We have been studying paragliding. We mapped the paraglider’s GPS track in ArcGIS Explorer, calculated the vertical and total distance traveled, measured its velocity, and then used a web resource to obtain the total distance in the horizontal and vertical dimensions. Now, let us consider an even more accurate measure of the distance traveled.

Because the Earth is an oblate spheroid, calculating a more accurate distance between two points requires spherical geometry and trigonometry. The Great Circle Distance Formula is one method, which uses double-precision (about 15 digits of accuracy) and the conversion of latitude and longitude values from decimal degrees to radians. Divide the latitude and longitude values by 180/pi, or 57.29577951. After conversion to radians, use:

3963.0 * arccos[sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1)]

If you do not first convert the latitude and longitude values in the database to radians, you must include the degrees-to-radians conversion in the calculation. Substituting degrees for radians, the formula becomes:

3963.0 * arccos[sin(lat1/57.2958) * sin(lat2/57.2958) + cos(lat1/57.2958) * cos(lat2/57.2958) * cos(lon2/57.2958 -lon1/57.2958)]

or

r * acos[sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1)]

Where r is the radius of the earth in whatever units you desire. r=3437.74677 (nautical miles)
r=6378.7 (kilometers)
r=3963.0 (statute miles)

If the software application or programming language you are using has no arccosine function, you can calculate the same result using the arctangent function, which most applications and languages do support:

3963.0 * arctan[sqrt(1-x^2)/x]

where

x = [sin(lat1/57.2958) * sin(lat2/57.2958)] + [cos(lat1/57.2958) * cos(lat2/57.2958) * cos(lon2/57.2958 - lon1/57.2958)]

Compare the result of this formula with the results that you obtained using the first manual calculation method and with the easier method on the web. How much do the three methods differ in meters? Which do you think is the most accurate? The length of the track is also stored in the GPS receiver. Do you think the track length stored in a recreational grade GPS is more accurate than your calculated values?

GIS and GPS can serve as excellent bridges between disciplines, in this case, mathematics, physics, and geography. What other phenomena could you map and analyze with these tools?

- Joseph Kerski, Education Manager