# Cubic splines better than quadratic splines?

I have read in a number of places that cubic splines are of more practical use than quadratic splines in general (there are exceptions of course). Anyone know specifically why they are more applicable/better?

• Cubic spline has continuous second derivative, while quadratic spline only has continuous first derivative. So cubic spline is smoother. – KittyL Apr 5 '15 at 17:33
• Yeah, but would a fourth order spline be even smoother or does it get worse if you go even higher order? If not, and it also gets better, i.e. it is always just a tradeoff between complexity and accuracy, why does cubic seem to be enough for most applications? – connorwstein Apr 5 '15 at 18:01
• I believe you can get higher smoothness but as you said there has to be a point to stop. People stop at cubic spline because it has smoothness at second derivative, which means it minimizes the curvature of the function. And that is enough for most of the applications. – KittyL Apr 5 '15 at 18:40
• Also, in the regression spline/least squares sense, it has some (statistical) optimality properties, probably connected to the fact of minimizing curvature. See the book Hastie, Tibshirani, et.al "Elements of Statistical Learning" for this viewpoint. – kjetil b halvorsen Dec 7 '17 at 9:59

## 1 Answer

A cubic curve can twist in space (i.e. it can be non-planar). A quadratic curve is just a parabola, so it's always planar.

In real applications like graphic arts, engineering, manufacturing, nobody cares much about derivatives, they only care about curvature. And it's possible to get continuity of curvature without continuity of second derivatives (so-called G2 splines, versus C2 ones). So the C2 argument for cubics is a bit fragile.

For some applications, like design of car bodies or cams, cubic splines are not good enough, because you need continuity of the derivative of curvature (G3 continuity). So, in these applications, higher degree curves are often used.

In many situations, splines are used as approximations of more complex functions. If you use cubics, then you'll need fewer polynomial segments for a given approximation tolerance. But each segment will be more complex, so more difficult to handle in subsequent calculations. That's basically the trade-off -- number of segments versus complexity of segments.