The mathematical beauty of hyperbezier curves
linebender.org
linebender.org
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.
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...
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.
Yes, there are regions of the parameter space where moving a control point has vanishing effect on the curve shape. I'm not thrilled about this and have explored alternatives, but this is the best I could come up with. Pretty much all approaches based on solving for optimal curve fit have discontinuous "flipping" behavior. Another approach would be to split
We haven't yet done the work wiring this up into a spline like the older pen tool draft. The "auto point" is not a form of subdivision but is a way of achieving G2 continuity.
Yes, hyperbeziers can be split and subdivided without much trouble at all, and being closed under subdivision is an important mathematical property that was lacking in the earlier draft. This follows directly from their formulation as polynomials. If you look into the code, you'll see that there's subdivision at the near-cusp to make the integral robust.
The P/C naming evokes Hermite but there wasn't a lot of thought into it.
Math is heavier, but this is more nuanced than you might expect. My feeling is that computers are fast, so if we can draw higher quality curves with less human effort, it's worth it. But the seeming simplicity of Béziers is deceptive; you often have to solve inverse arc length problems (for example to compute dashing), and curve fitting is really hard (I have a blog post in the pipeline on that). So I think overall the math is only slightly heavier.
The goal is human-driven interactive design, originally motivated by font design. Splitting into more Béziers obviously gives much finer control, but it's hard to avoid them getting lumpy. In most fonts you have a minimum number of Bézier segments and accept whatever fine details the Bézier math gives you. That's even more so when making variable fonts, where the control points are interpolated.
I haven't yet done a rigorous comparison with NURBS. While they're hugely popular in CAD, they're basically unused in things like font design.
Thanks for these questions, they obviously show a deeper understanding.
I have some modest experience with the polynomial curves but very little experience with the kind of non-polynomial curves you post about. I’m always curious but haven’t spent the time studying that I’d like to. I don’t have a lot of experience with NURBS either; it was just the rational in the hyperbezier curvature formula that made me think maybe there’s a relationship, and/or the rational in NURBS has something to do with it’s greater level of control versus the uniform B-spline.
First, for hair rendering you've got a 3D trajectory, which Béziers can handle just fine, but I'm not sure about these spirals. I do have a chapter in my thesis and there has been a bit of followup from others, but I can't say anything with confidence.
That said, because of the specular reflections, applications like hair really would benefit from higher degrees of continuity. Cem Yuksel has a recent SIGGRAPH paper on how to tweak Béziers to get more continuity. In fact, seeing that was one of the motivations to take this work off the back burner.
I'm not sure about maps. But I do know of applications representing the centerline for autonomous vehicles that use polynomial spiral representations, specifically because of the extremely high degrees of continuity you can attain. For that, you don't have to represent these high-tension curvature regions.
I do think there is a specific application for drawing smooth connecting lines in autogenerated diagrams. These tend to be S-shaped, and with Béziers you tend to get curvature peaks near the endpoints. You're much better off with monotonic curvature and with spirals that's easier to achieve. And I also believe being able to get those squircle/superellipse shapes would help as well.
Oh this is interesting. So FWIW - I’ve done a bit of CG hair in production (in a previous life, for DreamWorks) - and now that everyone’s ray tracing, I like to advocate for considering quadratic B-splines for CG hair over cubics. In practice, I don’t personally think you can see anything in hair specular reflections that is based on degree or continuity beyond C1. Catmull-Rom is only C1 and people have been using Catroms for hair for a long time (and interesting side note - I’ve found that Catmull-Rom curves are pretty lumpy and ugly, even lumpier than quadratic B-splines. This is possibly very similar to the issue of designing with subdivision you mentioned).
