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)]2d(x) ≤ ∫[f”(x)]2d(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.