Is there a cubic spline interpolation with minimal curvature?

Question

I came across the term "cubic spline with minimal curvature". However, I am not able to find any documentations/explaination on its computation method.

Can anyone help me by advising how I can go about finding more information (maybe there is another more common name)?

Thanks a lot!

Ryou

Answer 1

In general, the cubic spines are piecewise cubic functions passing through the given points [edit: with continuous first and second derivative] with minimal curvature as measured by the second derivative.

So either your question is about functions where the "true" geometric curvature is minimized.

Or it is in contrast to spline interpolations where one assigns a slope to the end points of the sample interval. Then the minimal curvature measured as the integral over the square of the second derivative is a function of these slope values, and has itself its minimum where the second derivative at the end points is zero.

Answer 2

Minimizing true curvature (as opposed to the second derivative) is equivalent to finding steady state solutions to the problem of "elastica." Even between two points this is a hard problem involving elliptic integrals. And the answer isn't necessarily unique.

I have a paper which contains many references: http://faculty.missouri.edu/~stephen/preprints/springs-straight.html. And probably one could solve the equivalent dynamic problem with damping, and thereby find a stable solution.