Measure of curve smoothness

Question

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness?

I was thinking that since curvature measures how fast a curve changes, should we not be integrating the square of curvature? Basically why are we ignoring the denominator from the definition of curvature before even checking if first derivative is small enough.

This is also used in Smoothing Splines so I guess there is something to it then just being a mere approximation.

Answer 1

Using integral of squared second derivatives allows the functional to be minimized to become a quadratic form of the unknowns (i.e., control points of the spline), which eventually will result in a linear equation set, which is easy to solve. This is actually similar to the famous "least square" method which minimizes the sum of the squared errors, instead of the sum of errors, so as to result in a linear equation set.

Answer 2

Indeed, it is analogous to root mean square computations. When profile changes, curvature and smoothness change and the former is a good differential computable index of the latter parameter.

If such a powerful indicator as curvature is anyway taken as a feel factor, the denominator with its first degree is not anymore considered as a better physical indicator of smoothness retaining its inclusion, as any how next derivative change is reported into the numerator. Result is practically same for roughness levels normally encountered.

From Mechanics of Materials Beam theory an example can be given. For low slopes of beam center line when $ \frac{dy}{dx}<<1$ no practical extra advantage is seen when curvature is defined as $ \frac{d^2y}{dx^2}$ or $ \frac{d^2y/dx^2}{(1+ (dy/dx)^2)^{1.5}}.$

Answer 3

You are correct. Second derivatives are not really the right thing to be considering. The main issue is that second (and other) derivatives are dependent on the parameterization of the curve, whereas "smoothness" is a geometric property that is independent of parameterization. We should really be considering parameterization-independent quantities like curvature and derivative of curvature with respect to arclength. But curvature is a nasty non-linear function, and using derivatives is much easier.

In fact, you might even say that the whole idea of cubic splines (enforcing continuity of second derivatives) is wrong. We ought to be enforcing continuity of curvature, instead. This has been done -- there are splines that have so-called "geometric continuity". This is an old idea, going back to Even Mehlum's KURGLA system in the 1970s. Again, it's more correct, but the computations are much more difficult, and the results often don't justify the effort.

The answers to this question have some further discussion.