The distance puzzle, part one: The running track

Last minor revision on October 16, 2020.

In a typical distance running competition, I find a mismatch between official course distance and actual distance covered, as measured by wearable sensors. Usually, course distance falls short. Sometimes the difference is small, but occasionally it is quite significant. In a recent half marathon, I had no less than 22.14 km on the watch. 

That's not only an issue for stats nerds. Since speed displayed on the head-unit is based on sensor-measured distance, pacing strategies may fail big time, especially in long races. This can be frustrating.

I was wondering what's behind the mismatch, for a practical reason. If course distance can be assumed correct, then the problem could in principle be resolved by using reliable sensors and proper calibration. There are well-known issues with GPS that render it very unreliable for distance measurement in the vicinity of trees or tall buildings. 

But I was sceptical about the "course distance is correct hypothesis", because I already have been using the most reliable sensor on the market for quite some time. The Stryd pod (v4) uses accelerometers, gyroscopes and magnetometers to measure distance (and other metrics) from foot motion trajectories. It is extremely well calibrated and accurate out of the box, and by far superior to GPS-devices. I tested it. I trust it.

So, what's up with the other side of the mismatch? How is official course distance actually determined? I digged a bit into the World Athletics (formerly IAAF) rules. Long story short: a marathon is rarely a marathon. 

But before going into the messy details of road running courses (I'm doing this in a series of follow-ups to this post), it's useful to start with the running track. Wait, for real, isn't the whole point of a track that it is standardized to a 400 meters lap? Actually, no.

The World Athletics Standard Track

First of all, not all tracks are standardized to the specifications of World Athletics, the international governing body formerly known as the International Association of Athletics Federations (IAAF). Bust most are, especially those built since the 1980s.

Section 2.2 of the IAAF Track and Field Facilities Manual defines the Standard Track as follows.

Shape and dimensions (in meters) of the World Athletics Standard Track.
Shape and dimensions (in meters) of the World Athletics Standard Track.

The Standard Track is an oval track consisting of two semi-circles, each with a radius of 36.50m, which are joined by two straights, each 84.39m in length. By simple geometry, this gives us an inside edge (which must have a white-colored kerb of well-specified dimensions) of 

$$ 36.5 \mathrm{m} \cdot 2 \cdot \pi + 84.39 \mathrm{m} \cdot 2 = 398.116 \mathrm{m} $$

with \( \pi \) denoting Archimedes' constant.

The Standard Track has at least four and at most nine lanes with a width of 1.22 ± 0.01m each, numbered by from the inside to the outside. The "theoretical line of running" on the inside lane (lane 1) is defined at a constant distance of 30cm from the kerb, which therefore has a length of 

$$ 36.8\mathrm{m} \cdot 2 \cdot \pi + 84.39\mathrm{m} \cdot 2 = 400.001\mathrm{m} $$

The theoretical line of running on all other lanes is set at a constant distance of 20cm from the adjacent inside lane. By this definition we can calculate the length of the theoretical running line on each lane, respectively:

Lane Straights Semi-Circles Total
1 168.78m 231.22m 400.00m
2 168.78m 238.88m 407.67m
3 168.78m 246.56m 415.33m
4 168.78m 254.22m 423.00m
5 168.78m 261.88m 430.66m
6 168.78m 269.54m 438.33m
168.78m 277.22m 446.00m
8 168.78m 284.88.m 453.66m
9 168.78m 292.54m 461.33m

Actual distance covered in a one lap run on a given lane can differ, depending on how close the runner follows the theoretical line.

The implications

I was a bit surprised about the magnitude of the differences between lanes. There are at least three takeaways.

First, if in a race you cannot run on the ideal line for some reason, which is common if running in the pack, you can indeed end up running significantly farther than nominal race length. A 10,000m race is nominally defined by 25 laps on the ideal line (the theoretical line on the first lane). If I'm forced to run on lane 2 for, say, 10 laps (and the remainder on the ideal line), then I'm actually running 10,076,7m. That's almost a whole straight longer than nominal. At a pace of 4.7 m/s (about 17 km/h) that's a time cost of 16.3 seconds, which is quite a lot in a 10K race.

Second, in a race, never pass a runner on the outside lane in the bends, use the straights. This will save extra distance.

Finally, the lane differences are also relevant for training on the track. Many running training plans define intervals in terms of distance. I don't recommend that anyway, because the metabolic pathway dynamics depend on time, not on displacement. The above table highlights another source of error. A common speed workout involves multiple reps of 400m or 800m intervals. The intuitive assumption is that one rep is one or two laps, respectively. That's apparently false if you are running on the outer lanes, which is what you should be doing according to track training etiquette (to keep the "racing lanes" in good condition). If you run the reps on lane 7, your intervals will be 46m or 92m long, respectively. A workout with, say, 20 reps will be sizably off plan. 

Theme image derived from a photograph by He Junhui, processed with Graphite Sketchbook.

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