A. Map Scale
Every map is a simplification of reality. This simplification comes as a result that maps typically show the earth at a smaller size that it truly is. However, note that in some cases, maps (e.g., maps of human DNA sequences) are exaggerations of the true size of objects they are trying to represent. In order for maps to be used for measurement or cartometric activities, a map scale must be known. Map scale is defined as the distance on a map compared to that same distance on the earth’s surface. This relationship is expressed by Equation 1.
where map_{d }is equal to the distance measured on the map (e.g., in inches, centimeters, etc.), earth_{d} is the same distance a measured on the earth’s surface in the same units as found in the map_{d}, and MapScale is reported as 1/x.
MapScale is reported as a nonunit based fraction, also known as a representative fraction. The representative fraction is commonly reported as the following 1:24,000 (but also can be written as 1/24,000). This fraction reads as follows: 1 unit as measured on the map is equal to 24,000 units as found on the earth’s surface. For instance, if I measure one cm. on the map, then according to the map scale, this corresponding distance equals 24,000 cm. on the earth’s surface. Once the representative fraction is known, various measurements can be made on the map. For example, assume I have a map where the representative fraction is stated as 1:62,500 (a common map scale that was used by the USGS for its topographic map series). On such a map, I know that one cm. on this map measures 62,500 cm. on the earth’s surface. Therefore, if I measure a distance of 6.5 cm. between two points, I will arrive at a distance on the earth’s surface as 406,250 cm. (see Equation 2)
Obviously, this value can be converted into a more logical unit such as kilometers. Since there are 100,000 cm. in one kilometer, I can take 406,250 cm. and divide by 100,000 cm. which produce 4.0625 kilometers, which is the distance between these two points on the earth’s surface (see Equation 3).
It is important to note that not all maps have stated scales. Moreover, representative fractions found on maps that are photocopied or otherwise have their dimensions altered can no longer be considered valid. In this case, you must compute the representative fraction. Assume that I have a map (as shown below) that does not have a stated representative fraction. I wish to determine the map scale. You should begin by selecting an object on the map in question such as a straight border or two point locations. In this case, assume I select to measure the distance between the town of Lawrence and Ottawa (see Figure 1). I measure the distance and report it as 12.4 cm. This is my map distance and is placed in the numerator in Equation 4. Next, I locate a separate map on which these same two locations are found that also has a representative fraction. On this separate map, I measure the distance between the two towns. I compute that these two towns are 31 km. apart. This is the earth distance and this value is inserted into the denominator in Equation 4.
Solving for x I first convert the 31 km. into cm. Multiplying 31 km. by 100,000 I obtain 3,100,000 cm. (see Equation 5)
Now, I divide the values and in so doing, I obtain 4.0 x 10^{6}. In this case, x is reported as the decimal equivalent of the representative fraction. For example, the decimal equivalent of the fraction ½ is 0.5. One can change the 0.5 into the denominator of the fraction by raising 0.5^{1}. By raising any value to the x^{1} power, the denominator of the representative fraction can be found (e.g., 1/x). So, taking (4.0 x 10^{6})^{1} power, we obtain 250,000. The unknown representative fraction of my map is found to be 1:250,000 (or 1/250,000). Equipped with this value I can carry out measurement activities across the map.
View this!Figure 1 displays a map of eastern Kansas with an unknown representative fraction. The two red circles are the point locations between which I measure the map distance. U.S. Geological Survey, 1:250,000 topographic sheet, Lawrence, KS. (Note: this screen capture will open in a new window.) 
B. Scale Factors
The issue of scale is an important concept as most maps, even on those showing the largest scale. On all maps, the stated representative fraction is only valid at the standard point or line(s). The scale at this location is often referred to as the principal scale. At all other locations, the scale varies and is referred to as the local scale. The amount that this local scale varies at any given point depends on the type of map projection and the placement of the standard point or line(s).
Standard point or line(s) are those locations where there is no distortion resulting from the map projection process. In conceptual terms, the standard point or line(s) is the location where the developable surface (cylinder, cone, or plane) comes into contact with the reference globe (the conceptual earth reduced to the same size as the final map scale). At any point away from the standard point or line(s) the scale will change. Fortunately, since the map projection process is mathematicallybased, we can measure the amount of departure from the principal scale at any given location and compute the scale factor. By definition, the scale factor is the ratio between the local scale and the principal scale at any given point. The scale factor is computed through Equation 6.
where the local and principal scales are stated as representative fractions.
Again, at the location of the standard point or line(s), there is no distortion. Here, the scale factor is equal to 1.0. In other words, the ratio between the local and principal scale factors is the same. At all other locations on the map, the scale factors will be greater or less than 1.0. The rate at which the scale factors change across the projection is a function of the type of map projection upon which the map is based.
Figure 2 illustrates 3 lines drawn on the Earth’s surface. The first line (line A) is drawn along the equator between 105° W and 120° W. The second line (line B) is drawn between 30° N and 45° N along 45° W longitude. The third line (line C) is between 60° W, 15° N and 75° W, 30° N. We can easily compute the distance between these locations.
Figure 2. An illustration of three lines drawn on the earth’s surface.
To compute the distance on the earth’s surface requires the knowledge of the length of one degree along a line of longitude or a line of latitude. Tables illustrating the various values of these lengths are readily available in many cartography textbooks.
