GG2470: Intro to G.I.S.
Fall 2002, September 4, Wednesday
Dr. Rudnicki

360 degrees (of arc) = complete circle = angular distance of a parallel or meridian
1 degree of arc contains 60 minutes of arc
1 minute of arc contains 60 seconds of arc
1 minute of arc (angular distance) = 1 nautical mile (linear distance) along or astride the equator
1 degree of arc = 60 nautical miles
1 nautical mile ~ 6,000 feet (just a little bit more, really -- 6,076.11549'),  along or astride the equator
1 foot = 12 inches

Map scale can be expressed as -- map distance over earth distance, or md/ed

NB: Units are the same in numerator and denominator, and the fraction is reduced so that '1' is in the numerator. This is one way to express map scale.   This method is called the RF (representative fraction)

Using the Graticule to Compute a Map's Scale.

A nautical mile is defined as one minute of arc at the equator. The U.S. Navy defines the nautical mile (and one minute of arc) as equaling 6,000 feet at the equator. While not as accurate as it could be, the relationship is 'close enough for government work'.

A brief, useful definition of nautical mile can be found at:  http://www.howstuffworks.com/question79.htm
A very interesting and more elaborate explanation can be found in the following on-line dictionary under the letter 'N':
http://www.unc.edu/~rowlett/units/dictN.htm

As one moves from parallel to parallel between the equator and the poles, the distance between two meridians one minute of arc apart, along a parallel, is no longer equal to ~6000 feet (or one nautical mile). The linear distance gets shorter although the angular distance stays the same. The reason is, of course, that the parallels describe smaller circles and thus the meridians (lines of longitude) converge toward the poles. The linear distance between two meridians is equal to the linear distance between them at the equator (~6,000 feet) times the cosine of the latitude at which the measurement is made. For example, the distance between two meridians one minute of arc apart at the equator is ~6,000 feet. (In terms of spherical trigonometry, that distance is 6,000 feet x cosine of 0 degrees. Since the cosine of 0 degrees = 1, then 1 x 6000 = 6000...feet.) The distance between the same two meridians at latitude 60 degrees N or S, is 6,000 multiplied by the cosine of 60 degrees (0.5), so the linear distance at that latitude is 3,000 feet.

As a result, scale and distance measurements are much simpler to compute along a meridian, between two parallels, because that linear distance stays pretty much constant between the equator and the poles.

Since there are 60 minutes of arc in a degree, then there are 60 nautical miles in a degree. If the parallels on a map are plotted at 10 degree intervals from the north pole (90 degrees N) to the south pole (90 degrees S), then the distance between each pair of neighboring parallels is 600 nautical miles (60 nautical miles per degree times 10 degrees).

Map scale is equal to map distance divided by earth distance. If map scale is expressed as RF, then these two values need to be in the same units, i.e. if inches are used in the numerator (map distance), then inches need to be used in the denominator (earth distance). This form of map scale, the RF, can be written as a fraction, 1/20000, or as a ratio, 1:20000. In this example, one unit (inch in this example, but it could be centimeters or some other linear unit) on the map equals 20000 inches on the earth. Because the same units were used in the numerator as in the denominator, no units symbol (inch, centimeter, etc) needs to be given. Thus a map reader can use any linear unit because the ratio will remain the same.

If the distance between two parallels 10 degrees apart on a map is measured to be 2 inches, that means 2 inches = 10 degrees. Ten degrees is equal to 600 nautical miles. Each nautical mile consists of 6000 feet and each foot contains 12 inches. Thus, 2 inches on the map = 6000 x 10 or 60,000 feet. 60,000 feet x 12 inches = 720,000 inches on the ground. Because the units are now the same in the numerator and denominator, the scale might be expressed as 2/720,000, but it's not, because the fraction can be reduced, and scale is written so that '1' is always in the numerator. Thus, in this example, the map scale, expressed as RF, is 1:360,000.   The fraction was reduced by dividing 2 into the numerator and denominator.