Work the Shell - Calculating the Distance between Two Latitude/Longitude Points

Last month, I closed this column with a script that can return latitude/longitude values for two addresses, with the intent ultimately being to have the script calculate the distance between those two points. As an example:

$ farapart.sh "union station, denver co" \
      "union station, chicago il"
Calculating lat/long for union station, denver co
= 39.75288, -105.000473
And calculating lat/long for union station, chicago il
= 41.878658, -87.640404

The formula to calculate distance actually is pretty complicated. Here's a JavaScript implementation of the math I showed last month:

var R    = 6371;       // kilometers
var dLat = (lat2-lat1);
var dLon = (lon2-lon1);
var a    = Math.sin(dLat/2) * Math.sin(dLat/2) +
           Math.cos(lat1.toRad()) * Math.cos(lat2.toRad()) *
           Math.sin(dLon/2) * Math.sin(dLon/2);
var c    = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a));
var d    = R * c;

This is going to be a wee bit tricky to convert into a shell script, needless to say, but because we have to use a more sophisticated math tool than the built-in capabilities of Bash anyway, this also means we have a number of options to work with, including Perl, awk and bc. For that matter, we also can just write a quick C program that solves this equation given four variables, but really, why make it easy when I can make it complex? If I wanted easy, I would whip out some Perl, right? Last month, I promised some bc, so let's see if we can make that rusty old app do the heavy lifting.

The first step mathematically is to convert the lat/lon values we get from the mapping system from degrees to radians. This turns out to be straightforward:

Radians = degrees * ( pi / 180 )

Pi, of course, is 3.1415926535897932384.

Given values like:

41.878658, -87.640404

The radians equivalent of those is then:

0.7309204767, -1.529613605

To warm up with bc, here's a simple command-line way to calculate one of these values:

echo "scale=8; -87.640404 * ( 3.14159265 / 180)" | bc

That's all well and good, but it turns out that the different equations I explored for calculating the distance between two points requires the atan2() function, which isn't part of bc.

Rather than beat my head against the old-school wall until the bits are bloodied, I'm going to throw in the towel and admit that this might just be a bit too complex a mathematical problem for a shell script and bc.

Having spent way more hours than I want to admit trying to get this to work properly in bc, I'm going to “cry uncle” and switch temporarily into a different programming language. I'm going to jump into C for a few lines and whip out a simple program that, given two lat/lon pairs in degrees, calculates the distance between them in miles (Listing 1).

#include <stdio.h>
#include <math.h>
#include <stdlib.h>

#define EARTH_RADIUS       (6371.0072 * 0.6214)
#define TORADS(degrees)    (degrees * (M_PI / 180))

main(int argc, char **argv)
{
   double lat1, long1, lat2, long2;
   double dLat, dLong, a, c, d;

   lat1  = TORADS(atof(argv[1]));
   long1 = TORADS(atof(argv[2]));
   lat2  = TORADS(atof(argv[3]));
   long2 = TORADS(atof(argv[4]));

   dLat  = lat2 - lat1;
   dLong = long2 - long1;

   a = sin(dLat/2) * sin(dLat/2) +
       cos(lat1) * cos(lat2) * sin(dLong/2) * sin(dLong/2);
   c = 2 * atan2(sqrt(a), sqrt(1-a));

   printf("%g\n", EARTH_RADIUS * c);
}

Does it work? Let's find out:

$ distance 39.75288 -105.000473 41.878658 -87.640404
917.984

That seems reasonable. The great circle distance between those two points is 917 miles. Google Maps, of course, shows about 10% greater distance, but perhaps that's because there is no direct-as-the-crow-flies route via roads?

Of course, there also are errors with this formula too, because Earth isn't a perfect sphere but rather an oblate spheroid that has a different diameter depending on where you're measuring. But for our purposes, let's just gloss over that problem. You can Google it to learn about things like the Vincenty formula, but that's beyond the scope of this ridiculous sidetrack.

Now we have all the pieces we need: location to lat/lon and distance between two lat/lon points. Let's put it all together.