Approximation of total curvature

[fishingspree2]

Hello, I am trying to find an interpolating curve between a few points that has minimal curvature. That means, as close to a straight line as possible.

Reading a document about cubic splines, they say that

$$\kappa \left ( x \right )=\frac{|f''\left ( x \right )|}{\left ( 1+\left [ f'\left ( x \right )^{2} \right ] \right )^{\frac{3}{2}}}\approx |f''\left ( x \right )|$$

Why are they able to say that? Is there any proof or explanation? Thank you very much

 

[Eynstone]

That's just the formula for calculating curvarture.
I'd rather fit a curve with as few 'humps' as possible (a low total curvature doesn't mean that a curve is nearly straight). Lower the degree, fewer are the roots of its derivative in the range & thus, humps.