Some features of a function may make it unsuitable for ordinary polynomial approximation. Corners, closed curves, data errors.[br]Consider the following [br][b]Assertion:[/b] [i]The order 3 spline offers no advantage over the order 3 polynomial.[/i] (-[i]me[/i])[br][br]And a simple[br][b]Counterexample: [/b] Circular arc: [math]{\small f(x) = \sqrt{1-x^2}} [/math][br][br]Here is a spline:

1. The ordinary 3rd degree polynomial through A, D, and sharing the tangents, is omitted. (why?)[br][br]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?[br][br]3. Why is the midpoint approximation better than the curvature condition?