**boston one day wrote:**

I don't believe your explanation is correct. If the ends are not circular, we need more information to calculate lane lengths.

To see why, let's say that the ends of the track are elliptical. I looked up the formula to calculate the circumference for an ellipse, and it is quite complicated. In fact, there may not be an exact "formula."

Specifically, the formula is not linear in r, so when calculating lane lengths, the different radii do not subtract out, as they do when calculating lane lengths under the assumption of semi-circular lane ends.

This is a layup for me.

If the track shape is elliptical it doesn't matter. An ellipse is a specific kind of curve defined by two focal points. The fact that to solve the circumference of an ellipsis requires a more complex integration, rather than a simple 2*Pi*r is irrelevant, because you are physically measuring the circumference of this hypothetical elliptical track. It is 400m.

You are going around that ellipse once. That is 360°. While each point is defined by those two focus points it can also be represented by an imaginary radius, but is doesn't matter what that imaginary radius is, the line in Lane X is 2*Pi*r longer.

Secondly, if you were to add lanes on this ellipical track you wouldn't do so by merely increasing the ratio between those focal points (which means that the lanes would not have a constant width) you would simply measure the lanes and equal distance from the inside lane (line) of the ellipse.

In other words, only lane 1 would be an ellipse, by definition. Lanes 2, 3, 4, ... would not be. You don't need calculus to measure lane 2, 3, 4 etc. you need 2*pi*r. You don't need calculus to measure lane 1 because you are going to physically measure it. (see note at the bottom of the page) Lane 2 would be 2*pi*r more than the distance on the inside lane.

Another way of looking at an ellipse is that it can not only be described as a ratio between to focus points it can also be described as a constantly changing radius. Each infinitesimally small point along that path has a radius. As long as you make one revolution, no matter what your path, the distance traveled L meters from that inside path is always 2*pi*L more.

If you walked around the world your head would travel exactly 2*pi*r more than your feet.

NOTE. Now if you were going to measure out lanes 2, 3, 4...etc as an ellipse then that is another story. But what sense does it make to have 1.5m wide lanes at two points on the track and .35m wide lanes at the tightest curves?

RWIW, this question has been answered before by two physicists.