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 ❱ https://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 ❱ https://u3d.as/1W37
📈 Using Mathfs, my math library ❱ https://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: https://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