The Continuity of Splines

Freya Holmér
13 HN recommenders · recommended in 15 months, 2022–2026
YouTube
1.9M views · 3 years ago · explainer · expert · evergreen

Freya Holmér gives a long, animated tour of splines, covering the different kinds, the equations behind them and why derivatives and continuity matter when joining curves. It goes well beyond Bézier curves to explain how smooth paths are built.

HN rates it as one of the most informative math videos around, the one that finally made spline derivatives and Bézier splines make sense.

“It's one of the single most informative and useful math videos I've ever seen”
Explorable explanations & visual explanationComputational & parametric design
#splines#continuity#béziercurves#animatedmath#curvedesign

25 comments from Hacker News

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@virgil_disgr4ce2026 · in “A History of Curves and Surfaces in Computer Aided Geometric Design [pdf]”
I'll take any opportunity to spread the word about Freya Holmér's incredible overview of splines, "The Continuity of Splines": ▶ this video It's one of the single most informative and useful math videos I've ever seen, and taught me so much stuff I had no idea existed. I've been using beziers since I was a teen learning a pirated version of adobe illustrator 6 or 7 and I had simply assumed beziers were all you needed. NOPE!!!! Not only that, it allowed me to realize what was wrong with a game prototype I was working on at the time, which had exactly to do with the continuity problems discussed in the video. Cannot recommend it enough.
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@viggity2025 · in “Bezier-rs – algorithms for Bézier segments and shapes”
Anytime béziers are mentioned on HN, I feel compelled to share these absolutely incredible videos from Freya Holmér ▶ this video (73 minutes) link (24 minutes)
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@ixwt2024 · in “Hobby's algorithm for aesthetic Bézier splines (2020)”
I didn't really understand Bézier splines until I watched this excellent video [0] from Freya Holmér. It talks about various kinds of splines, and the various equations behind them. Explains the importance of the derivatives of splines (as other comments here have pointed out). [0]: ▶ this video
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@aaronblohowiak2024 · in “Look, ma, no matrices”
the splines yt video (▶ this video) is (IMHO) one of the best educational videos on a programming topic, full stop.
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@viggity2023 · in “Interpolation methods (1999)”
If this paper is interesting to you, you may also be interested in this amazingly well done video on the continuation of splines. It isn't exactly the same thing, and delves into some deeper concepts, but I've watched it like 3 times over the last year. ▶ this video
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@zxexz2023 · in “Why can't you multiply vectors? [video]”
Oh lovely, can't wait to give this a watch. Freya dives deep, and gives wonderful talks. Her video "The Continuity of Splines"[0] is my favorite watch of the past year. [0] ▶ this video
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@gadgetoid2026 · in “Curvature Beziers: Improving on a timeless recipe”
Well that’s wild because I had a disagreement with beziers recently and did a similar thing! [1] albeit with more madness and less polish. Driven by the a/s/d/f keys with find edges snap to reference images and curve fitting to construction geometry. Feel I should also call out Freyr’s videos for anyone who has missed them - continuity of splines [2] and the beauty of Bézier curves [3] [1] link [2] ▶ this video [3] link
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@junon2025 · in “G0-G3 corners, visualised: learn what "Apple corners" are”
If anyone wants a good primer into curves, Freya Holmér has an amazing deep dive into continuity. ▶ this video
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@WillAdams2025 · in “Ask HN: What cool skill or project interests you, but feels out of reach?”
For me, it's Bézier curves --- Freya Holmér has a wonderful video on them: ▶ this video and I've been trying to read through _METAFONT: The Program_ and link and I keep wondering if I shouldn't just try scripting Inkscape.... I want to do a couple of different things, and not sure if they all fit in one project or no: - implement a single line font in my current project: link - implement a way to convert arbitrary curves into smooth arcs (for DXFs or G2/G3 arcs for G-code) - work up an interactive version of METAFONT/METAPOST which allows both programming and drawing
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@skydhash2025 · in “Dijkstra On the foolishness of "natural language programming"”
> I just need to find the perfect GUI toolkit which allows Bézier Curve representations in a way which clicks and makes sense to me, and suits my next project. Not a proper answer, but here is a very good video on splines: ▶ this video
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@viggity2025 · in “Jacksonpollock.org (2003)”
If splines/beziers are of any interest to you, I could not recommend these videos from Freya Holmer enough. ▶ this video link
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@WillAdams2025 · in “Desmos Animated Graphing Calculator”
A wonderful example of this is: The Continuity of Splines ▶ this video and The Beauty of Bézier Curves link by Freya Holmér link While not animated, I find: _Euclid's Elements_ (Joyce's Java Version) very helpful link Since my current project has pretty much devolved to putting a UI on various trigonometric functions and allowing the user to program them, I'd be very glad of other resources in this space (currently working through Hilbert's _Geometry and the Imagination_ and the book which matches link and just finished the _Make:Geometry/Trigonometry/Calculus_ series, and have had recommended Curves and Surfaces for CAGD: A Practical Guide by Gerald Farin)
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