My opinion here does not apply to surface patches - with surfaces, the continuity can be visible in the reflections. But at my work among the researchers there was a very strong assumption that cubic B-spline curves would be visibly better than quadratic, and that quadratic would suffer from reflection artifacts due to the C1 continuity, which just doesn’t seem to be true (and ignores the history of Catmull-Rom). Quadratic curves are faster and more accurate for ray intersections and bbox calculations, and they offer some interesting advantages like having an analytic rotation-minimizing frame (with the Frenet frame, even…). Of course with procedural curves, I recommend only the quadratic B-spline, not Bézier. With a uniform B-spline, the degree of the curve is essentially nothing more than a smoothing factor. Production hair/fur/sim data tends to be over-sampled already, so the smoothing factor doesn’t buy you much.
BTW, yes I know Cem and I’m familiar with his work. He also has published on some non-polynomial curve types, so your post reminded me of him as well.
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.
I’d argue that talking about Bézier is absolutely sufficient in this context; B-splines and Catmull-Rom splines and a few other types are all piecewise polynomial splines (cubic specifically, at least in the case of Catmull-Rom). You can convert B-splines and Catmull-Rom splines to Bézier directly and analytically; therefore B-splines and Catmull-Rom splines cannot do anything beyond what Bézier curves can do. There are some minor usability / interface / intuition differences, but there is no difference in mathematical control.
It is my understanding that only a couple of very expensive font editors use B-splines where there are three control points which ripple as I noted, however it seems that FontForge supports order2 (quadratic, TrueType) or order3 (cubic, PostScript).
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.
I don’t doubt something Q3 related might have popularized it. I’m familiar with the fast inverse sqrt approximation that was popularized by Q3A. That’s not what you’re thinking of, is it? (https://en.wikipedia.org/wiki/Fast_inverse_square_root) But animators and most people in graphics and rendering (the Pixar, ILM, and Siggraph types) as well as all font designers and mathematicians and car designers and CAD people all knew about Bézier curves long before Quake 3 Arena.
You're right, I'm referencing to inverse sqrt.
Bezier curves were the main talking point about Quake 3 engine throughout all PC magazines when Quake 3 demo came out.
Because it was like nothing ever seen before, and noone ever expected it to run ~20-30FPS on period HW.
It wasn't a talking point for too long as with full game, Quake 3 became top multiplayer game, and graphics behind it were not important anymore.
Why do you say that? I was in graduate school studying computer graphics when Q3A was released. Before that, at a different school as an undergrad, there were entire courses on Bézier splines and NURBS and multiple textbooks that had written about them. All the universities in the US with computer science departments that taught any graphics in the 80s & 90s covered Bézier curves, and that’s not to mention anyone studying design, drafting, CAD, architecture, engineering, etc.
Ah I looked it up. Bi-Quadratic Bézier patches were used in Quake 3. They were tessellated into triangles for the HW.
The first quote I found from Carmack about this: “Well, this is technology that's been done since the late '60s as far as the different geometric representations go.” So it wasn’t new in general, but was new to real-time game engines, and there is a pretty big group of gamers and aspiring game programmers who heard about Bézier patches for the first time with Q3A. https://www.reddit.com/r/quake/comments/16w5g1i/the_quake_3_...
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.
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.
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.
For reference, drawing lines for a wooden boat is still made with actual wood splines on the drafting table, held in place by heavy "ducks" or "whales". Here's what is looks like:
https://dataphys.org/list/physical-splines/
https://www.core77.com/posts/55368/when-splines-were-physica...
That's especially true when taking the paper plans and drawing them (lofting) at scale 1:1 on the shop floor.
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.
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.
There are other Bézier shapes that the hyperbezier can't match, including two cusps on the interior. As you say, I consider these not very useful in actual 2D design, and am happy to give them up for smoother curves and the increased range of superellipse shapes.
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.
Wasn’t your PhD thesis on attack-resistant trust metrics? At least I remember reading it.
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
Hindsight is really something. Of course we usually write the denominator as |x'|^3 but once you see it, it makes sense.
>August 8, 2026
I would guess not.