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

$\begingroup$ 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. $\endgroup$– S. Carnahan ♦May 7, 2010 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 wellknown example: It is a rectangle capped by semicircles. See https://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 Loskutov, Alexander; Ryabov, Alexei, Particle dynamics in timedependent stadiumlike billiards, J. Stat. Phys. 108, No. 5–6, 995–1014 (2002). ZBL1124.82310.
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.