A Primer on Bézier Curves (2013)

although as a Springer book it's a bit pricey =( (I'll be trying to add rational B splines and NURBS to the primer soon..ish... there's a few other topics that still need expanding first)

The tutorial doesn't work for me. It provides a thorough explanation of the polynomials, but it fails to explain the very basics of how the curves are laid out on the plane.

It does a nice work of explaining how a circle is drawn, but its explanation of how x and y are calculated for a Bézier curve doesn't make sense to me: "Bézier curves use the "binomial polynomial" for both x and y." If the same function on "t" is used for both x and y, they would always have the same value, and the "curve" is always a diagonal with 45 degrees. What am I missing? I can't infer it from the tutorial.

Also, it introduces the idea of a "control point" and starts using it, but never manages to define what it is, or how it is represented in the equation or graphically. It merely labels some curves as "P1", "P2"... in the graphs and telss that these somehow change the shape of the curves, and tells us that they have an associated linear "wi" weight; but again, the way in which the (x1, y1) pairs are connected to the binomial and polynomial term is never spelled out, and left as an exercise to the reader.

Things are further complicated by the mysterious "strength" parameter "S" in the graphs, which is also unexplained, never used again, and appears right when we're trying to understand the difficult concept of a control point to further confuse things. I had followed the tutorial quite well up to that point, but that section lost me completely.

nice, file it and I'll see how easily I can get on that.

Quadratic curves get very "pointy" with increased curvature, so you're trading "easy to draw a single smooth curve" (fairly meaningless in vector work) for "a genuine hassle to line up the consecutive curves that you'll need for more complex curve shapes" (pretty much the meat of vector work).

Of course, I say "complex" but and those aren't actually all that complex at all: you'll already need two quadratic curves just for a quarter circle (and by extension any quarter ellipse), for instance. While it sounds appealing, once you can actually draw quadratic curves you end up needing so many of them that refining your shapes (which takes up the majority of time for curve work if you're playing human data processor) becomes an incredibly hard job.

The most obvious example of a common technology that uses quadratics is OpenType fonts with TrueType outlines, which only uses straight lines and quadratic Bezier curves, although most tools I know of will give you cubics there, too, and simply save to a correct quadratic representation upon export/font generation.

Outside of those, you can use visual programming languages like Processing for your own purposes if you absolutely need them.

I used quadratic curves to render strokes in TabletDraw, although that's not user level.

Another advantage of quadratic curves is that they can be converted to a parabolic function plus a rotation. That opens the door for all sorts of optimization that is much harder with cubic bezier curves, which are not guaranteed to be functions.

Most fonts are created with quadratic Bezier curves, I think.

Sheetmetal fabrication is hard, and knowing what you can and can not make out of a single sheet is not obvious. For that reason car companies were very early in the game to use computers to gain an edge over their competitors by making others 'impossible' shapes with repeatable accuracy and resulting material thickness. Deformation results in variable material thickness and if you want to input curves into a computer (so your numerically controlled mill can make the tool-and-die used to stamp the part) then it helps if you can do so easily. That's where the Bezier curves (and splines) come in, they allow you to specify a complex curve using only a few points.

I used them extensively for a computer program that cut sails for sailing vessels, the problems are roughly similar (only you can't really deform cloth all that much so you're going to end up cutting panels, the end result looks much the same as a 2 dimensionally deformed sheet of metal though).

Of course it’s not a coincidence. Car companies were one of the first places where it was reasonable to do computer-aided design, back when computers were rare and expensive.

3d Beziers have been a thing for a long time already though! In modelling, NURBS surfaces are typically used (the non-uniform rational B-spline, where the rational B-spline part is a "rationalised" bezier, which is a whole section that I've not had the time to write yet...)