Natural Cubic Splines
If you want 2nd derivatives at joints to match up, the resulting curves are called natural cubic splines
It’s a simple computation to solve for the cubics' coefficients. (See Numerical Recipes in C book for code.)
A real-time implementation with mouse-based moving, insertion, and deletion of control points feels a lot like a physical spline
Finding all the right weights is a global calculation (solve tridiagonal linear system)