I love bezier curves. In community-college, it was the first time I ever encountered a subject that made me want to go do more research on my own. One of my core memories is toiling away for multiple nights when the rest of the house was asleep on my 2015 Macbook Pro, writing janky p5.js code and pressing refresh on the browser page over and over until suddenly, I started see real, beautiful curves blossom from my control points.

I just went back to dig up some old sources[0], and I can't believe this post is almost a decade old now. This guy's explainers and animations were leagues beyond any other resource I could find through Google searc at the time.

[0] https://jamie-wong.com/post/bezier-curves/

Related: Last year I had the pleasure to work with Bezier curves to make a very specific UI for an app I was working on.

I wrote down a bunch of posts on how I achieved the UI primarly to explain the process to my future self[0]

As a mobile developer it was so much fun to do something other than building a CRUD app.

[0]https://www.krtkush.com/computer-graphics-basics-with-compos...

So Raph, how do you draw these things? I’m not sure exactly where I’d start given an equation of the curvature or tangent (I assume?) angle. The pen tool link talks about “auto points”, giving the impression that it’s a sort of Bezier subdivision with some constraints, is that accurate? And your article mentions you’ve since fixed some things in the math - so this is a bit different from the pen tool, not using auto points?

Playing with your demo, I see the curve lock shape in certain configurations and stop moving even when I’m dragging one of the control points around. Is this a temporary or numerical stability issue, or is this a property of hyperbezier curves? For example if I arrange the control points in a square with P0 bottom left, C1 bottom right, C2 top left, P3 top right, and then drag C1 to the right, the shape locks up quickly. Eyeballing, it’s maybe when the length of P0C1 is ~1.5 times the length of C2P3? Moving C1 further right and up/down, there’s a huge area where moving C1 doesn’t affect the shape of the curve.

I was slightly curious about the P/C naming as well - is this a Hermite-like idea where P[03] are points and C[12] are similar to tangent vectors? Why not use P[0123]?

Probably one really obvious thing worth mentioning - Beziers are indeed very versatile, but it has to be said that one of the reasons they’re so popular is because the math is so simple, right? Decomposable into linear steps, integer exponents, with a couple multiplies of only your parameter you can evaluate them with a matrix multiply. They’re trivial to draw. Add in the controllability and intuitiveness and simplicity of computing other properties of the curve (such as finding roots & inflections & bounding boxes, splitting curves, etc.), and it makes sense they’re very popular. Hyperbeziers look interesting, and I don’t have intuition on their uses yet, but I assume their rendering involves heavier math than Beziers - since the curvature & angle equations have a divide & square root? Can they be split or subdivided easily?

You’ve been in search of a better curve for a long time. I’m wondering how you think about and where you stand on subdividing curves like Bezier vs finding a more perfect interpolation. Obviously there are tradeoffs, but are you aiming at problems that can’t be solved in practice using subdivision/approximation? I’d also be curious to hear your thoughts on hyperbeziers vs NURBS.

One unintuitive thing I noticed about these curves, at least from the demo on the page, is that they have both areas where their behavior quickly 'snaps' from one configuration to another, and areas where they saturate such that changes to the control handles no longer produce any movement. Standard Bezier curves don't have either of these; I wonder if that's essentially coupled to the higher expressivity they offer?

Playing with the example, the controls points don’t feel any more intuitive to work with than traditional cubic Béziers. They still exhibit annoying behaviors near the endpoints, though they don’t tend to “explode” like traditional cubics.

I find it curious that the author makes no attempt to compare his solution to other splines like B-splines or Catmull-Rom splines, given their popularity in computer graphics and CAD.

One big aspect that made Bezier popular was local support, i.e. moving control points resulted in expected changes in the same direction. It's interesting that this is the result of a search for 'a curve family better suited for interactive design than cubic Beziers.' I think it's better for achieving curve quality in the sense of curvature changes, but not so much in interactivity.

Trying the example from TFA, I think that it is easy to acquire a good intuition about how the curve moves when you pull a control point and the shapes of the curves that you can obtain are more beautiful and more interesting than what you obtain with cubic Beziers.

So I believe that if for you the "interactivity" does not seem better, that is more likely to be caused by being much more familiar about how cubic Beziers behave, from past experience, than because these hyperbezier curves were really less suited for interactivity. Also, the current implementation in JavaScript seems buggy.

For me, the poor approximation by cubic Beziers of some important curves, like conics and the Euler spiral, is an extremely serious defect, so I like these hyperbezier curves much more and I intend to investigate how efficient can they be, from a computational viewpoint.

Ran into this exact thing switching from cubic beziers to G2 splines in a font tool a while back. Muscle memory fights you for maybe a day, then dragging a point and getting curvature continuity instead of a kink at the join just feels obviously correct. Hard to go back after that.

Yeah, editing TrueType fonts w/ B-splines which have a shared off-curve control point between two on-curve control points is nightmarish as a small change will ripple through the entire outline.

There are two separate questions here. One is how much moving control points creates expected changes in the same direction. Béziers nail this, as the position of a point at t is a linear combination of the control points, with the Bernstein polynomials as weighting functions. So it always feels like direct control. With my mapping, you get this for a nice big chunk of the parameter range – small to moderate angles and control point distances. But this property does fall apart when pushing to extremes.

