The Beauty of Bézier Curves

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

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)

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

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

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 .

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)

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

Maybe not exactly what you're looking for, but this video is excellent. And her other video on Splines is also great. ▶ this video

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)

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

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

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.