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A Primer on Bézier Curves (2013) (pomax.github.io)
243 points by acdanger on Aug 20, 2019 | hide | past | favorite | 34 comments

Wow, every time I see this site I am blown away by how well it is done.

I am going to put a couple of book titles out there in case somebody doesn't know about them and could benefit:

1.) "The NURBS book" -- Anybody got this one on their bookshelf? (authors Les Piegl and Wayne Tiller) I can't recommend it enough, if you are into rolling your own traditional B-spline lib with expert help!

2.) Then, for a gentle intro to multiresolution (especially because it makes for a good jumping off point from traditional B-splines) there is "Wavelet's for Computer Graphics" by Eric Stollnitz, Tony DeRose, and David Salesin. Very good for a "matrix transform / projection-prolongation picture of splines, and related entities.

Anyway, these helped me on my (continuing) Spline/CAD journey. Cheers.

This really shouldn't need the (2013) suffix - it's constantly getting updated, and the most recent update was from only a few days ago. (it's probably about 50 pages longer than it was back in 2013, if not more)

I didn’t know it was under active development, but I use this reference a lot. Thank you!

What are you adding these days?

You know what’s on my wish list? A simplified uniform B-spline treatment specific to cubic & quadratic curves. Everywhere I look online, the B-spline explanations are crazy long and complicated with recursive summation diagrams and math about knots and non-uniformity and N-degree evaluation all at once. An open uniform cubic B-spline, though, is really simple, and that’s probably all I’d ever use in CG in practice. But it’s difficult to distill from what’s online right now. Converting a cubic B-spline to Bezier, for example, is a simple 4x4 matrix, and recently I needed it and went looking... it took a while to find.

That does sound useful to talk about, could I convince you to hit up the issue tracker to for that as a section request?

I just added a weirdly missing short section on plain rational beziers (not b-splines, just single beziers), and I'd like to still add a section on turning hand drawn paths into polybeziers.

>I just added a weirdly missing short section on plain rational beziers

That's great! Thanks for writing and maintaining this document over the years!

Coincidentally, I was referencing the section on offsetting [0] just last week and read the following,

>There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. ...We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

and thought, something like: "That's correct because Bézier curves in the context of this document, SVG, HTML canvas and fonts means polynomial (i.e non-rational) Bézier curves."

But if you're now considering rational Bézier curves as in-scope for the document that changes things doesn't it ? E.g Circles represented as rational Bézier quadratics, and Pythagorean Hodographs [1].

[0] https://pomax.github.io/bezierinfo/#offsetting

[1] http://demonstrations.wolfram.com/G1HermiteInterpolationWith...

Rational Beziers are no longer polynomials, though. If the basis value used for the division was a constant, then it would be, but instead it's a function of t.

Yes, I’ll request it via the issue tracker. Thanks!!

I open sourced Bezie (https://github.com/jperler/bezie), a library I wrote three years ago for MIDI automation curves. It includes some good examples of how to interpolate cubic and quadratic bezier curves given drag points on the curves. I used Pomax's bezier write ups as a starting point, which really helped out a lot (thanks Pomax!). If anyone is interested, take a look in `bezie/app/utils`, specifically the `getControl` methods that allow you to derive control points.

Always cool to see what people end up making!

The bezier.js library associated with this is great

Thanks! I had to update it for the most recent update (it had no concept of rational bezier curves) so hopefully I can update its API page soon with a short bit on how to specify curve ratios and have that be available to everyone, too.

Previous discussion (2017): https://news.ycombinator.com/item?id=14191577

People keep wanting to read stories about bezier curves. It's the law of spline demand.

Splines is short for “sparkling lines”, which is the correct term when they don’t come from the Bézier region of France

Beautiful! This joke has a very "XKCD" quality to it.

Not gonna lie: that was great.

Thank you for this pun.

I constantly add new sections. I might not be "every week" or even "every month", but if the last time you saw it was 2017, there's a whole lot of content that you haven't seen before.

This is a living online book, and there's more to write on the subject. Every time you see a link for it on HN, it's probably because I wrote a new section, and a lot of people discover the internet that weren't on it one or two years ago. This content is evergreen =)

It's almost as if I keep adding new material to it every half year or so =)

On HN, even without you having added to it, reposts are acceptable after about a year. What you're replying to is meant to connect past conversations, not to claim it shouldn't have been posted :-)

Noted, thanks =)

I enjoyed reading it, but I wouldn't call it a "primer", since I can hardly think of what else there is to know about Bezier curves! It's a rather comprehensive exposition.

Here's a primer on Bezier curves if you had to program them in 30 minutes. You won't need more than this:

1. Bezier curves are just taking weighted averages, repeatedly. A weighted average of two points, A and B, is a linear combination P(t) = (1-t)A + tB. As t goes from 0 to 1 with constant speed, P(t) goes from A to B with constant speed.

A weighted average like this gives you one point out of two. Now, say you have four points, in order: A1, A2, A3, A4. For a number t between 0 and 1, take weighted averages of consecutive points: A1 and A2, A2 and A3, A3 and A4. That gives you a sequence three points. Repeat this procedure until you have only one point left. That's your Bezier curve defined by A1..A4 at time t. This is called a 3rd degree curve, and generalizes to sequences of any length (you can figure out how!).

