I wonder why we choose a cubic over any other polynomial type? Even degree polynomials seem interesting enough? Perhaps having a higher degree makes things more complicated, too much calculation necessary? Though a cubic does give you access to both the odd and even degree polynomial.
@philippe-eecs To my understanding, a cubic polynomial is the bare minimum needed to interpolate two points with their derivatives set to some value (e.g. if both slopes on the 2D points were positive, we can't use a quadratic, but we can use a cubic function to fit it). Also I believe that you're correct in that high degree polynomials aren't quite necessary due to higher computation costs, but the tradeoff isn't quite clear. This short paper goes into how quartic Hermite interpolation can be better than cubic Hermite.