# Bézier curve limits

Can be any curve of any shape (without sharp edges) described by Bézier curve with unlimited (but finite) number of control points?

The answer to the question above would probably be no, because I have already found out that circle cannot be described exactly by a Bézier curve. (http://en.wikipedia.org/wiki/B%C3%A9zier_curve)

But, are there any other limits of Bézier curve?

• Unlimited but finite number of control points, right? I'd say that there are many curves which cannot be exactly represented by (cubic) Bézier curves. Most cubic rational curves would fall in that class, and most curves of higher degree as well. You can approximate them, and given an arbitrary number of points you can probably approximate them arbitrary well, but it's still not exact. – MvG Apr 16 '14 at 7:28

The Weierstrass theorem tells us that any continuous function can be approximated arbitrarily closely by polynomials (which is what Bézier curves are). And, in fact, Bernstein's proof of this theorem uses polynomials expressed using the Bernstein basis (i.e. Bézier curves). So, by using a Bézier curve of sufficiently high degree, you can approximate any curve as closely as you like, even without the "no sharp corners" restriction that you mentioned.

This approximation technique (in the form used by Bernstein) isn't really practical because huge degrees are required to get decent accuracy.

You can approximate any curve by a (cubic) spline, too, and a cubic spline is just a sequence of cubic Bézier curves joined end to end. This approximation technique certainly is practical, and is very widely used.

If you want exact representations, rather than approximations, there are many curves that can not be handled. As you mentioned, circles are the simplest ones.