Bézier Curves and the logic behind them
blog.richardekwonye.com
blog.richardekwonye.com
Long interactive read from Bartosz Ciechanowski [2]
https://ciechanow.ski/curves-and-surfaces/
His blogs always hit the top of the front page for good reasons.
Quadratic bezier: constructed by a velocity vector and an acceleration vector -- strictly 2D (planar curve). If you can draw a "V", then you can extend it into a parallelogram.[0]
Cubic bezier: constructed by velocity, acceleration, and jerk vector -- generally non-planar 3D curves (since there are three vectors involved), only special cases are planar.
[0] https://www.desmos.com/calculator/enz2qqc2ov
IMO Rational Quadratic Beziers are much more interesting since it defines 3D perspective, the entire field of Computer Vision can be derived from it alone.
Also the use of polynomial isn't exactly just due to being "easier to fit", rather it's that it's a fundamental primitive of the derivative operator, it being easier to fit is more like a "perk" that arises. Kinda like saying that "we use integers because it's easier to represent with bits" -- not wrong, just that the cause and the reward is kinda backwards.
People run away with their math woo... You're just talking about an eigenspace of a linear operator. It's true polynomials are closed under d but they're not special - sinusoids are the most obvious other example. There are countless other examples as soon you start building slightly more complicated differential operators.
A single curve segment is indeed planar, but it might be too strong to say strictly 2D? You can certainly make 3D quadratic curves. I use cubic and quadratic Beziers mostly for fur and hair, and some people do mention to me their concerns about quadrics being planar, but I think it’s not an issue in practice at all. (And quadratics are simpler and have some big advantages over cubics.) When you piece together multiple quadric segments and match tangents properly, it’s a 3D spline and you can’t really even see the planarity, each segment is a different plane. 2 quadratic segments do a pretty good job of approximating 1 cubic segment.
I like thinking about B-splines more that individual segments, and for a B-spline, you can think about the difference between quadratic and cubic as more like a smoothing factor and little else.
I can't remember the reference precisely but I believe in the film Dune some missiles follow Bezier curves to an estimated target position because it gives a more interesting flight path. Or something like that.