## Cubic Spline and Curvature

In comparing the parametric cubic spline to the cubic Bezier spline, you may notice that the parametric spline has a tendency to produce a ‘tighter’ or ‘less curved’ fit through the knot set. There is a technical reason for this observation that is based on the derivation of the parametric spline from the natural cubic spline, discussed in this TechNote.

Suppose the cubic spline, S, is to fit a set of knots in the interval [a,b]. Suppose the knots have some functional representation, f (which is unknown), and that the second derivative of f is continuous in [a,b]. By interpolating f with S at the knots, we exactly represent f at each knot and approximate f in between knots. It can be shown that

∫[S”(x)]^{2}d(x) ≤ ∫[f”(x)]^{2}d(x)

where the intgeral is over [a,b]. Since the curvature of a function is given by

|f”(x)|[1 + f'(x)^{2}]^{-3/2}

|f”(x)| is an approximation to the curvature, which is minimized by the natural cubic spline. As a result, this spline has a tendency to take a ‘less curved’ path through the knots.

Looking at blue/red curves from previous post left-to-right along the curves, I have this impression:

vertex max.curvature

1 red

2 blue

3 blue

4 red

5 red

6 blue

7 red

well, ok, 4 counts of red vs 3 counts of blue, but I’m still not convinced.

Cumulative curvature across the *entire* spline, not just at the knots. You can see even just from the one example, that the parameteric spline takes a generally less roundabout path through the knots. As I alluded to in the post, it’s a ‘general’ tendency towards lesser overall curvature as a result of the cubic spline on which the parametric spline is based.

regards,

– jim