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 💖

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
🐈 our children, Thor, Salad & Toast, for helping me out during recording

• How was this video made?
🔨 I made it in Unity, the game engine
...along with a bunch of hacky, partially broken, procedural animation tools I made specifically for this video
🎥 Final editing in DaVinci Resolve

• Links

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

[temporaltoast]

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

[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 YouTube

[mirkoforastiere]

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.

[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!

[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.

[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 💖

[stijnvandrongelen]

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.

[fersahahmet]

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

[bennettgardiner]

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

[damionmurray]

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.

[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.

[marklandgraf]

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.

[JohannesSchmitz]

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

[LasseVågsætherKarlsen]

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 Youtube. 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!

[georgecorney]

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!)

[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!

[cstiger]

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.

[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!

[kdbin-]

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

[RobFulwell]

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