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#beziercurves#splines#vectorgraphics

The Beauty of Bézier Curves

Freya Holmér

291K subscribers • 6 videos on Hallway

YouTube
  • Recommended by 17 people: 16 on Hacker News and 1 on Lobsters.
  • A classic. People kept recommending it from 2021 to 2026.
  • Presented by Freya Holmér, game developer
  • Linked in the descriptionFreyaHolmer/Mathfs

Beginner. No background in the topic needed.

2.2M views5 years ago17 recommendations

Graphics and audioComputer scienceRenderingCreative codingMath for programmers

Freya Holmér explains the mathematics and visual construction of Bézier curves and their applications in computer graphics and animation.

Bézier curves - how do they do? They're used for animation, text rendering, and all sorts of curved shapes! But how do they actually work? well, like, that's what the video is about, so, watch it to find out etc!! • Lots of love to 💛 Jazz "queenjazz" Mickle for making the music ❱ queenjazz.bandcamp.com/ 💙 Grant "3Blue1Brown" Sanderson for pushing me to finally do this ❱ youtube.com/3blue1brown 🐦 Pomax for their wonderful writeup on Bézier curves ❱ pomax.github.io/bezierinfo 🐈 Thor the cat for.. hanging out.. when I was trying to record audio with no interruptions 🇫🇷 French subtitles by @damqui on Twitter • How was this video made? 🔨 Created in the Unity game engine ⭕ Using Shapes, my vector graphics plugin ❱ u3d.as/1W37 📈 Using Mathfs, my math library ❱ github.com/FreyaHolmer/Mathfs ...along with a bunch of hacky, partially broken, procedural animation tools I made specifically for this video • Did you know that "osculating" isn't pronounced that way? ✅ yes I do now in fact please stop commenting on it ;-; ❱ twitter.com/FreyaHolmer/status/1429472399389507591 • Links 💖 Patreon ❱ patreon.com/acegikmo 🐦 Twitter ❱ twitter.com/FreyaHolmer 📺 Twitch ❱ twitch.tv/acegikmo 💬 Discord ❱ discord.gg/v5VWuga 🌸 Instagram ❱ instagram.com/freya_holmer Intro Use Cases of Bézier Curves Lerp Quadratic Béziers Cubic Béziers De Casteljau's Algorithm Bernstein Polynomial Form 1st Derivative (Velocity) Tangents & Normals 2nd Derivative (Acceleration) 3rd Derivative (Jerk/Jolt) Curvature Bounding Box Arc Length Arc Length Parameterization & Uniform Animation Outro Credits Thor helping me record this at 2am

Why it's here

17 people recommend this

What people say. Highly recommended as one of the best visual and educational explanations of Bézier curves and spline mathematics.

friendlysockon Lobsters • Recommends

The Beauty of Bézier Curves

progbitson Hacker News • Recommends

On “SoME1 results”

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 link to this video

eveningtreeon Hacker News • Recommends

On “A Primer on Bézier Curves”

Another beautifully explained piece on bézier curves is: The beauty of bézier curves by Fréya Holmer link to this video Found this via a contest by Grant Sanderson, and it absolutely blew me away(and most people I forwarded it to)

armchairhackeron Hacker News • Recommends

On “Ask HN: Higher order derivatives in everyday life?”

…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: link to this video .…

viggityon Hacker News • Recommends

On “Jacksonpollock.org (2003)”

If splines/beziers are of any interest to you, I could not recommend these videos from Freya Holmer enough. link link to this video

viggityon Hacker News • Recommends

On “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 link (73 minutes) link to this video (24 minutes)

amtunlimitedon Hacker News • Recommends

On “An interactive guide to SVG paths”

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 link to this video

jedimasterton Hacker News • Recommends

On “A Primer on Bézier Curves – So What Makes a Bézier Curve?”

Related, I think Freya Holmér's "The Beauty of Bézier Curves" is in the running for one of the best educational videos on YouTube. link to this video

nrjameson Hacker News • Recommends

On “Manim: An animation engine for explanatory math videos”

While not Manim generated, I came across this video explaining Bezier curves yesterday and thought it was pretty awesome. link to this video

ud_0on Hacker News • Recommends

On “Chaikin’s Algorithm: Drawing Curves”

Freya Holmer made a very nice visualization of bezier curves by showing the "helper" lines here: link to 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).

BoppreHon Hacker News • Recommends

On “A layman’s guide to recreational mathematics videos”

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: link to this video

jbluepolarbearon Hacker News • Recommends

On “Curves and Surfaces”

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: link to this video

thecrims0nchinon Hacker News • Recommends

On “Cubic Bézier: From Math to Motion”

Grant Sanderson (3blue1brown) had a video submission contest and the winner that blew me away did a video on bezier curves. link to this video

an1sotropyon Hacker News • Recommends

On “Ask HN: Why are quadratic Bezier curves described so obtusely?”

…unctions. 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 link to 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.…

joshuon Hacker News • Recommends

On “When Splines Were Physical Objects”

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. link to this video even less relatedly, here is a drawing i did by torturing a clothoid optimizer link

zoogenyon Hacker News • Recommends

On “An Interactive Guide to SVG Paths”

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. link to this video