Posted by Dave on July 16, 2013 | Comments Off
If you’ve ever looked closely at a topographical map, you might notice something odd. It’s not actually odd, because there is no other way to make the map, but it does cause some interesting problems. Take a look at this close-up of the USGS map of Half Dome, in Yosemite National Park.
Half Dome’s peak is at an elevation of 8,836 feet, and to the northwest from the peak, there’s a sheer vertical drop of over 2,000 feet. It takes place in just about the width of the “H” in the word “Half Dome” on the map. If you look at the legend, you’ll see that that’s perhaps only 200 feet of horizontal distance representing a 2,000-foot drop!
When slopes get steeper on a topographical map, the contour lines get closer together — but that doesn’t necessarily mean that the physical distance a person covers is actually less. When you climb Half Dome, you’ve climbed 2,000 feet, not 200!
Obviously a runner isn’t going to tackle anything as steep as the face of Half Dome, but even on slopes that are actually runnable, the same problem exists. When you run up a steep hill, you cover more distance than what is represented on a conventional map — even if your entire run is represented as a straight line on the map. So when your GPS trainer tells you how far you ran, does it count the vertical distance you ran, or just the horizontal distance?
The thought occurred to me when I ran a race a couple weeks ago in Squaw Valley, California. The run covered 2.88 miles and climbed over 1,700 vertical feet. But did I actually run farther when you count those vertical feet? My Garmin could conceivably have been off by 1,700 feet — a third of a mile! Did I get credit for the climb?
Unfortunately, it’s not an easy question to answer on most running routes, because roads — especially roads on hills — aren’t usually perfectly straight. Even when they are straight, they don’t often have a consistent incline for their entire length. The rough dirt road I ran up at Squaw Valley was anything but straight!
Ideally, to know whether the GPS gives you credit for those vertical feet, you’d record something like an elevator ride. But I don’t have an elevator handy right now, and I’m not sure I’d get GPS reception in any of the buildings that are nearby. But I do have a GPS record of a ski trip I took a couple years ago. I recorded the whole day of skiing, including the lift rides up. Ski lifts travel in a straight line, and their runs are often at a fairly consistent incline as well. Here’s the map of my day’s skiing:
As you can see, there are several straight lines on the map — they represent my lift rides. The lifts go straight up the mountain with very little change in the grade of their ascent. So we can use our old friend Pythagoras to figure out how far I went on the lift:
The Pythagorean Theorem states that for any right triangle labeled as above, the lengths of the sides a, b, and c are related by the formula a2 + b2 = c2. I picked one lift ride (ending at the marker on the map) and looked at the data for that ride:
So the question is, does the 0.79 miles the Garmin recorded reflect only the horizontal distance covered, or does it take into account the 1,074 feet of elevation gain in the lift ride?
I measured the pixel length of the ride (using the Pythagorean Theorem to calculate the length of the diagonal) and got 314 pixels. The 350-meter scale itself measured 94 pixels (the image has been reduced in size for this post). Using simple proportions, I came up with a map length of 1,170 meters for the lift, or 3,838 feet. Remember, this represents only the horizontal distance covered!
So how does that compare to my Garmin data? Garmin gives the length of the lift ride as 0.79 miles, or 4,171 feet. That’s longer than 3,838, but it’s not 1,074 feet longer. But since the lift ascends at a fairly constant slope, its length should be less than the sum of the horizontal and vertical length.
We can use the Pythagorean Theorem to calculate the shortest possible distance between the base of the lift and the top, assuming the lift went in a perfectly straight path from the bottom to the top. We know from the map that the horizontal distance (a) is 3,838 feet. We know from my Garmin that the elevation gain (b) is 1,074 feet. The length of the lift should be the square root of a2 + b2, which works out to 3,985 feet.
At a minimum, Garmin should have given me credit for traveling 3,985 feet, and it did! It claimed my distance was 4,171 feet. But why the discrepancy? We know that the lift doesn’t travel in a perfectly straight line because the lift cable sags between poles. This sagging might account for the entire discrepancy, or there could be other sources of error.
The takeaway from this is that there’s no way that the Garmin should have measured my ride as shorter than 3,985 feet, even though I only covered 3,838 feet horizontally. It didn’t do that, which suggests it is taking vertical distance into account. If the lift cable sag or a varying rate of climbing were a significant factor, they’d only increase the difference between my Garmin-reported distance and the calculated straight-line distance, and that’s what we see in the numbers. Take a look at this image to convince yourself that in any real-world case, the lift is going to travel farther than the straight-line distance covered:
So it looks to me like the Garmin is indeed giving credit for the actual distance travelled, in three dimensions. To truly verify this, we’d probably need a Garmin plot of an elevator ride or a long rappel, which would be the ideal, straight-line, vertical case.
In the end, though, as you can see, the vertical component doesn’t add much to a typical run. It only added about 300 feet to my very steep 0.79-mile lift ride. Let’s see what kind of an impact it would have on a run.
We know my Squaw Valley race covered 2.88 miles and gained 1,719 feet. 2.88 miles is 15,206 feet. That means I climbed more than a foot for every 10 feet I ran, or better than a 10 percent grade, a very steep run! When we calculate the horizontal distance travelled using c2 – b2 = a2, the result is just 15,108 feet. All that climbing would only result in an extra distance traveled of 98 feet, assuming the pitch of the run was consistent from start to finish. In my case, I’m quite certain it wasn’t — there were even a few downhill stretches — but it’s still clear that the vertical component of the run didn’t add much to the total distance travelled at all.
In a typical road race, 200 feet per mile would be considered a very steep hill. If that hill was perfectly graded, then it would add only 4 feet to the horizontal component of the run. But rest assured that your GPS does appear to take this distance into account (though as we have discussed, there are many other sources of GPS error!).