Spherical Coordinates and the GPS

In this module we develop simplified formulas for finding the distance between two points on the earth. The coordinates of each of the two points are given in the form

This is the form used by many Global Positioning Systems. We use a simplified model of the earth. In this model the earth is a sphere whose radius is 6367 kilometers. Because the earth is not a sphere this model is somewhat inaccurate. For our purposes, it is a good first approximation. See the module The Earth is Round -- Most Maps are Flat for more details.

The first step is to convert the measurements of latitude and longitude into a more usable form. Since there are 60 minutes in a degree, the number of degrees is given by

We use spherical coordinates in this module. The figure below compares spherical coordinates (in degrees) with latitude coordinates. In spherical coordinates we measure the angle phi from the north pole. Thus, the north pole corresponds to phi = 0; the equator to phi = 90 degrees; and the south pole to phi = 180 degrees.

       / 90 - latitude         if latitude is North
phi = {
       \ 90 + latitude         if latitude is South

In spherical coordinates we measure the angle theta starting at the prime meridian (longitude 0) and moving east. Thus

         /  longitude          if longitude is East
theta = {
         \ -longitude          if longitude is West

Both phi and theta as described above are measured in degrees. It is mathematically much better to measure angles in radians. The conversion formula is

                    angle in degrees * 2 * Pi
angle in radians = ---------------------------
                               360

We want to express the location of a point in Cartesian coordinates with the origin at the center of the earth, the north pole at the point

and the positive x-axis going through the prime meridian. The conversion formulas are:

x = 6367 (cos theta) (sin phi)
y = 6367 (sin theta) (sin phi)
z = 6367 (cos phi)

Using these formulas we can determine the Cartesian coordinates of any point from its latitude and longitude. For example, I was recently in San Luis Opisbo, California and my trusty GPS told me the location of my hotel was

The location of the Carroll College Fountain in Helena, Montana is

The Cartesian coordinates of San Luis Opisbo, CA and Helena, MT are

---------------------------------------
   San Luis Opisbo, CA      Helena, MT
--------------------------------------- 
x =      -2650                -1641
y =      -4471                -4055
z =       3678                 4626
---------------------------------------

The distance between these two points is 1,446 kilometers or 897 miles, but this distance is the straight line distance through the earth.

                          -1   straight line distance
surface distance = 2 R sin   (------------------------)
                                       2 R