The extra distance run per lap for running in a lane other than lane one is given by 2π∆r, where ∆r is the increase in the measure line radius of the lane you are running in over the measure line radius of lane one. For tracks with raised curbs to calculate this one has to be aware that the measure line in lane one is 30cm out from the curb, and for all other lanes it is 20 cm out from the inside lane line of the lane in question. Thus, ∆r is given by ∆r=2π((n-1)w-.1), where n is the lane number you are running in and w is the width of the lane in meters. For a track with no raised curb ∆r=2π(n-1)w. This formula is true for any track, including so-called broken back tracks, as long as the track is composed of segments which are straight lines and circular arcs joining smoothly, i.e., no discontinuities in the first derivatives at the junction of segments, so that the sum of the circular arcs add to 2π.
This formula would not work for for an elliptically shaped track (if there were such a thing) since the circumference of an ellipse C is given by C=4aE(e), where E(e) is a complete elliptical integral of the second kind and e is the eccentricity of the ellipse.
Sorry, but this ^ is incorrect. What malmo explained is correct.
Whether the track is completely circular, a square with curved corners, an ellipse or any other random closed, non-overlapping, curve, the difference in the length of adjacent lanes is ALWAYS 2 𝝅 Δr (where Δr is the width of the lane).
The extra distance run per lap for running in a lane other than lane one is given by 2π∆r, where ∆r is the increase in the measure line radius of the lane you are running in over the measure line radius of lane one. For tracks with raised curbs to calculate this one has to be aware that the measure line in lane one is 30cm out from the curb, and for all other lanes it is 20 cm out from the inside lane line of the lane in question. Thus, ∆r is given by ∆r=2π((n-1)w-.1), where n is the lane number you are running in and w is the width of the lane in meters. For a track with no raised curb ∆r=2π(n-1)w. This formula is true for any track, including so-called broken back tracks, as long as the track is composed of segments which are straight lines and circular arcs joining smoothly, i.e., no discontinuities in the first derivatives at the junction of segments, so that the sum of the circular arcs add to 2π.
This formula would not work for for an elliptically shaped track (if there were such a thing) since the circumference of an ellipse C is given by C=4aE(e), where E(e) is a complete elliptical integral of the second kind and e is the eccentricity of the ellipse.
Sorry, but this ^ is incorrect. What malmo explained is correct.
Whether the track is completely circular, a square with curved corners, an ellipse or any other random closed, non-overlapping, curve, the difference in the length of adjacent lanes is ALWAYS 2 𝝅 Δr (where Δr is the width of the lane).
CORRECT!
I don't know why this is so hard for people to get. On this thread, perhaps they are forgetting that when you add a fixed lane width to an ellipse you no longer have an ellipse.