How do maps work?

January 10, 2010 7:58 AM

How do maps work?

Say I start at point A and drive up a mountain and down the other side to point B. The number of times the wheels have turned is greater than if the mountain wasn't there and I drove from A to B on flat ground.

On my map, it looks like there's 5 miles between A and B, which is fine because it's two-dimensional. On my odometer, it indicates that I've traveled 5 miles up and over the mountain. Had I made the drive on flat ground, my odometer would also have read 5 miles. Please explain this to me.

(Anonymous because of my job and colleagues on mefi)

posted by anonymous to travel & transportation (16 answers total)

Point A and B on a map are connected by a straight line that is shorter than the route you actually must travel: highways curve, zig, and zag around natural features. What you describe is the same phenomenon in the third dimension: altitude. Most maps feature elevation contours to help you estimate the extra distance.
posted by fatllama at 8:09 AM on January 10, 2010

It sounds like your question is more like "how do odometers work" than it is "how do maps work". I think distance in maps is measured more in an "as the crow flies" sense rather than measuring the physical length of the actual ground traversed -- which, yes, would be different if you were going up and over a mountain. But distance on maps isn't done by measuring the ground, it's done in an "as the crow flies" sense.

But your point about the odometer is a good one, and it sounds more like your question is about why the odometer doesn't seem to be tied into the revolution of the wheels. That I'm not sure about either, now that I think of it; but that's a separate question from maps themselves and how distance is calculated thereon.
posted by EmpressCallipygos at 8:11 AM on January 10, 2010

Your odometer would have probably read slightly less than 5 miles had you made the drive on flat ground. How much less? It depends on how tall your mountain is. Let's say you are driving straight through the mountain, and we approximate your mountain as an isosceles triangle (two of its sides are equal in length) with a height of 1 mile. Let's assume your odometer is the correct measurement here, so the two equal sides of the triangle add up to 5 miles. The length of the base of the triangle would then be sqrt((5/2)^2 - 1^2)*2 = 4.58 miles, or about 8.5% less than 5 miles. It would be easy for you to misjudge that distance on a map unless you were measuring very precisely, and the map itself could be a little off. If your mountain was less than a mile tall, the difference would be even smaller (for example, if your mountain was only half a mile tall, the difference would be about 2%, very easy to misjudge).
posted by Behemoth at 8:13 AM on January 10, 2010 [1 favorite]

The difference between the long side and the hypotenuse of a very long triangle is negligible.

Most maps that deal with distances that are more than a blip on your odometer, have roads whose grade is at most 10% (steeper sections of road are generally short segments). 10% grade is 10 feet up for every 100 feet horizontally. This corresponds to an angle of (arctan(0.1)) 5.7 degrees. The distance on the road itself, i.e. the hypotenuse, is (100*sec(5.7)=100.49 feet. So the map reads 100 feet, your odometer reads 100.5 feet - well within the margin of error.
posted by notsnot at 8:13 AM on January 10, 2010 [7 favorites]

Addendum: in Behemoth's example - one mile of rise and then fall over five miles - is 40% grade. That's steeper than the parking ramps they put into the basements of warehouses that are converted into lofts. (I use that example because that's the steepest thing upon which I've ever driven a vehicle).
posted by notsnot at 8:18 AM on January 10, 2010

You're right, that the altitude change makes it a longer distance, but it's not very much additional distance at all. It's all about the road's grade. Interstate highways in the US are limited to a maximum 6% grade, and even that won't be reached most times.

A 6% grade is a rise of 6 feet for every 100 feet traveled horizontally. Form a right triangle with a width of 100 and a height of 6. The "map distance" will be 100, while the actual distance traveled (the "wheel distance") is the hypotenuse of that triangle. Sqrt(6*6+100*100) = 100.18. So the difference between "map distance" and "wheel distance" is about a fifth of one percent.

If you were to somehow tackle a 50% grade (a 45 degree angle with horizontal), then the up-and-down would make a noticeable difference. But your wheel distance would actually go down substantially because your car would die.
posted by whatnotever at 8:19 AM on January 10, 2010 [9 favorites]

A 12% grade (maximum for recreational traffic according to this) -> 0.7% inaccuraccy.

