[dahart]

That’s fair, compelling uses of quadratics are legitimately hard to find, and usually it’s not significantly more costly to go to cubics. Here are few more reasons I’ve collected: the quadratic is easier to explain and understand, useful from a pedagogical perspective; quadratics, unlike cubics, have an analytic rotation minimizing moving frame (a normal and binormal that minimize twist along the curve); quadratics can be analytically sampled in steps of equal change in tangent angle (very handy for drawing, path rendering, tessellating for GPUs); and quadratics are much easier to ray trace (using a swept-circle or swept-sphere formulation). Ray tracing a swept-sphere cubic is solving an order 10 polynomial, while the quadratic is order 6 - but you can reduce the order to 4 with some constraints, which suddenly allows it to be evaluated analytically (in theory), while the cubic is permanently stuck in iterative numeric methods. One of the reasons I liked investigating the quadratic B-spline is because it suddenly makes clear that the order of a B-spline is nothing more than a smoothing factor. Quadratic and cubic and quartic are all the same curve, more or less, just successively smoother. Maybe that’s unsurprising, but from a practical point of view, it can allow you to reduce or increase the order without changing the high level behavior of your curves. Anyway sorry to geek out, I was just curious - I’ve done some work in ray tracing of hair, which is what led me down this path. I did not use quadratics when I worked in games or films, but looking back on it, I think there might have been good reasons to consider it especially on lower powered hardware, maybe for simulation and procedural animation more than hand-animated stuff…