Calculating map distance

[abdon gonzalez]

When calculating distances using a map, how do you factor in for elevation ups and downs? Is there a formula to adjust for the extra distance that gaining/loosing elevation entails?

[Dylan Skola]

I generally worry more about the ups and downs themselves than the extra distance they introduce.

However . . . if you can obtain a linear elevation profile and divide it into sections of approximately continuous slope, these become right triangles to which you can apply the Pythagorean theorem (length of hypoteneuse is the square root of the sum of the squares of the other two sides, that is, the elevation gain and the map distance (in same units of course)).

For hiking this is probably a waste of time. Consider that even for a strenuous 20 % grade (about 1000 feet gained in a mile), you only travel an additional 94 feet, less than 2 % of the total. Insignificant additonal effort compared to the effort needed to get you up that 1000 vertical feet. In addition, most trails, even relatively “straight up/down” sections have a fair amount of “micro-elevation gain/loss” e.g. up 10 feet, down 10 feet, up 30 feet, down 5 feet. In fact, the total elevation change increases as the granularity of measurement becomes finer. Another way of saying this is that it possess a fractal dimension greater than 1. If you could measure down to the individual grains of sand the total would be astronomical, but of course the finest scale relevant to our concerns is the vertical distance between human strides, which is still much finer than most published figures seem to be based on. I’ve seen altimeter totals (with ~ 2 ft resolution) nearly double published figures for some sections of trail. Granted, some of this discrepancy may be due to altimeter drift being recorded as gain / loss, but a large component seems to be this micro elevation change. So this calls into question the relevance of published figures for this purpose (or map-calculated ones, most 24k topos for mountainous regions have a 20 meter contour distance).

However, since you ARE posting this to the mountaineering / alpinism forum, I’ll assume you are talking about slope angles much greater than a 20 % grade. In that case obviously the elevation gain is a much higher component of your travel across the map. Also the more vertical routes seem to be generally more continuously up/down, without so much “micro.” Here I’ll defer to someone with actual experience in this area, since I have zero. But, IMHO it seems that, if the purpose of this exercise is to approximate energy consumption / strenousness of different routes, then again, the vertical component will be far more significant than the “diagonal” component you want to calculate.

Hope this helps.

[abdon gonzalez]

I’m in Japan where english info on trails is somewhat limited. I’m trying to compile some data for others so I’m trying to be as precise as possible. I don’t care to go overboard with this but the closer the better.

For the most part, this is for hiking volcano trails. A 1,000m+ elevation gain over a mile is not uncommon. As it is you already get some distance gain because of the switchbacks that don’t show up on the map, so adjusting the distance up will get it closer to the truth regardless.

A simple formula to account for total elevation gain+loss over distance would get things a hair closer to what they should be.

[Dylan Skola]

Yikes, 1000+ m / mile? It does sound like you are going straight up. So, if you are reasonably sure that the elevation gain is pretty constant try the following formula:

MD = map distance (in feet, 5280 feet / mile)

VD = vertical distance (loss or gain) (in feet, 3.28 feet / meter)

then

Distance Traveled = √(MD^2 + VD^2)

But, I have a feeling that those unmarked switchbacks add quite a bit of distance. 1000 m / mile gives you about a 32 degree slope, which is practically scrambling terrain. Maybe Japanese trail designers are more sadistic than the ones over here but my guess is that a trail on that slope is going to be heavily switched, adding enough distance to bring the grade to a somewhat more managable figure. Which then increases your total distance much more than the angle would. By the way, plugging those numbers (1000 meters, 1 mile) into the above formula would give you 6216 feet traveled, or 1.17 miles.

If you have segments of constant loss or gain, you can break up the trail and apply the formula to each segment. But those switchbacks are probably going to account for much more distance than the angled slope.

Good luck, it sounds like fun!

[abdon gonzalez]

We have a good number of volcanoes here that are pretty much conical. If the Japanese can get away with it (erosion being the biggest obstacle), they would send you in a straight line for the top, at a 45 degree angle towards the end.

Doing a quick search through my maps, I found a confirmed 750m gain over a mile, no switchbacks.

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