The other question is whether the control is local. As others have remarked, this has much more to do with the way the curve is embedded into a spline than the curve family itself. In particular, it is deeply affected by the continuity constraints. As payment for the local support, cubic Béziers only give G1 continuity. Euler spiral splines, by contrast, are G2 but changes do cause those ripples.

The pen tool prototype linked in the blog post suggests giving designers more choices. If you specify all control points, you get G1 just like a cubic Bézier. But it also gives the choice of specifying one control point on a smooth endpoint, and solving the other for G2 continuity. In my experience, it feels more like local control than Euler spiral-like splines. When you want smoother curves, you do that, and when you're willing to sacrifice continuity for local control, that's also possible.

Fine at t=0.5 in the demo. Someone drags a control point to a weird angle in prod, curve does something nobody predicted, and now it's your bug ticket at 3am.

One big aspect that made bezier curves popular was Quake 3 Arena. That was literally first time general public even heard the name.

Bezier curves were invented to design physical objects (cars), but they were an idealized and simplified solution to enable CAD at the time computer were much less powerful. Ideally, CAD should put elastica curves as the first choice - but they are difficult to compute as they can diverge drastically when control points get too close, and there is sometimes more than one solution for a set of constraints.

Spiro curves were supposed to solve it but I am surprised to read here that even spiro fails to approximate elastica. Have you mapped where the limitations are?

For information, I am drawing boat hulls, and elastica would model the natural bend of wood/plywood/metal much better than bezier... Amateur naval designers really lack good tools to bridge the gap between ideal hydrodynamic forms and ease of construction, and there are still a lot of tiny but annoying adjustments when one planks the first hull.

Elastica curves have only three parameters, compared with four for cubic Béziers, Spiro curves, and the new hyperbezier. The easiest way to understand that limitation is to consider the parallel curve of an Euler spiral. That has a built-in asymmetry, one end has higher tension than the other. But the math for elastica locks in odd symmetry around the inflection point.

I think hyperbezier would be a natural fit for boat hulls, but it's not a mathematically precise approximation to elastica either. I'd say to carefully evaluate it, and I'd very much like to hear how that goes.

ETA: "parallel curve of elastica" is an intriguing curve family to consider for this application, as it has the correct number of parameters and ticks a lot of the other boxes. However, the math for this is hard mode.

Very cool demo, The curve does sort of go screwball when the control points are close together.

After playing with it some more. Not close together, but more side loaded. On reflection while many of these forms would be awkward to find in a drawing program, I think this is the source of the superior spiral the author likes.

Cusps not useful in type design, yet somehow half of comic sans is one.

It can’t do a loop (or a cusp, in the limit) like cubic Beziers can, and may instead "lock" to a weird hook-like shape, I’m not sure whether it’s just an artifact of the way the demo tries to match the curve to a Bezier.

Loops and cusps are of course not very useful in 2D graphic or typeface design, and even if you’re animating a plane or designing a rollercoaster you’d probably use more than one Bezier to make up a looping shape.

These are an exciting new curve. I’d like to hear about how they will handle stroking. The inside and outside stroke of a Bézier curve can’t be represented with a Bézier curve. How do hyperbeziers fare?

This is future work, but I expect it to go fairly well. One encouraging sign is that there are parts of the parameter space where you do get an exact solution: circular arcs and circle involute. Another encouraging sign is that it's easy to get bounds on curvature, which is important for figuring out whether there's a cusp in the offset curve.

I have a wooden banana hanger that I designed using bezier curves, twenty years ago. Actually I made two; one of pine and another one of oak. I have the files somewhere, I hope; it was made using an abandonware vector graphics program called Dia.

It is made of two pieces: the base and the hook support. If you look at it from certain angles, you can perceive continuity between the curves of these two parts, because of the way the bezier curves project. I feel that this gives it elegance even when viewed from other angles.

“Polynomial spirals, or Spiro curves, the subject of my PhD thesis, fail to achieve this goal.”

Wasn’t your PhD thesis on attack-resistant trust metrics? At least I remember reading it.

That was my first attempt at a PhD thesis. The one that actually got me over the line was the second attempt, on spirals and splines. Of course this was all years ago.

This is awesome.

Geometric topology makes my brain itch. I refuse to engage with it, I am certain it would drive me mad.

I do agree though, those elastica (elasticas? elastici??) are fine as hell

>Perhaps this is not surprising, as the exponent of 1.5 resembles the equation for curvature of a parametric curve : k = x' × x'' / (x'•x')^(1.5)

Hindsight is really something. Of course we usually write the denominator as |x'|^3 but once you see it, it makes sense.

This is a really cool demo, but I don't necessarily see these superceding traditional beziers so much as being another option for designers

Has anyone used those for a real life project that solves some issues or whatever?

>In this post, I present the curve family to the world.

>August 8, 2026

I would guess not.

Not yet.

Same story as splines in Postscript/Metafont days, or NURBS in CAD — curvature-continuous parameterizations always sound great in the paper, take years before anyone bets production code on them. Font tools especially graveyard these. Ask again in 2030.

Curvature-continuous splines aren't new territory — Mehlum's work on curvature splines and the "log-aesthetic curve" literature from Miura and others tackle similar aims. What's notable here is treating tangent angle as the primitive rather than fitting curvature directly; that choice seems to be what tames the numerical instability the older methods struggled with.