Spline approximation of a circle.
Sometimes ordinary polynomial approximation an unwise choice. Suppose I said,
[math]\;\;\;[/math][i]There is generally no reason to prefer the spline approximation over the ordinary order 3 polynomial.[/i]
[math]\;\;\;\;\;\;[/math] (I did: [url]http://www.geogebratube.org/material/show/id/145572[/url])
Counterexample: [i] the circle.[/i]
Any polynomial approximation can be made arbitrarily bad by including tangents sufficiently close to the x-bounds of the circle. Spline approximation however, can be made very good. For that matter, if we are careful, we make make a good polynomial approximation of a circle. How does one define "careful" for an unknown function?
What conditions can alert us when a specific implementation of polynomial approximation is a poor choice?
