There's been a lot of partial truth on this thread.
Here's how I see it.
The formula "new lane=old lane+2pi x lane width"
is true if the track is convex (not kidney shaped).
Here is a proof that does not require calculus (though it uses the ideas of calculus).
Imagine the inside curb of the track is made of short sticks laid end to end.
We place one end of a measuring rod (of length d) on one of the sticks, perpendicular to that stick to the right. We'll find our new lane line by moving this rod around the track, keeping it perpendicular to the sticks. When we get to the corner between two sticks, the left end of the rod stays in the corner and the right end pivots until the rod is perpendicular to the next stick, and we continue parallel to the next stick. The new lane line consists of segments parallel to the sticks of the same length, plus circular arcs at each corner, where we pivoted the rod. As long as you always pivot counterclockwise (which you do if the track is convex), the total of the arcs will add up to a circle when you go all the way around.
So the new length is the old length plus the length of this circle. The length of the original sticks was not specified. They could have been one atom long, so the stick track would not be measurably distinguishable from the curved track. hence the formula holds for any convex track.
However, a weird thing happens if the track does not always curve to the left:
You have to run backwards to stay in your lane!
For example, imagine the track looks like a thick L. If you're going along the top side of the foot, you run into the wall when your rod hits the corner. Then your rod pivots clockwise 90 degrees, so you run backwards, until you are facing parallel to the wall, and can go forward again.
Other comments:
1. Malmo's "harvard physicist" made no sense to me.
For one thing, R2 is stationary from R1's reference point, not going around in a circle.
Also, he (as did dukerdog) mixed up "velocity" with "speed" in the integral calculation.
The arclength is the integral of speed, which is the length of the velocity vector.
Adding two vectors does not result in adding the speeds. However, the integral can be done, as follows.
Let's travel around the curb at a constant speed of 1 mph.
Let p(t)=(x(t),y(t)) be our position at time t.
The velocity vector is p'(t)=(x'(t),y'(t)),
and |p'|^2=(x')^2+(y'^2)=1 since the speed is 1.
We go once around the track when as t goes from 0 to L (L=length of inside curb).
Now travel d units perpendicularly away from p(t), and move parallel to the curb.
This position is q(t)=p(t)+d n(t), where n(t) is a unit vector perpendicular to p(t).
Also q'(t) is parallel to p'(t). The length of the new lane is the integral from 0 to L of |q'(t)|.
Unlike |p'|, this new speed is not constant, so we have some work to do.
Since p' and q' are parallel, and |p'|=1, it follows that |q'| equals the dot product p'.q'
of p' and q'. Since q=p+dn, this means
|q'|= 1+d(p'.n') . (*)
Now p'.n=0, so if we differentiate, using the product rule for dot products,
we get
p''.n+p'.n'=0, so p'.n'=-p''.n . (**)
Doing the same thing starting with p'.p'=1 gives p'.p''=0,
so p' and p'' are perpendicular. Therefore
n=-p''/|p''| (***)
(the minus sign is because n point outwards).
Putting (*), (**),(***) together says that
|q'|=1+d |p''|,
so the length of the new lane is L+d integral_0^L |p''|.
Now comes the clever part. Since |p'|=1, and the curve always turns left, the path p' parametrizes the unit circle. The velocity of the path p' is p''.
Therefore the integral of |p''| gives the arclength of the circle, which is 2pi.
2. Ellipses are tricky. There is no simple formula for its arclength, and no easy way to evaluate the integral. The online "calculator" posted by dukerdog is a very rough approximation that gets worse the flatter the ellipse.
Anyway, it is not possible to build an elliptical track! The reason is simple. If you take an ellipse, and draw a lane around it with constant width, the new curve is no longer an ellipse. It looks oval, like an ellipse, but it is not a true ellipse (defined as the collection of points whose distances to two fixed points have constant sum). You could make elliptical lanes by scaling the original ellipse, but the lanes would have varying width.