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Gmap and Hills (Read 916 times)

Here's a geeky question. Does Gmap (or a Garmin, for that matter) take three dimensions into account when it measures distance? If a rolling course is measured as if it took place on a two-dimensional plane (like a computer screen or the surface of a globe), then the measurement will come up short because it won't take into account the distance traveled in the third dimension (up and down). I know that it can chart this third dimension with the elevation toggle, but it does not do this automatically. I can imagine several possible responses to this question: 1) Gmap doesn't take the third dimension into account when it measures distances, but it's not a big deal because the distance traveled in the third dimension is so minuscule that it's not worth worrying about. or 2) Gmap does take it into account. or 3) Oh my god, you're right! We've all been running farther and faster than we thought!!! Hold the presses!!! Adjust the pace bunnies!!! or 4) Aren't you supposed to be a running purist? Why the heck are you worried about the fractions of a mile you might or might not be running? Just get out there and RUN! Any other responses?

Or... 5) What if the third dimension really only exists in your head 6) What if the third dimension is really the 5th dimension and there are two whole other dimensions that we haven't discovered yet? 7) What if Garmin knows about all these other dimensions, yet they are holding back on the press release so they can push as many 2/3 dimensional products as they can before releasing a new product that measures the next, then the next...and they are really the same company that makes/markets the I-Pod? I'm going to take a nap now, Lynn B

""...the truth that someday, you will go for your last run. But not today—today you got to run." - Matt Crownover (after Western States)

The Year of the Monkey

#4. And No, gmap does not take this into account when calculating distance. Neither does RA. And No, it does not really make much difference to include this in the distance calculations. --------------------------------------------------------------------------------------------------------------------------- Doing the math: Grade is defined as rise over run, multiplied by 100. So a course that rises 100 feet over a mile is (100 / 5280) * 100 = 1.8% To determine the distance run on the hypotenuse of this route, you can apply the Pythagorean Theorum, which states that A^2 + B^2 = C^2 (where the notation ^2 means squared). So in a triangle formed on one side by the distance 5280 feet and on another side by the distance 100 feet, your hypotenuese = sqrt(5280^2 + 100^2) = 5 280.9. Catch that? A 2% grade, which is just over a 100 foot climb over a mile adds one foot to your distance. That is less than 0.1% error per mile, which is insignificant. That is less than a foot of error per mile. This may matter more on a nasty run, such as the Pikes Peak Marathon. The Pikes Peak Marathon includes a 13 mile climb up more than 7500 feet, with an average 11% grade. Assuming you actually do climb 13 miles in 7500 feet, let's determine the error. 13 * 5280 = 68 640 feet. Applying the Pythagorean Theorum, we get sqrt(68 640^2 + 7500^2) = 69 048.5 feet. That is a difference of 408.5 feet over 13 miles (31 feet per mile), or a 0.6% error. I think that with these mapping tools, you are more likely to introduce more feet to your expected distance run by your mapping (im)precision rather than by the effects of elevation change.
Any other responses?
No. 1 and 4 were my first instincts.

Runners run.

Jeff, I've wondered about this as well. The answer is no, GMAPS does not take elevation change into account when calculating distances. The is from the FAQ section of GMAPS: "The google maps API does not expose data about elevation, which means that elevation changes are not taken into account in calculating the distances. Some users have pointed out that there are services available on the web which can return elevation given a certain latitude and longitude point. I experimented a bit with adding functionality to the site to show that, but wasn't thrilled with the fact that the data is only available for the US, and seems to be intermittent in its reliability. Maybe some day it'll be added, if enough folks indicate that want it, and they're comfortable with the fact that it may not be all that reliable." Hope this helps
#4. And No, gmap does not take this into account when calculating distance. Neither does RA. And No, it does not really make much difference to include this in the distance calculations. --------------------------------------------------------------------------------------------------------------------------- Doing the math:
Well, if you travel a horizontal distance x and a vertical distance y, then the total distance (meaning the length of the curve) s is only _lower bounded_ by \sqrt{x^2+y^2}. You'd need a complete elevation profile to measure the actual curve length, but it'd be somewhere between \srqt(x^2+y^2} and x+y. You could always bike the route and measure it with your odometer. (Which I think is the standard way of measuring courses... or not?)
Jeff, "but wasn't thrilled with the fact that the data is only available for the US, and seems to be intermittent in its reliability."
RA seems to be quite precise here in Austria, as opposed to some other sites that I tried.
I like olethros' response--you could either write a function for the curve and do a line integral over the function or you could ride the course with a bike and an odometer. It's funny to think about different modes of inquiry and their relative advantages. Especially if you're a philosopher. I was actually thinking about lynn's #6 on my run (I know). What if the difference between slower and faster runners is that the slower runners are just spending more energy in those other dimensions? Can you tell that I'm procrastinating. Damn, I gotta get to work--anybody got any good ideas for a dissertation prospectus?

