# What is the stadium curve, and how differentiable is it?

What is the definition of a stadium curve and does it have a curvature that is defined and continuous at each of its points?

• Based on the responses below, it seems that the curvature is not defined at each point. There are four points where the curvature has a discontinuity, namely the points where the semicircles contact the straight sides. – S. Carnahan May 7 '10 at 5:52

## 2 Answers

I do not know if there exists a general definition of a stadium curve, but a Bunimovich stadium is a well-known example: It is a rectangle capped by semicircles. See http://en.wikipedia.org/wiki/Dynamical_billiards#Bunimovich_stadium

"... the classical stadium [curve] with the boundary that consists of two semicircles and two parallel segments tangent to them ..." from

http://www.springerlink.com/content/m046766h2k175122/

It's continuous, so is its derivative, but its second derivative is (I believe) not continuous. So I'd say it was in differentiability class C^1.