Some features of a function may make it unsuitable for ordinary polynomial approximation. Corners, closed curves, data errors.
Consider the following
Assertion:The order 3 spline offers no advantage over the order 3 polynomial. (-me)
And a simple
Counterexample: Circular arc:
Here is a spline:

1. The ordinary 3rd degree polynomial through A, D, and sharing the tangents, is omitted. (why?)
2. The spline approximation can be made quite good. What is a practical maximum value of θ? For example, try the midpoint condition, and drag D to different positions. What is the maximum absolute error along the curve? Relative?
3. Why is the midpoint approximation better than the curvature condition?