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https://github.com/dkogan/inaccessibility


https://github.com/dkogan/inaccessibility

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README 

          
* Overview



So I was out hiking with a friend, and a question came up about where the least

accessible point of the San Gabriel Mountains was, with "accessible" defined as

having a road or trail nearby. This repository answers that question.



This is called the [[http://en.wikipedia.org/wiki/Pole_of_inaccessibility][Pole of Inaccessibility]]: a point that is as far away as

possible from a given set of objects. Locations of such poles are known for the

most landlocked spot on earth or most far away from land. Here we limit

ourselves to the San Gabriel Mountains, and try to stay away from roads and

trails.



* Approach



** Input data processing



[[http://www.openstreetmap.org][OpenStreetMap]] has open data I can use to map out all the roads and trails. This

is the input dataset.



For 2D geometry, the best approach to compute the Pole of Inaccessibility

appears to be to construct a [[http://en.wikipedia.org/wiki/Voronoi_diagram][Voronoi diagram]] of the geometry we're trying to

stay away from, and to find the Voronoi vertex corresponding to the

furthest-away point.



Our world is not 2D. Instead, it has varying elevation sitting on top of an

ellipsoid. The grand purpose here is to compute a location that hardy people can

visit and to tell everybody they did it, so extreme accuracy is not required.

Thus I claim that assuming the world is locally-flat and using the

Voronoi-diagram-based method is sufficient. So I construct a plane that best

describes my query area and project all my input points to this plane. I use a

plane that is tangent to the Earth's surface at the center of the query area.

This clearly wouldn't work if trying to find the pole of inaccessibility of

something as large as an ocean, for instance, but it works here.



To compute the tangent plane, I assume the Earth is spherical. As I move along

the tangent plane away from the point of tangency, the elevation error grows:



 E = sqrt(R_earth * R_earth + d * d) - R_earth



The San Gabriels are about 80km across, and the tangent plane sits in the

middle, so at worst d = 40km and the error is about 125m. That's plenty good

enough.



I ignore the ellipsoid shape of the Earth outright. I ignore the topography as

well, since including it in my distance metrics would require a fancier

algorithm than making a Voronoi diagram, and it would make the notion of

"inaccessibility" more ambiguous.



** Pole of Inaccessibility computation



I want to use the most basic Voronoi algorithm, so I represent my input as a set

of points only; no line segments. To get reasonable accuracy, I make sure to

sample each road at least every 100m.



Now that I have my set of dense-enough points in 2D, I construct the Voronoi

diagram. Without constraints the furthest-away point would be infinitely far off

to one side, so generally people constrain the solution to lie within the convex

hull of the input points. This means that the Pole of Inaccessibility lies

either on a Voronoi vertex or at an intersection of a Voronoi edge and the

convex hull of the input. In my case there are generally more roads at the edges

of my query area that in the interior (less stuff in the mountains than in the

flats), so I simply assume that the Pole of Inaccessibility is not on the convex

hull. This simplifies my implementation since I simply ignore all the Voronoi

vertices that are outside of the query region.



So I need to look at every Voronoi vertex, check the distance between it and an

adjacent input point, and return the vertex with the largest such distance.



* Implementation



Each step in the process lives in its own program. This simplifies

implementation and makes it easy to work on each piece separately.



** Data import



First we query OSM. This is done with the =query.sh= script. It takes in corners

of the query area, constructs the query, sends it off to the server, and stores

the result. =query.sh= takes 4 arguments; lat0, lon0, lat1, lon1, and stores its

output in a file called =query_$lat0_$lon0_$lat1_$lon1.json=. The query uses the

[[http://wiki.openstreetmap.org/wiki/Overpass_API/Overpass_QL][OSM Overpass query language]]. By default I simply look at all the roads, trails

(everything with a =highway= tag):



#+BEGIN_EXAMPLE

[out:json];



way ["highway"] ($lat0,$lon0,$lat1,$lon1);



(._;>;);



out;

#+END_EXAMPLE



If I want to only consider roads in my computation (allow trails), then I can

exclude trails from the query:



#+BEGIN_EXAMPLE

[out:json];



way ["highway"] ["highway" != "footway" ] ["highway" != "path" ] ($lat0,$lon0,$lat1,$lon1);



