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

AAA degrees, BB.BBB minutes

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.

Missing figure

Thus, the following formula converts from latitude expressed in degrees to phi also expressed in 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 = ---------------------------

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

north pole = (0, 0, 6367 kilometers)

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

35 degrees 17.299 minutes North and 120 degrees 39.174 minutes West

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

46 degrees 36.003 minutes North and 112 degrees 02.330 minutes West

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.

Missing diagram

You can use a bit of trigonmetry to determine the distance along the surface of the earth from the straight line distance through the earth. The formula is

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

where R is the radius of the earth. Verify this formula.

Using this formula, we see that the surface distance between Helena, Montana and San Luis Opisbo, California is 1449 kilometers or 898 miles.


This work is copyrighted c 1996 by Carroll College, Helena, MT 59625.

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Department of Mathematics, Engineering, Computer Science, and Physics
Carroll College, 1601 N. Benton Avenue, Helena, MT 59625