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 💛 Jazz “queenjazz“ Mickle for music & sounds ❱ 🐈 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 ❱ 📈 Using Mathfs, my math library ❱ ...along with a bunch of hacky, partially broken, procedural animation tools I made specifically for this video 💡 Striped studio skybox by ProAssets: 🎥 Final editing in DaVinci Resolve • Links 🐦 Twitter ❱ 📺 Twitch ❱ 💬 Discord ❱ 🌸 Instagram ❱ 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