A 20% grade (maximum for 4-wheel-drive traffic according to this) -> 2.0% inaccuraccy.

Of course, if you're getting your routes online (e.g. from Google Maps) it would be possible for them to give you distances which account for that already - don't know if they do or not though.
posted by Mike1024 at 8:30 AM on January 10, 2010

Two things are happening here - most hills with a road on them are less steep that they look and most odometers are less than 5% accurate.

If we assume a worst case 1:3 hill - in one mile horizontal length it would rise by 1/3 mile
Applying Pythagoras theorem the road is √(1² + 0.3²) = 1.04 miles long
but your 5% accurate odometer could read anything between 0.99 and 1.09 miles
posted by Lanark at 8:33 AM on January 10, 2010 [1 favorite]

one mile of rise and then fall over five miles - is 40% grade. That's steeper than the parking ramps they put into the basements of warehouses that are converted into lofts

I used to drive a Unimog 404 and can report that 100% grades are very unnerving, especially when it is a side-slope. I was certain of my impending doom almost the entire time.

The Unimog could also handle infinite slope discontinuities of up to about 1 meter...
posted by autopilot at 9:04 AM on January 10, 2010 [1 favorite]

The fundamental difficulty here is that the surface of the earth exists in three dimensions, but the map can be only two. The map-maker does the best job they can, but it will never be perfect. To think of the exact same problem in more extreme conditions: imagine projecting an image of a house onto a wall using a normal movie projector, and then trying to estimate the number of steps it would take you to walk around the house. You will be able to do this fairly well if the picture of the house is taken from directly above, and you won't be able to do it at all if the image is taken from the front.

The process of going from three-dimensional reality to a two-dimensional image necessarily throws out one dimension of information -- and your measurements will therefore be less accurate by an amount given by how much information you've thrown out. When you're going from point A to point B, most of your travel is in the horizontal two directions, so map-makers make the intelligent choice of discarding the vertical direction -- it's the one that contains the least information that you need.
posted by wyzewoman at 10:18 AM on January 10, 2010

Roads do not go straight over steep hills. Instead they use switchbacks to wiggle back and forth, generally attacking the ascent at an angle. The extra trip distance due to these curves is much, much greater than the extra trip distance due to elevation changes.
posted by ryanrs at 10:19 AM on January 10, 2010

Map distances are often sourced from official documents. A highway department will measure the laying of pavement using something like a measuring wheel (at least, my city street department does), perhaps vehicle-mounted. (They do this, among other reasons, to verify the contractor invoice.) Essentially they will measure the road surface as it is, not from a map, and the maps will reflect the highway department's measurements, instead of the other way around.

Thus an odometer-read distance should come out close to the nominal distance regardless of slope.

I suppose it's possible that maps might measure the length of the lines they use, but that would be highly inaccurate if the roads had more than a few slight curves, especially at higher scales.
posted by dhartung at 10:54 AM on January 10, 2010

The error from the grade will be swamped by the error in your speedometer. Your speedometer's calibration is affected by things like what brand/model of replacement tire (or custom wheels) you choose and the amount of air pressure you maintain in the tires. It is nearly impossible for a speedometer to be absolutely correct given these variables, and most laws only require them to be within 5%, which is much larger than the fraction of a percent that the road grade accounts for.
posted by Rhomboid at 11:12 AM on January 10, 2010

50% grade (a 45 degree angle with horizontal),

A 45 degree angle is actuall a 100% slope - you gain 100 feet in elevation for every 100 feet of horizontal distance.
posted by LionIndex at 12:14 PM on January 10, 2010

Map distances are often sourced from official documents. A highway department will measure the laying of pavement using something like a measuring wheel (at least, my city street department does), perhaps vehicle-mounted.

GPS surveying measures positions accurate to a few cm. It's how a lot of surveying is done nowadays. It captures both longitude/latitude and height information.

It's likely the map measurements are accurate.
posted by Mike1024 at 2:15 PM on January 10, 2010

A 45 degree angle is actuall a 100% slope - you gain 100 feet in elevation for every 100 feet of horizontal distance.

You're absolutely right, as I realized soon after posting. If only Metafilter had an "edit comments to reduce shame" feature.

posted by whatnotever at 4:47 PM on January 10, 2010