 Synergos Technologies, Inc. - Great Circle Distance

# Great Circle Distance

### Getting Your Arms Around "Great Circles"

I remember back in grade school, when we were taught a simple method for measuring objects, especially round objects. Simply take a string, wrap it around the item, then measure the length of the string required to circumscribe the item. Viola! You had your measurement. Unfortunately, this process isn't quite as handy when measuring objects larger than you can comfortably get your arms around, like, say, the earth.

To measure a distance on the earth we must delve into an area of spherical trigonometry called "great circles." A great circle is a circle around a sphere, whose radius is equal to that of the sphere. When measuring a distance between points on a sphere, the measurement is taken along the arc of a great circle. It is worth noting that on spheres with a significantly large enough radius, the arc distance closely approaches the same distance as that measured by a straight line. In reality, the earth is not a true sphere, but is in fact considered a geoid. Fortunately, the differences in measurements is so small, that the calculations for great circle distances are still appropriate and indeed this is the mathematics used by MapInfo® in its distance calculations.

Before preceding into the mathematics, a brief understanding of the basic units of earth measurement might be interesting. The following outlines some basic facts:

Earth mean (avg.) radius = 3959.740 miles

1 degree on a great circle = 69.1105 miles

1 degree on the equator = 69.186 miles

1 geographical mile (1/15 equatorial degree) = 4.613 miles

The actual formula (with example) for calculating a great circle is as follows:

x_1 (longitude) = -30.3
y_1 (latitude) = 59.9

San Francisco
x_2 (longitude) = 122.4
y_2 (latitude) = 37.8

miles = 69.1105

distance = acos(sin(y_1) * sin(y_2) + cos(y_1) * cos(y_2) * cos((x_1 - x_2))) * miles
distance = acos(sin(59.9) * sin(37.8) + cos(59.9) * cos(37.8) * cos((-30.3-122.4)) * miles
distance = acos(.86515 * .61291) + .50151 * .79016 * -.88862) * 69.1105
distance = acos(.17812) * 69.1105
distance = 5510.836 miles

Please note many computer programs require that the degree information be converted into radians before computations begin (the radian conversion value is 0 .017453293).

Are you starting to feel like it would have been much easier just to get a really BIG ball of string?

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