Help with distance running formula

[Glenstr]

In the winters, I run on an indoor track, and use a footpod to calculate my distances with my GPS watch.

The facility has three lanes, inside #1 is 181.6 meters, middle lane #2 is 187.2 and outside #3 lane is 194.3 meters long. The track is more less rectangular shaped as opposed to a circle, but with circular corners.

I try to calibrate my foot pod with roughly 5 laps to 1 KM, so I run where a 4th lane would be, if it existed. My simple calculations have the imaginary 4th lane at about 203 meters. I arrived at this figure by taking the percentage increase of each lane - lane #2 is 103.0837% of #1 and #3 is 103.7927% of #2. The difference in % increase between #2 and #3 is .7090%.

So I arrived at 203.0469 meters for #4 by multiplying #3 distance of 194.3 * (103.7927% + 0.7090%) - in effect adding the percentage difference from #2 to #3 to #3's percentage increase compared to #2. To clarify more I've attached a screengrab of my calc's in excel. (Column C is difference in meters from #1 to #2 and #2 to #3, and was not used in calc's for #4, column D is each lane divided by the lane inside it, formatted to percentage)

Did I use to correct formula to arrive at the distance of a 4th lane?

 

[mfb]

There is no way to tell without a sketch of the lanes. In particular, lane 4 could be everywhere.

 

[Glenstr]

There is no way to tell without a sketch of the lanes. In particular, lane 4 could be everywhere.

I don't have a sketch on hand, but basically picture a square track with rounded corners, all lanes follow the same path, including the imaginary lane 4

Here is an image of part of it, just picture 3 more corners like this, and another blue lane on the outside that would be lane 4

 

[Mentallic]

If we make the assumption that your measurements for lanes 1,2,3 are correct (within 0.1m error) and that the circular part of the track is completely circular, then we run into the problem that the difference between the distances in lanes 1 and 2 should be equal to lanes 2 and 3. Since they are not, then either or both of these assumptions are incorrect.

You'll need to give us more detail about the circular part of this track, such as an accurate, to scale drawing. Or, you could measure out the distance across the track between each lane 1 straight and the distance from the imaginary centre of the "circle" that creates the semi-circular part of the track to lane 1 at the middle of the circular part between the straights. If it were truly a semi-circle, then this length would be half of the distance between each straight.

edit: I just caught the part about the track being square with rounded corners. Any more finer details on this? If you figure out how long the curve of any corner of the track is, then we can figure out the entire track's dimensions - assuming the corners are circular.

 

[Glenstr]

I'll see what I can find in the way of drawings, there is a to scale (I think) drawing on the wall to show fire escapes etc. which shows the general shape, but as far as can recall there's no dimensions on it. However I'm running there tonight and will have a look.

I'm also not sure how accurate their existing measurements are, I assumed that the percentage increase should be identical as you said. My guess is they wavered a bit while walking around on one of the lanes with the wheel measure.

I'm not sure how I'd measure the length of the curve, but I think if you just draw a square on a piece of paper, then use a compass from the center to round the corners inside you'd have a pretty accurate small scale rendition of it. In any case I'll see if I can snap a pic of the drawing tonight and post it later.

I may even just see if I can borrow a wheel measure and check their accuracy as well as measure the imaginary 4th lane - but I thought if even only the inside lane was measured one should be able to calculate accurately the rest of them, given that they're the same width - whether it's a circular, ellipse or rounded square/rectangle shaped.

 

[Mentallic]

I'm also not sure how accurate their existing measurements are, I assumed that the percentage increase should be identical as you said.

Well I never said that. In fact, an identical percentage increase isn't possible because this problem isn't geometric (each lane can't increase at an equal rate).

I'm not sure how I'd measure the length of the curve, but I think if you just draw a square on a piece of paper, then use a compass from the center to round the corners inside you'd have a pretty accurate small scale rendition of it. In any case I'll see if I can snap a pic of the drawing tonight and post it later.

If you have a square with rounded corners, and the way this was constructed was by using a compass from the centre of that square and cutting the corners with it, then the corners would be cut pretty jaggedly. There wouldn't be a smooth transition from straight to curve while running. Also, this doesn't help us solve the problem either because we don't know how much of the square is cut.

I may even just see if I can borrow a wheel measure and check their accuracy as well as measure the imaginary 4th lane

That would probably be a good idea ;)

- but I thought if even only the inside lane was measured one should be able to calculate accurately the rest of them, given that they're the same width - whether it's a circular, ellipse or rounded square/rectangle shaped.

Sadly, no. Your approach would give a good estimate, but it would really start to break down if you wanted to calculate the -10th lane (a very small lane), or the 100th lane for example.

 

[Glenstr]

I must have misunderstood what you meant when you said the difference between the distances in lanes 1 and 2 should be equal to lanes 2 and 3

 

[mfb]

I must have misunderstood what you meant when you said the difference between the distances in lanes 1 and 2 should be equal to lanes 2 and 3

Like this:
lane 1: 181 meters
lane 2: 187 meters (6 meters longer)
lane 3: 193 meters (6 meters longer)

If the lanes have the same width, their absolute length difference should be the same (like 6m in this example). To be more precise, the length difference should be 2 pi times the width of the lanes. I would expect large uncertainties in the measured values given in post 1.

 

[Glenstr]

Ok, here is a picture of the sketch of the track - there is no measurements but it gives you the shape of it.

 

[Mentallic]

Ok well with that shape, each lane should still increase at a linear rate. You're going to need to accurately measure at least 1 lane, and the width between each lane. With this info and that drawing, I could give you the details about the rest.