Problem Description

Here's a little program that estimates the surface distance between two points on earth defined by their latitude and longitude.

Background & Techniques 

A viewer recently requested this facility which happened to pique my curiosity.  A usual, there is lots to learn but the basic concepts are not too difficult.   

Latitude and Longitude represent the angle portion of a point in space defined in polar coordinates.  Recall that in 2 dimensional space a point may be defined either by Cartesian or polar coordinates.  A Cartesian coordinate defines a location as the horizontal and vertical distance from some fixed point, the origin.  Polar coordinates define a point by the length and angle of a line segment from the origin to the point.  There is a Math-Topics page illustrating the relationship between the two systems.   Now, if we imagine a point on a flat 2-dimensional page defined by the two numbers and then we want to move the point into 3-dimensional space by lifting up off the page, we need one more number.   For Cartesian coordinates, the third number will be the height of the point above the page.  For polar coordinates, we'll rotate the line segment up off of the page and measure the vertical angle from the page up to the line.     

Latitude and Longitude are the  two polar coordinate angles.  If we imagine ourselves at the earth's center with the top our head pointing to the North pole and out feet toward the South pole, the equator will be at 90 degrees from North - straight out in front of us.  .  This imaginary line around the earth at a 90 degree angle to the North pole is defined as 0 reference for measuring  Latitude, the North-South angular coordinate of a point.  Another imaginary line running north and south along the surface of the earth and passing through the poles and Greenwich England is the 0 degree reference for measuring East-West angles, the Longitude.  Latitudes range from 90 degrees North to 90 degrees South with South angles treated as negative.  Longitudes range from 0 to 180 degrees East and West with West angles treated a negative.    

The actual math to calculate the distance between two points is straightforward and derived in many places on the web, namely:  angular distance =   arccos(Sin(lat1) * Sin(lat2) + Cos(lat1) * Cos(lat2) * Cos(lon1-lon2)) and assuming that the angle is given in degrees,  distance = distance per degree*angular distance = (2*Pi*radius/360)*angular distance.    This is the spherical distance estimate produced by this program. 

Flat Earth?   

The above explanation considers  the third parameter of the polar coordinate defining a point, the length of the line segment, to be constant, namely the radius of the earth.  But in fact the earth is not spherical.  Our rotation has caused the earth to bulge slightly at the equator and flatten at the poles.    Sir Isaac Newton in the 1600's calculated the radius at the poles to be 0.9967 of the radius at the equator.  The current value is about 0.9966  so Sir Isaac didn't do too badly.    The difference of the two circumferences (polar and equatorial) is about 83 miles, so spherical coordinates for two points on on the equator but 180 degrees apart should produce the maximum error, about 40 miles.    By the way, these distance estimates are "great circle estimates" meaning they lie on the path defined by the plane through the two points plus the earth's center and the surface of the earth.   The great circle route is the shortest surface distance between the two points.  Counter-intuitively, if you are north of the equator and want to get to a point directly east of your current location, don't head east - the shortest way is to start out traveling northeast and arrive traveling southeast!  

Calculating distances taking into account the elliptical shape of the earth is quite complex. There are approximations that are at least close to the true distance as those that assume a spherical earth.   The one used here is from http://www.codeguru.com/Cpp/Cpp/algorithms/general/article.php/c5115/

Addendum October 3, 2008:   After 4 years, it must be time for an update.    A viewer wrote recently asking how to calculate an end point given a starting point, a bearing, and a distance.  After some searching, I found Vincenty's algorithm and equations.  In 1975, geodesist Thaddeus Vincenty published equations that produced very accurate results by iterating intermediate results.   It was one of the early applications to geodesic measurements during the period when applications were moving from desktop calculators to computers.  Versions for both the "distance between points" problems and the "endpoint from start, bearing and distance" problem.  Both are included in  Version 2 of LatLong Distance program posted today.   

Running/Exploring the Program 

• Download source
• Download  executable

Suggestions for Further Explorations

Most accurate calculation of surface distance reportedly requires numeric integration of over the path between the points.  I assume that these are spherical trig estimates with the radius adjusted for the "flatness" at the current location.  Might be interesting to try.