The Continuity of Splines
1,380,335
Published 2022-12-07
if you like my work, please consider supporting me đ
www.patreon.com/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
âą Lots of love to
đ Jazz "queenjazz" Mickle for music & sounds â± queenjazz.bandcamp.com/
đ our children, Thor, Salad & Toast, for helping me out during recording
âą How was this video made?
đš I made it in Unity, the game engine
â Using Shapes, my vector graphics plugin â± u3d.as/1W37
đ Using Mathfs, my math library â± github.com/FreyaHolmer/Mathfs
...along with a bunch of hacky, partially broken, procedural animation tools I made specifically for this video
đĄ Striped studio skybox by ProAssets: assetstore.unity.com/packages/2d/textures-materialâŠ
đ„ Final editing in DaVinci Resolve
âą Links
đŠ Twitter â± twitter.com/FreyaHolmer
đș Twitch â± www.twitch.tv/acegikmo
đŹ Discord â± discord.gg/v5VWuga
đž Instagram â± instagram.com/freya_holmer
Timestamps:
00:00:00 Intro
00:01:17 Examples of Paths
00:02:16 CHAPTER 1 - BĂ©zier Curves
00:02:20 Lerp
00:03:16 Quadratic BĂ©zier
00:03:54 Cubic BĂ©zier
00:07:41 n-Degree BĂ©zier
00:09:37 CHAPTER 2 - BĂ©zier Splines
00:10:00 Cubic BĂ©zier 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 BĂ©zier
00:14:28 Tangent Points
00:15:07 CHAPTER 3 - Continuity
00:15:28 Câ° (Positional Continuity)
00:16:12 Spline Derivatives
00:17:32 CÂč (Velocity Continuity)
00:20:25 CÂČ (Acceleration Continuity)
00:21:48 Cascading Loss of Local Control
00:23:46 CÂł (Jolt Continuity)
00:24:49 Câ Continuity
00:25:19 Parametric Continuity Summary
00:26:20 CHAPTER 4 - Geometric Continuity
00:26:50 GÂč (Tangent Continuity)
00:27:46 GÂČ (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 GÂč (Algebraically)
00:33:27 GÂČ (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 â BĂ©zier 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 CÂČ 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
All Comments (21)
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âAs you can see, it yeets off to fucking whereverâ caught me off guard. But like in a good way
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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
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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.
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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!
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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.
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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.
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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
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absolutely amazing. incredibly good animations, sense of aesthetics, and teaching style. and "yeets off" at 22:50 caught me completely off guard đand cracked me up. thank you
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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.
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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 đ
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Not only is this material fascinating, the visualisations are absolutely world class! Outstanding work! â
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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.
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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.
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OKAY. I already thought the animation was gorgeous, and then we entered 3D reflective surfaces. This is absolutely incredible and you deserve more views.
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This might be the best video on youtube!
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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.
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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.
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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 â which would be awesome as a future video in more detail! Thank you so much for going through all the research!
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Your skills in coding, math, animation, knowledge sharing and keeping viewers focused are amazing! Thanks for inspiration :)
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I take my hat off. This is one of the best videos I have ever watched on YouTube. Excellent explanations. As a mathematician and numerical analyst, I will certainly recommend this video to my students.
