What is the proper geometrical name for a a rectangle with a semi-circle at each end?

Solution 1:

Obround, apparently. I don't know Wiktionary's source. This definition of obround does not appear in OED, for example. Googling indicates that this definition is commonly used for machine parts having this shape.

Solution 2:

I'm pretty sure they actually use four clothoid arcs joined together in practice, e.g. this. This has a lot to do with the fact that the clothoid is the curve whose curvature is directly proportional to its arclength; an abrupt variation in curvature would equate to an abrupt variation in centripetal force, which can be bad for the racehorses (or even racecars, for that matter).

Here's a simulated clothoid track drawn in Mathematica:

Just to show that the bends are honest-to-goodness clothoids, I drew the clothoid corresponding to the lower right portion of the track in full (the dashed gray one).

The parametrization used is

$$(x\qquad y)=\left(\sqrt{\frac{\pi}{2}}C\left(\sqrt{\frac{2}{\pi}}s\right)\qquad \sqrt{\frac{\pi}{2}}S\left(\sqrt{\frac{2}{\pi}}s\right)\right)$$

where $C(x)$ and $S(x)$ are the Fresnel integrals; I leave you to verify using those expressions that the curvature of the clothoid is indeed directly proportional to the arclength.

Solution 3:

It's called a stadium. See http://en.wikipedia.org/wiki/Glossary_of_shapes_with_metaphorical_names or http://mathworld.wolfram.com/Stadium.html