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.
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.
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.
That's great! Thanks for writing and maintaining this document over the years!
Coincidentally, I was referencing the section on offsetting  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 .
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 =)
Threads with same title but different URLs:
2009 (a bit): https://news.ycombinator.com/item?id=558066
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. 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).
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.
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 =)
>"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.
>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: 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).
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!)