I don't think this does a great job of building up an intuition of how these curves are derived. Instead, I'd recommend starting with quadratic (and noticing th

I don't think this does a great job of building up an intuition of how these curves are derived. Instead, I'd recommend starting with quadratic (and noticing the constraints on the tangents) and then realize how the cubic is built up from that. Like all (later discovered) splines it's a really elegant and simple thing!