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I have this hunch that I could travel no more than 50 miles from the most remote spot in the Eastern United States (east of the Mississippi River), in the direction of the nearest road, and find a road.

Definitions:
Most Remote: Spot furthest from a road.
Road: Google Maps definition of a road.

Q: How could I prove or disprove this claim?

Q: Where is the most remote spot in the Eastern US?

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migrated from math.stackexchange.com Aug 22 '11 at 16:19

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This has nothing to do with topology in the mathematical sense. I am closing and migrating to the Geography SE –  Willie Wong Aug 22 '11 at 16:18
    
"Any direction" or "some direction"? Good luck finding a road if you go north from the southern shore of Lake Michigan. –  user4016 Aug 22 '11 at 16:29
1  
How is 'remoteness' defined? business.otago.ac.nz/sirc/conferences/1999/23_Dunne.pdf (population, road type, islands) Remoteness Classification. –  Mapperz Aug 22 '11 at 17:03

3 Answers 3

up vote 12 down vote accepted

A fast and informative way is to create a distance grid based on the roads. This is usually done in a projected coordinate system, which necessarily introduces some error, but by choosing a good coordinate system the error will not be too great (and can be corrected).

The following example defines a "road" as a US Interstate or US or state highway of comparable magnitude. These roads are shown as red polylines. It uses a Lambert Conformal Conic projection. Although its metric distortion can readily be corrected in terms of latitude, that's not really necessary in this example because the distortion is less than 0.6% except in Florida, where it grows to 2.3%: good enough for this illustration.

Road distance map

The distances are color coded from dark cyan (short) through yellow (long) and hillshaded to emphasize the local maxima. A glance shows the greatest distances are attained in central Wisconsin and the North Carolina coast. The GIS tells me the maximum distances attained are 194 km and 180 km, respectively. (The maximum attained in Michigan is 120 km, less even than the maximum in central Mississippi, 137 km.)

Using any raster GIS (such as ArcGIS, GRASS, Manifold, etc.) one can perform a similar computation using any roads layer desired (such as Census TIGER streets features). Straightforward post-processing will find all local maxima of the distance grid (seen as peaks on this map), thereby identifying all points that locally are as far from a road as you can get. Very simple post-processing will identify all points exceeding a distance threshold such as 50 miles (about 80 km).

A variant uses a "costdistance" calculation, instead of Euclidean distance (as a proxy for spherical distance), to determine points that are (say) a maximum travel time from the nearest road. This is not an onerous task: typical computation times are a few seconds (at most) at the 1 km resolution used here.

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Nice suggestion - I think this would be a lot quicker than buffering also. –  djq Aug 22 '11 at 17:23
    
@celenius You're right. Buffering works but it's less informative and is not flexible enough to answer the questions a distance map invites us to ask, such as "where are the almost remotest points" and "how do I adjust the calculation to preclude travel over large bodies of water," etc. –  whuber Aug 22 '11 at 17:31
    
This is a wonderful place to start. I have no knowledge of GIS applications so some of the technical jargon lost me. (SE Math sent me here), but this response has pointed me in the right direction. From here, I'll get a Delorme map of Wisconsin, Mississippi, and North Carolina and use a compass and colored pencils. –  zundarz Aug 26 '11 at 15:12

It is certainly not topology. I would suggest geography.

To prove the claim, I would create a map of the area of interest, then color each point within 50 miles of a road. After your favorite list of roads, see if there are any uncolored points. Then go to Google maps, for example, and look if there is a road you missed. Of course, a "road" is not well defined, so the result will depend on what you consider a road.

For question 2, (assuming the result of 1 was that you proved the claim), try the same process using 40 miles and see if there are points that far away.

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I'm defining road as a road that appears on Google maps. So I'd be using Google's definition of a road. <p> Also, I thought this was question of topology. Would someone with edit rights please remove 'Topology:' form the question. –  zundarz Aug 22 '11 at 17:00
    
Google's definition of roads in the western US leads people to take shortcuts that ends up stranding and sometimes killing them. Google may call it a road, but that doesn't mean that it's passable in the summer without 4 wheel drive or in the winter at all. –  thursdaysgeek Aug 23 '11 at 2:52

How could I prove or disprove this claim?

Take the road network (TIGER data?) and buffer it with 50 mile radius. You'll see if any land masses are not within the buffer zones.

Where is the most remote spot in Eastern US (spot furthest from a road?)

Iteratively increase the buffer radius until you've narrowed it down.

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