For spline interpolation, it looks the degree 3 cubic spline is accepted as the better way and in my understanding it requires 1st and 2nd derivatives at the joints to be the same.
It looks to me the second order spline can provide smooth continuity at the joints, so it can satisfy the same value and smooth continuity at the joints.
Then why cube is preferred or better where the 2nd order can satisfy the needs of Spline interpolation?
Is it because cube with two derivatives can easily solve the equation or are there other advantages?
- Cubic Spline Interpolation
- Python for Fiance 2nd edition - Chapter 11. Mathematical Tools - Interpolation
The basic idea is to do a regression between two neighboring data points in such a way that not only are the data points perfectly matched by the resulting piecewise-defined interpolation function, but also the function is continuously differentiable at the data points. Continuous differentiability requires at least interpolation of degree 3—i.e., with cubic splines. However, the approach also works in general with quadratic and even linear splines.
Why "ontinuous differentiability requires at least interpolation of degree 3—i.e., with cubic splines"?
- Interpolation with Spline Functions
Cubic splines are most common. In this case the function is represented by a cubic polynomial within each interval and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end-points, which are commonly taken as zero; this gives the natural cubic splines.