The Beauty of Bézier Curves on HackerTube: recommended by 16 Hacker News commenters across 13 separate months, 2021–2026.
explainer · intro · evergreen
Freya Holmer explains how Bezier curves work, from the helper-line construction to their uses in animation, text rendering and shapes. It uses visuals from her Unity Shapes plugin and was a SoME1 contest winner.
HN repeatedly calls it one of the best math and graphics explainers on YouTube, praised for bridging game dev and math.
“it's genuinely one of the best math/graphics/explainer videos ever made”
20 Hacker News comments link this video. Each one links back to its thread.
Related, I think Freya Holmér's "The Beauty of Bézier Curves" is in the running for one of the best educational videos on YouTube. ▶ this video
in “A Primer on Bézier Curves – So What Makes a Bézier Curve?” · View on HN
Anytime béziers are mentioned on HN, I feel compelled to share these absolutely incredible videos from Freya Holmér [link] (73 minutes) ▶ this video (24 minutes)
in “Bezier-rs – algorithms for Bézier segments and shapes” · View on HN
I'm really glad the article links to Freya's "Beauty of Bezier Curves" videos, it's genuinely one of the best math/graphics/explainer videos ever made ▶ this video
in “An interactive guide to SVG paths” · View on HN
If splines/beziers are of any interest to you, I could not recommend these videos from Freya Holmer enough. [link] ▶ this video
in “Jacksonpollock.org (2003)” · View on HN
Bezier curves. Cubic bezier curves (generally the most common type) are represented by cubic polynomials x(t) and y(t) (or just one polynomial P(t) where the coefficients are vectors). The first derivative gives you the tangent and normal. The second derivative gives you the direction the normal will gradually become as t increases. The third derivative gives you the change in this change (the fourth derivative is zero). Curvature is also computed from the first and second derivatives: K = det(P', P'')/||P'||^3 Other kinds of bezier curves are similar. In quadratic ones the first and second derivatives are normal and normal change but the third derivative is zero. In higher-polynomial bezier curves the third derivative changes over time as the fourth is non-zero, and so on. Source is a really interesting video on bezier curves which I highly recommend: ▶ this video .
in “Ask HN: Higher order derivatives in everyday life?” · View on HN
Another beautifully explained piece on bézier curves is: The beauty of bézier curves by Fréya Holmer ▶ this video Found this via a contest by Grant Sanderson, and it absolutely blew me away(and most people I forwarded it to)
in “A Primer on Bézier Curves” · View on HN
Probably "The Beauty of Bézier Curves" [1], one of the choices by 3B1B, comes closest. It is already a relatively large channel although usually focused on game dev tutorials rather than math explanations. This video bridges those two areas and is absolutely beautiful. [1] ▶ this video
in “SoME1 results” · View on HN
Maybe not exactly what you're looking for, but this video is excellent. And her other video on Splines is also great. ▶ this video
in “Bezier-rs – algorithms for Bézier segments and shapes” · View on HN
A wonderful example of this is: The Continuity of Splines [link] and The Beauty of Bézier Curves ▶ this video 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)
in “Desmos Animated Graphing Calculator” · View on HN
If you are looking for solid content on Bezier curves then Freya Holmér's video on The Beauty of Bézier Curves [1] is well worth the time it takes to watch. 1. ▶ this video
in “An Interactive Guide to SVG Paths” · View on HN
i think - but am not sure - they use clothoid definitions to characterize jerk, as it is more restrictive regarding the maximum rate of turn change. you can make quite a sharp turn with a mathematical spline. but i can see how they would use the physical spline to solve this… Googling “railroad spline clothoid” got some intriguing hits — and this HN comment. [link] semi-relatedly, this is a fun watch. ▶ this video even less relatedly, here is a drawing i did by torturing a clothoid optimizer [link]
in “When Splines Were Physical Objects” · View on HN
It is not the case that cubic Beziers are 3D. There’s a good chance that the font you’re using to read this (especially if you print it out) defines the outline of (2D) letters with a sequence of cubic Beziers. Lots of 2D computer graphics uses cubic Beziers because they ~balance the needs of having lots of DOF for expressivity, while not being so complicated that they’re a pain to evaluate. And the control points that define a quadratic Bezier could be in any dimension, including 3, or even 1, which is good for plotting the basis functions. Where you are shapes how you make sense of things. As you learn to see Bézier curves from different viewpoints, the particular physical viewpoint you have now will coexist with others. It’s always that way learning math. That feeling of being disoriented is something to treasure, btw; you won’t always have it. My hunch is that this video ▶ this video comes from someone who’s path through learning Bézier curves was non-traditional, but they do a beautiful job of explaining and illustrating things. It showed me that I have more to learn too.
in “Ask HN: Why are quadratic Bezier curves described so obtusely?” · View on HN
Grant Sanderson (3blue1brown) had a video submission contest and the winner that blew me away did a video on bezier curves. ▶ this video
in “Cubic Bézier: From Math to Motion” · View on HN
Curves and Surfaces was my favorite class in college and I’ve tried to apply it where ever possible. I recommend this video from Freya Holmer on Bézier curves. Really cool: ▶ this video
in “Curves and Surfaces” · View on HN
I'm also a big fan of Freya Holmér. She's the author of one of my favorite VR games, Budget Cuts, and her Youtube channel has videos on game development math and shaders [link] Her latest video is on Bézier Curves, using beautiful imagery from her Unity plugin Shapes: ▶ this video
in “A layman’s guide to recreational mathematics videos” · View on HN
While not Manim generated, I came across this video explaining Bezier curves yesterday and thought it was pretty awesome. ▶ this video
in “Manim: An animation engine for explanatory math videos” · View on HN
Freya Holmer made a very nice visualization of bezier curves by showing the "helper" lines here: ▶ this video Worth a watch for a quick introduction/recap on how these curves work in the first place (with a focus on gamedev applications).
in “Chaikin’s Algorithm: Drawing Curves” · View on HN
The fundamental question here is one which I don't think there is an agreed-upon answer for: >What does an algorithm look like? I am working on a rather visually-oriented tool, and while I did a fair bit of early development using BlockSCAD, and also have access to OpenSCAD Graph Editor, it hasn't made sense to show the algorithms visually because it's a lot of work making them suitably expressive. One almost wishes that the specialized library used for: ▶ this video was published and widely used. Similarly, why can't we have more things such as: [link]
in “Literate programming: Knuth is doing it wrong (2014)” · View on HN
It's the same creator, but longer video. ▶ this video I wonder if the beginning can be skipped if the first video was already seen?
in “The Continuity of Splines – Video Essay by Freya Holmér” · View on HN
In the last 3 days I've worked on this little side project. I got the idea when I saw this video: ▶ this video . I just had to agree with her. Bézier curves are truly beautiful. While coding this project I sometimes found myself playing around with the curves for 10-20 minutes. Something about them is just hypnotising.
in “Show HN: Interactive Visualisation of Bézier Curves” · View on HN