I've got a fever...

A slightly different question is (and I haven't found this out yet), is whether Google Earth takes 3-D into account with distance traveled, since it is a 3-dimensional rendering of the earth. Anyone know? I'm guessing no since vertical is negligible in almost all cases. The following link didn't answer my question explicitly (unless I missed it), but there's a very interesting and geektastic overview of the challenges of distance measurement over a large scale spherical surface. Google Earth and Geographic Distances

On your deathbed, you won't wish that you'd spent more time at the office.  But you will wish that you'd spent more time running.  Because if you had, you wouldn't be on your deathbed.

The Year of the Monkey

a line integral over the function
Right. But it would still likely make an insignificant difference, far less of a difference than the imprecision of plotting a map or the wavering of a GPS signal over a run, both of which can cause errors of 1-2%, or even more. Also, while gmap API does not account for elevation, RA does. Eric measures elevation from USGS every 1/10 mile, which means that any small undulations that occur in under that distance go unmeasured. In Nashville's Percy Warner Park, with all the abrupt ups and downs, I believe that this accounts for about a 30% negative error in the estimated elevation change over 11.2 miles, especially when compared to GPS data.

--anybody got any good ideas for a dissertation prospectus?
How about... Correlations of "Up, Up and Away", Fartlek and IPod Marketing, or An Inquisition Into the Matter of "^" ? Lynn B

""...the truth that someday, you will go for your last run. But not today—today you got to run." - Matt Crownover (after Western States)

Bottom line: it's always farther than you measure. How *much* farther is open to debate but at typical running distances, you can ignore just about everything but straight lines on a plane. Curvature of the earth, altitude changes, microscopic fractal terrain, and the earth's mass and velocity effect on the space-time continuum are small enough for most folks to ignore in their daily run. So be happy. Just say. "I really ran farther than the measured distance, so I'm faster than it looks" Thesis: "The Dynamic Ontological Environment of Existential Running" or "I am, so I run, but why does this fricking hill feel so long today?"
The distance tracked on my Forerunner is usually shorter than what MB tells me once I've uploaded my workout depending on how hilly it is. A GPS the size of mine only calculates distance based on a flat plane. The difference is usually less than 1/4 mile unless I'm doing a lot of peaks and valleys in the mountains where I've seen it as much as 3/4 of a mile. It's fairly accurate once you factor in actual distance traveled. I know you can use the rise over run formula, but that only works on a consistent grade. There are some trails I run where you start out on top of a 150 foot hill then you go down, then back up 60 feet, then back down, then back up to 200 feet. Straight line might be 1 mile, but actual distance traveled comes out 1.5 miles.

The Year of the Monkey

Never use MB or your GPS as a gold standard distance. Ever. Unless maybe you have looked at the satellite image from MB up close and you are certain that EVERY step is exactly on the course you ran. Both the device and the website can misinterpret your location and add or subract distance incorrectly. The rise over run works for all routes. I provided a simplistic example in which you run on a flat plane. But even the most rolling ups and downs are really just a series of rises and runs, one after the other. You could figure it out if you had the time and energy. I provided the extreme example of Pikes Peak, which still only was off by a margin of 0.6 % (30 feet in a mile). For a one mile horizontal course to be 1.5 mile due to climbs and drops, you'd have to spend a lot of time climbing on your hands and knees or rolling down. More likely, I suspect that the GPS is incorrect.

The Year of the Monkey

Using the example, as you suggest, of a one mile horizontal route that drops 150 feet, then climbs 60 feet, then drops 60 feet, then climbs 200 feet, and assuming that each of these changes is a continuous plane occurring over 1/4 mile (they are not likely doing this, and this would MINIMIZE the effect of the change), the total distance run is 1 mile and 26 feet. So you are off by just 26 feet, a 0.5% error. Taking the same course but assuming that each change in elevation is a vertical drop or climb (which would MAXIMIZE the effects of the change), the total distance is 1 mile and 470 feet. That is an 8.9% error, still far less than the 0.5 miles (i.e., a 50% error). My guess is that the truth is somewhere between the 0.5 and the 8.9% estimates. (thanks olethros!)
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