JimFiore, We've always been talking about an ellipse in the sense of your second definition. I thought it was a given to everyone that any configuration would be consistant with equal width lanes? I believe I'm speaking for dukerdog correctly when I say that his example of a ellipse was used to demonstrate that the track needn't be constructed with smooth, circular curves.
BTW, this tread could hardly be described as being pedantic.
What follows is a mathematical proof from a Harvard physicist:
"The short answer is this: if D is the distance between the inside edge of lane 1 and the inside edge of the lane you are interested in, then that lane is as long as lane 1 plus this amount: 2*pi*D (2 times pi times D)."
"This answer is correct pretty much whatever the shape of the track. The curves don't have to be semicircles, for instance, and indeed the track can be as wiggly as you please, like a wandering path around a lake. The track may be circular, or oval, or wiggly. Call the further inside of the two lanes L1 and the other one L2. Call the (constant) distance between them D. All we assume is that it would be possible for two point-sized runners connected by a stick of length D to proceed (counterclockwise) around the track always keeping the stick perpidicular to their lanes (hence travelling the same direction at every moment).
Call our two point-sized runners R1 and R2. Viewing the world from the perspective of runner R1 (in the "frame of reference" that counts R1's location as the center of the universe), it appears that runner R2, on the other end of the stick, over the course of the lap completes a single counterclockwise orbit of radius D, with circumference 2*pi*D. Now, the difference in the lengths of L2 and L1 is the same as the distance travelled by R2 from the perspective of R1, around that circle counterclockwise, as we can show with some integral calculus. The key facts we need to know are:
- Distance travelled is always the integral over time of velocity.
- Integration factors out in subtraction.
- Since R1 and R2 are always travelling in the same direction, the velocity (v) of R2 from R1's perspective (in the counterclockwise direction around a circle of radius D) is always the difference (v2-v1) of the velocities of R2 and R1.
Then, we know that the difference in the lane lengths is the difference in the distance travelled by R1 and R2, which is (using "S" for the integral sign, signifying integration over the time it takes the runners to complete a lap):
S(v2*dt) - S(v1*dt) =
S(v2-v1)*dt =
S(v)*dt
...which is the total distance that R2 travels from R1's perspective counterclockwise arount the circle, namely, the circumerence of a circle of radius D, or 2*pi*D."