2. The above tells you how to draw this with a computer. Just pick, say, 100 values of t between 0 and 1, and connect the dots with straight lines. That is, you converted a curve into a poly-line, consisting of straight segments. This can be used to compute intersections (just see if any one segment intersects any other), compute tangents (extend any segment to get a tangent at that point), normals (in 2D: rotate a segment by 90 degrees clockwise).

3. Most importantly, why you would want to do something like this: Bezier curves are easy for machines to trace, and for humans to understand. Here are some properties of a (cubic) Bezier curve given by points A1, A2, A3, A4:

-it starts at A1, and ends at A4

-it is tangent to segments A1A2 and A3A4 at endpoints

-by moving A2 and A3, it is easy to make a C- and S- shaped curve.

Beyond that, things aren't as intuitive. This is why all graphics software gives you 3rd degree Bezier curves as a tool: it's flexible enough while easy enough.

In practice, Bezier curves came to replace the French Curve[1]. The irony is that there is little French about the French curve (nobody even knows why they're called that!), but Bezier curves are French through and trough: the two key people behind them, Paul de Casteljau and Pierre Bezier, have spent their entire lives in France, and came up with the curves for manufacturing at Citroen and Renault. Arguably, Bezier curves are the true French curves.

4. Bonus. In Photoshop/Illustrator/etc: the UI for Bezier curves is this: click and drag to define A1 and A2 (A1 is is where you click, A2 is where you release). Consecutive click-drag-releases define three points: e.g. A3 on mouse down, A4 on mouse up, and A5 such that A4 is the midpoint of A3 and A5. The curve you get is a union of cubic Beziers defined by A1..A4, A4..A7, A7..A10, etc.

The curve is smooth because the UI forces the tangents to be aligned (unless you do a single click without dragging: this makes 2 tangents of length 0, i.e. a vertex).

When drawing curves using this UI, you really are drawing tangents to the curve you want to get: draw _— to get an S, roughly.

That's it folks! For way, way more - read the article :) (Which is an interactive book, at this point).


> 2. The above tells you how to draw this with a computer. Just pick, say, 100 values of t between 0 and 1, and connect the dots with straight lines.

This is the obvious algorithm, but it has the problem that curvature is not uniform with respect to t - sometimes a straight line is a good approximation to a segment but sometimes it isn't. So instead you can draw a curve with a function that draws a straight line when that's good enough, and splits the curve in two then recurses on each half otherwise.

Good point! It's hard to make a call as to what to do include in a 5-minute overview, but your point fits in one sentence, and is easy to understand. Thanks!

"all" graphics software is quite the sweeping statement. One of the most common graphics languages used in every day life by everyone on a computer, is the truetype graphics language, used to define vector graphics for one of the two major flavours of openetype font. TTF only does quadratic curves, and is ubiquitous.

There are also plenty of CNC packages that don't even give you Bezier curves at all, and I would definitely consider those graphics software.

As for the book no longer being a primer: just because it's longer than a few pages, doesn't mean it's not a primer anymore =) It's currently about 100 pages, so essentially what you'd be expected to read "before the next class" in university, which is my standard for what a primer is: a primer should get you complete understanding at the fundamental level. Anything less and it's not going to prime the reader for further study. So it speaks to the subject of Bezier curves that you can have 100 pages and _still_ be just a primer on the subject. If I wanted it to be exhaustive, it'd probably be at least twice the current size (now... we might get there eventually, but not any time soon!)

As such I also wouldn't consider this description a "primer" but more of a short descriptiong: it correctly describes de Casteljau's algorithm, but the drawing algorithm is not the curve type, and doesn't really work in terms of getting people to understand Bezier curves themselves (unless they're scarily good at internalising just an algorithm description and then abstracting all the implications of that algorithm without further aid =)

Thanks again for the primer and the comment :)

>"all" graphics software is quite the sweeping statement.

I meant virtually all popular vector graphics programs with a GUI that let users draw curves. I am not aware of a single one that doesn't have a Bezier curve tool - even MSPaint has had it in Windows 3.11.

Can't say much for CNC packages, except that Bezier curves were literally made for CNC. I'd guess that they accept CAD designs which were made with software where Bezier curves exist, but are converted to poly-lines in the output.

Surely, specialized software without Bezier is out there; but that's how you draw curves in Photoshop and Illustrator :)

> It's currently about 100 pages, so essentially what you'd be expected to read "before the next class" in university

Ha! Spotted someone blessed with not having to teach an intro math class to 100 freshmen :D

FWIW Spivak's Calculus on Manifolds is about the same size, and it's a semester-long course[1].

>twice the current size (.. but not any time soon!)

..I'll keep hitting F5 in the meantime :)


On that note, a request. In the figure where you talk about curvature matching, you are effectively tracing out the Evolute[2]: the intersection points of nearby normal lines trace out the evolute in the limit (the locus of centers of curvature of the curve).

I think that actually drawing the evolute would be a good addition to the text (and would improve the reader's understanding of the concept of curvature).




Spivak doesn't use early high school maths, of course ;)

mentioning the fact that there is such a thing as an "evolute" is a good point, though - could I convinve you to file an issue on the tracker so I don't forget about that? (HN threads disappear all too quickly!)

What a great contribution.

Back in highschool I wrote a script using sagemath that could draw bezier curves when given the points and handles. It was really cool how simple the math behind it was.

The expression of a Bezier curve involves t^i (in the summation). Both t and i acquire values of 0 at some point, so you get a 0^0 in the summation, which is undefined.

But also understood to be 1 here.

Very readable. A fantastic contribution.


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