(._;>;);



out;

#+END_EXAMPLE



Sample invocation:



#+BEGIN_EXAMPLE

$ ./query.sh 34.1390884 -118.4944153 34.5020298 -117.5852966



  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current

                                 Dload  Upload   Total   Spent    Left  Speed

100 28.3M    0 28.3M    0   138  72803      0 --:--:--  0:06:48 --:--:--  138k



$ ls -lh query*



-rw-r--r-- 1 dkogan dkogan 29M May  6 04:12 query_34.1390884_-118.4944153_34.5020298_-117.5852966.json

#+END_EXAMPLE



** Data massaging



Next, I take the lat/lon pairs, map them to the tangent plane and make sure the

data is sufficiently dense. This is done by the =massage_input.pl= script. It

takes in the =query_....json= file we just obtained, and generates a

=points_$lat0_$lon0_$lat1_$lon1.dat= file that is a list of (x,y) tuples in my

plane. There's a small header of 4 values, representing the bounds of my data so

that I can reject outlying vertices, as described earlier.



Sample invocation:



#+BEGIN_EXAMPLE

$ ./massage_input.pl query_34.1390884_-118.4944153_34.5020298_-117.5852966.json



$ ls -lh points*



-rw-r--r-- 1 dkogan dkogan 4.3M May  6 04:20 points_34.1390884_-118.4944153_34.5020298_-117.5852966.dat

#+END_EXAMPLE



** Pole of Inaccessibility computation



Now we can compute the Voronoi diagram. I use [[http://www.boost.org/doc/libs/1_57_0/libs/polygon/doc/index.htm][boost::polygon]] to do this. I had

concerns that this step would be prohibitively slow, but the algorithm and this

implementation are quick-enough such that this "just works".



The =points_....dat= file is inputs on standard input. Note that this is

different from the other tools that read a file on the commandline instead.



For each Voronoi vertex I get an arbitrary neighboring edge, and an arbitrary

neighboring cell. The distance between the vertex and the cell center is

identical for any such edge, cell by definition of a Voronoi vertex. I keep

track of the cell with the largest distance between the vertex and the cell

center, and I report the vertex with the largest such distance as my Pole of

Inaccessibility.



Sample invocation:



#+BEGIN_EXAMPLE

$ ./voronoi < points_34.1390884_-118.4944153_34.5020298_-117.5852966.dat 



furthest point center, surrounding points:

25541 -78

25308 4259

21223 -543

26931 -4192

distance: 4342.873206

#+END_EXAMPLE



Bam! So the Pole of Inaccessibility is about 4.3 km from the nearest trail/road.

The coordinates here are in my 2D tangent plane, which isn't super useful. Now I

convert them to lat/lon and I'm done.



** Unmapping the planar coordinates



I do this with the massaging script as before simply by passing the coords in on

the commandline:



#+BEGIN_EXAMPLE

$ ./massage_input.pl query_34.1390884_-118.4944153_34.5020298_-117.5852966.json 25541 -78 25308 4259 21223 -543 26931 -4192



34.3206972918426,-117.761740671673

34.3597096140465,-117.764149633831

34.3165155855012,-117.808770212857

34.2837076589351,-117.746734473897

#+END_EXAMPLE



OK. Done.



* Results



I did this twice: once avoiding all roads, trails and again avoiding roads only.

Both Poles of Inaccessibility are above the East Fork of the San Gabriel River,

by Ross Mountain:



| pole         | lat,lon               | distance to nearest (m) |

|--------------+-----------------------+-------------------------|

| roads,trails | 34.3206973,-117.76174 |                    4343 |

| roads only   | 34.3107784,-117.74736 |                    5677 |



This is shown nicely on the map:



http://caltopo.com/m/4N7A



[[file:poles.png]]



Looks like the bounding spots for the roads,trails point are the road up to

South Mt Hawkins, the PCT on top of Mt. Baden-Powell and the trail at the Bridge

to Nowhere.



The bounding spots for the roads-only point is the same road up to South Mt

Hawkins, the Cabin Flat Campground and Shoemaker Canyon Road.

* License



All code Copyright 2015 Dima Kogan, released under the terms of the Lesser GNU

Public License (any version)