned it. It's so funny to watch and so many knowledges packed into it. Your sound, animation, visual presentation, chapters ...all perfect to me. Hope you recover from your burnout soon and your journey of continuity to be continued...

[@Pilikio]

How is this sooo goood??? I really appreciate the work you've put into making such a high quality explanation video. This is the content the internet needs more of! Keep doing what you're doing. I don't mind one hour long videos if they're this good. In the future please continue to create content. Obviously please don't burn yourself out...

[@Mister0Eel]

This is amazing, having used splines for a while now in different contexts/programs, I've wondered what that was all about. Thank you for this in-depth explanation that gave me soo many 'ooooooh that's why [thing] works that way' moments!

[@jayceez]

Je pense sincèrement que ton travail est exemplaire de ce que devrait être un cours concis. Et même si c'est "la surface du sujet", n'oublie pas que la seule mission d'un professeur est de créer une étincelle de curiosité chez les élèves. J'ai envie de me lancer dans des logiciels qui font des Splines, et c'est grâce à toi. Merci beaucoup ! xoxo

[@brandonhoughton]

I love that you made your own spline - I was trying to answer so many of the initial questions you posed with "just add another dimension" so it brought great pleasure you did in your spline"

[@tompotter8748]

Masterpiece! I have a feeling I will be watching this video multiple times over the years to review it and keep it fresh and well-understood in 'mah brain. .. In many moments watching it I was just amazed, like how all the information that is presented is itself so smoothly and expressively interpolated, and it just keeps going -- I'm sure it still takes a ton of work, but the fact that you are able to produce this much highly-optimal visual representation of what you're saying, moment-to-moment, in a continuous style with continuous interpolations, is seriously amazing and requires some level of mastery I'm sure. It was also amazing to me when you went from 2D to 3D, it's interesting to see that your stack is in Unity. Truly great work, and thank you very much for sharing such a very well-considered-and-detail-finessing-in-just-the-right-way smooth mental path through the conceptual realms, with just the right level of comprehensiveness-yet-lightness, as you charted these territories of the great mathematical world and traversed them with aplomb!!~ :)

[@paulbarton4395]

Great video! I stuck around to the very end. Don't be so hard on yourself or worry about complainers, there's no way anyone can cram topics like these into a video of this length and cover everything. This stuff takes up huge sections of books on differential geometry, numerical methods, graphics, etc, (as I'm sure you know). Your delivery was excellent: clear, not boring, and more fun than many lecturers out there

[@ant.robotics]

One of the greatest lectures of all time. It's like a 3B1B lectures, where one can go beyond just being able to solve equations but being able to understand it. Watching spline was simply a line with C ♾️

[@tacodiva7729]

I can't believe I just watched an hour long, fully animated beautiful video for free. Thank you for making this.

[@robbie7150]

This is so information dense, I'm gonna have to watch this at least 3 times before I can really appreciate the beautiful artwork, research, and explanations here on a deep enough level

[@alfcnz]

Amazing! Watched every single second of it. Truly mesmerising. And yes, I’ve just subscribed to your channel. Thanks a bunch for this work of art! ❤️❤️❤️

[@ammoniteonline5948]

I think that this is unequivocally the best video I have ever watched on youtube. I am going to go join Patreon just because of this. The amount of work is mind boggling. The whole topic is far beyond my realm of knowledge and yet was somehow accessible. I don't leave comments as a rule, but here I am. I can't help it. This is just so brilliant. Thank you for this and sacrifices you have made to create it. I am immensely grateful and now quietly obsessed with NURBS.

[@acegikmo]

thank you so much

[@Hippida]

Well said! I know I hardly understood half of it, but somehow enjoyed it, even the slight headache that followed

[@EliaCereda]

Love how you made a complicated math topic accessible, the world needs more content like this. Keep it up!

[@The_NSeven]

This makes so much sense!! If only I had known some of these things earlier. Thanks for providing this for free, absolutely beautiful animations

[@ChrisWilson1973]

This is an incredible piece of work. Fascinating, and very accessible, with clear, beautifully flowing visuals.

[@andreasmetz2438]

Freya I'm a GameDev-Teacher for Unity myself and can only pull my hat. I can't even begin estimate how many thousand hours you put into this but let me tell you, it was worth it! I'm not a huge math fan and tried to avoid it as much as possible in university but your lessons are just so on point that it's really a pleasure to watch. Absolutely looking forward to your next Animation.

