I realized that the cubic function fits very nicely with four information on two vertices. I wonder what it would look like if we do something similar for higher dimension curves with second+ derivatives or even surfaces.
One interesting place that this technique is used: robotics! When planning out paths for a robot to follow, this technique can be used to convert waypoints into a full path that can be tracked in real-time.
Wouldn't you have to discretize that anyway though (for the purposes of robot control update loop)