Curvature and the order of polynomial

I was wondering what the relation of curvature and the order of polynomial you select for regression. If the curvature is 0, it is better to use a straight line. If it is high, do we need to use higher order polynomial? I find the later statement does not hold. Very simple, the curvature depends on the point. Not every point on a higher order polynomial has higher curvature. Actually it is hard to tell where the curvature is high, it depends on the shape of polynomial. So there is no relation.

The regression method considers solely the error. Even if you do not have the best fitting curve, you still can get good fitting quality (if you are allowed to partition the domain).

What kind function to choose for approximating indeed depends on the data. If we can visualize the data, we can see that if it is a curve so that we can use polynomial or otherwise straight line. But if we can not, the only way is to try it out. I found a paper to decide the polynomial and the partition.