[@AimeePlaysMSM]

This is an amazing video, thank you so much for making it! About 'your voice', I agree that it shines through more in your streams, but honestly for videos like these I think the overall tone is spot on. The "yeets off" part caught me by surprise but in a good way! You touched on a few things that I hope you could go into a bit more in the future; You mention that the bezier curves approximate, but do not make, a circle ( 14:10 ) - is that something you could explain a little further, given that many vector programs, even when drawing using circle tools, are in fact using bezier curves (4 specifically, though I have seen 3 and even 2 can give a 'good enough' approximation) to do so? "Good approximation of circles by curvature-continuous Bézier curves", Dokken 1990 and "Circle approximation by G2 Bézier curves of degree n with 2n-1 extreme points", Ahn 2019 might be good references? Additionally: what type of curves, if any, *can* make a perfect circle? You mention the Apple icons shape, but never got back to it; I know there's great write ups about this one already and the shape you show later in the video ( 26:00 ) alludes to it, but might be good just to close that reference by applying that curve to the shape? At ( 17:00 ) you explain that the bezier curve is not continuous in velocity. With some animation programs where bezier curves can be used as paths for an object's position (among other), there's an option to force it to have constant velocity specifically to deal with it; without knowing the exact code a program might use, what do you believe goes on behind-the-scenes in those programs to make that happen? I know you're probably looking forward to talking about anything *but* curves at this point, but I for one would love to see more about 'math noodles' :)

[@acegikmo]

yes! like I mentioned during the credits, I want to make smaller videos ahead, and talking about splines and circles is one of the video ideas I'd like to touch on :) Long story short, you need a rational spline to solve this, the most simple is probably the quadratic rational bézier, though NURBS are usually mostly known for being able to make circles too. As for the app icon, there's lots of videos out there on squircles, if you want to read up on that! and about why rounded squares are generally not very smooth looking As for moving at a constant speed, that's called arc-length parameterization, which has no closed form solution for the cubic bézier, but I do talk about how to solve that in my bézier video!

[@boretrk]

As someone who spent a couple of weekends a while ago trying to make smooth splines that "yeets off" part really resonated with me.

[@angeldude101]

Squircles are cool and it's a shame they aren't better known. It's even possible to combine them with other structures like hyperbolas and metaballs. You can pretty much make any kind of equation and turn it into a cool picture in one way or another. Now I'm curious if there's a way to parameterise a spline to turn it into a level curve.

[@AimeePlaysMSM]

@@acegikmo Thank you so much! I look forward to future videos (and hop into a stream when I can!)

[@AdelaideBen1]

Having just watched this last night, and knowing there's a crap-tonne of comments already. I think you should be proud, but also don't think people want to be critical (even on the inter-webs). The important thing is not to expect too much of yourself, and not to depend too much on others reactions. You've always done a great job (and yes, it's a different dynamic to a live stream, but our lives are always coloured by the moments in which we live). You're personality is deeply embedded into this video, even if its spoken with something other than your external voice.

[@jayaprakash2050]

Thank you very much for making this video! Very helpful and fun way to learn Splines. It was very motivating! Definitely made the topic interesting and motivating me to learn more about it! The hard work and effort behind making this video is very evident and probably one of the best youtube videos I have come across. :)

[@acegikmo]

thank you ❤️

[@alexanderkruszewski7306]

As a high school physics teacher who is about to teach a semester of algebra and geometry, I CANNOT WAIT for the radians video! This film was absolutely mesmerizing and I was blown away by the thoroughness and elegance of your work. There is so much pedagogical potential for media like this in the education sphere and I’m excited to see what else is possible here (on a healthy and sustainable production schedule, of course).

[@beanie_baby_snail]

I LOVE this, it’s so interesting, the animations are beautiful and you explain it all so well! I also laughed a lot when you went from speaking with somewhat formal language to “yeets off into f*cking nowhere” and then back

[@Fruitysfaction]

"It yeets off to fucking wherever" Citing correctly is hard, I know.

[@RedStallion2000]

@@Fruitysfaction Not correcting someone for a minor error is hard, I know.

[@jensschmidt3687]

The work that has gone in to this...just wow !! Thank you so so much 😊

[@yuempek]

Wow! You are an incredible person! What an amazing video! You managed to combine art, math, and teaching all at once - that's impressive! The information you presented about this topic was very clear and informative. You worked very hard and it is truly admirable. Respect to you! I've watched thousands of videos on KZbin like this, but you've redefined some things.

[@OmegaFalcon]

No single human should be able to contain this much knowledge in their head

[@tbird81]

She wrote a lot of it down

[@acegikmo]

notion is one of my friends at this point

[@baksoBoy]

Holy crap this video is amazing!! The animations are beautiful, and the content of the video is super fascinating. After being completely absorbed by it, I got curious how long time was left for the video, and I find out that it is more than an hour long?! No way there is an entire hour filled with animation and commentary as good as this! That is insane! Edit: 1:09:35 holy shit I didn't expect you to be a part of that community!.. should of probably expected that concidering your headphones... Kind of related to that, when you talked about Knot Values & Knot Intervals I imagined you making the super specific joke of saying something like "to the furry reacting to me saying that, shut up" or something like that, and now that I know that you are a part of the community that honestly wouldn't be that big of a stretch... ignoring the fact that it is probably a really bad joke that like 1% of people will get ¯\_(ツ)_/¯

[@acegikmo]

oh the headphones? haha yeeeah I know, it's a little weird to not wear them on my ears, but like you say it makes me fit into the human community better

[@shreekararaghavan1471]

I just wanted to watch some video on what splines are, but once I started watching this video i was instantly hooked on, could grasp whatever you wanted to convey. You are a fabulous teacher. Its crazy how this content is free, especially with this amazing animation. Thanks for releasing it for all of us to learn about splines.

[@lucasvergara6524]

I love how thorough you are! I also appreciate how you address possible "um actually" critiques that people will have