# Derivatives of B-splines

1. Oct 20, 2015

### hunt_mat

Hi,

I have been playing with B-splines recently and I have had some curious results. As a rest, I had a set of points on the ellipse; $$\frac{x^{2}}{9}+y^{2}=1$$. I used the paramatrisation $$x=3\cos\theta ,\quad y=\sin\theta$$ and I computed the derivatives numerically (I had to come up with a numerical scheme for differentiation using non-uniform grids. It works okay for some simple functions but I am not very familiar with how these things are done)

I found that the first derivative (w.r.t. $\theta$) was very well approximated (to within 10^-3), plotting it on matlab, you could just about see where they were different. When I calculated the second derivative (w.r.t. $\theta$) (by applying my derivative formula again), the curves were a poor agreement. There were lots of oscillations in the calculation of the second derivative.

So my question is this:

Are the oscillations just a sign of my poor choice of numerical derivative, this phenomena an example of Runge's phenomena or something which is inherent within B-splines? I was using a degree 5 B-spline, the result was worse for a degree 7 curve.

Any suggestions?

2. Oct 20, 2015

### mathman

As a general rule, when approximating a function, the derivatives will differ, and each order of derivative is worse.

3. Oct 21, 2015

### hunt_mat

I understand that. The degree of the curve I am using is 5, so that implies $C^{3}$ continuity. What I don't get is HOW they're worse.