Check it out!For our example, we will use an online toolkit from the National Geodetic Survey that allows us to enter latitude and longitude values and compute the resulting distances. The web site is found at (http://www.ngs.noaa.gov/cgibin/Inv_Fwd/inverse2.prl). 
To use this software, simply enter the starting and ending latitude and longitude values into the corresponding textboxes. For example, I entered 0° and 105° in the latitude and longitude text boxes for the first station data and 0° and 120° in the latitude and longitude text boxes for the second station data. I pressed the compute azimuth and distance button. A new screen appears listing the distances between these two points. If I enter all three pairs of coordinate locations, I obtain the following distances:
Line 
Distance (in meters) 
A 
1,669,792.4 
B 
1,664,831.0 
C 
2,262,496.1 
Table 1. Distances on the earth’s surface of the three lines A, B, and C.
Thus, these distances between each coordinate pair are the true earth distances. In the next step, we will compute the scale factors at these locations on two different map projections. In so doing, we will demonstrate the effect that map projections have on the scale factors.
The first projection is an unknown conic projection whose principal scale is stated as 1: 410,758,336 and is shown in Figure 3.
Figure 3. The three lines represented on an unknown conic projection.
Since I know the true earth distances of each of the three lines, I will measure the map distance of each line to determine the local scale. After I measure each line, I arrive at the following values: Line A (0.45 cm.), Line B (0.4 cm.), Line C (0.6 cm.). I can now insert these values into our basic scale equation to compute the local scale of each line.
Converting the meter values into centimeters, we have the following:
The resulting local scales for each line are as follows:
Now that we have our principal scale and local scale factors, we can enter these into our scale factor equation and compute the scale factor for each of our lines on the unknown conic projection.
To compute the scale factor, we insert the local scale into the numerator and the principal scale into the denominator as shown by Equation 7. This equation shows the scale factor for Line A.
The computed scale factor for Line A is 1.1069. This means that this location has been exaggerated in scale to 106% of its original scale, which is not too great, but significant enough to warrant attention. The rest of the scale factor computations for Lines B and C are shown in Table 2.
(0°, 105°) to (0° ,120°) 
(30°, 45°) to (45°, 45°) 
(15°, 60°) to (30°, 75°) 

Principal Scale 
1:410,758,336 
1:410,758,336 
1:410,758,336 
Local Scale 
1:371,064,997 
1:416,207,749 
1:377,082,683 
Scale Factor 
1.1069 
0.9869 
1.0893 
Table 2. The scale factor computations for Lines A, B, and C on the unknown conic projection.
It is interesting to note that Line B, which is measured along a meridian on this conic projection, is very close to the principal scale. In fact, we can reveal that the unknown conic projection is really Euler’s equidistant conic projection. There are two standard lines on this projection located at 10° S and 30° N. This equidistant conic projection preserves scale along all meridians, which satisfies the definition of equidistance. We should have expected to see the scale factor along Line B exactly matching the principal scale; thus the scale factor at this location should have equaled 1.0. However, there was likely a measurement error while measuring the length of the line on the map in Figure 3 and for this reason the scale factor value was off by a small amount. While this may seem like a trivial fact, measurement accuracy is problematic in the paper environment. The GIS environment has made this task easier, but the important idea here is that the ability to carefully and accurately measure impacts considerably the ability to compute accurate scale factors.
The second projection is an unknown cylindrical projection whose principal scale is 1:433,243,423 and is shown in Figure 4.
Figure 4. The three lines represented on an unknown cylindrical projection.
As in the previous problem set, since I know the true earth distances of each of the three lines, I will measure the map distance of each line to determine the local scale. After I measure each line, I arrive at the following values: Line A (0.4 cm.), Line B (0.5 cm.), Line C (0.6 cm.). I can now insert these values into our basic scale equation to compute the local scale of each line.
Converting the meter values into centimeters, we have the following:
The resulting local scales for each line are as follows:
Now that we have our principal scale and local scale factors, we can enter these into our scale factor equation and compute the scale factor for each of our lines on the unknown conic projection.
To compute the scale factor, we inset the local scale into the numerator and the principal scale into the denominator as shown by Equation 8. This equation shows the scale factor for Line A.
The computed scale factor for Line A is 1.0378. This means that this location has been expanded in scale to 103.78% of its original scale. The rest of the scale factor computations for Lines B and C are shown in Table 3.
(0°, 105°) to (0° ,120°) 
(30°, 45°) to (45°, 45°) 
(15°, 60°) to (30°, 75°) 

Principal Scale 
1:433,243,423 
1:433,243,423 
1:433,243,423 
Local Scale 
1:417,448,099 
1:332,966,200 
1:377,082,683 
Scale Factor 
1.0378 
1.3011 
1.1489 
Table 3. The scale factor computations for Lines A, B, and C on the unknown conic projection.
It is interesting to note that in the case of Lines B and C, the scale factors suggest an increase in the scale (by approximately 130% and 115%, respectively). This cylindrical projection is the Mercator, which is conformal and has the standard line coinciding along the equator. Conformal projections maintain constant scale in all directions about individual points. At every point, however, scale varies widely. Because of this standard line location along the equator, the scale factor computation at Line A should have resulted in a scale factor of 1.0. But, measurement error on the map in Figure 4 produced a slightly inaccurate scale factor computation. On this projection, the scale factors increase away from the standard line. Note that the scale factor for Line B is considerably greater than along Line A. In fact, had we computed the scale factor along the parallel at 60° N, we would have found that the scale factor here is 2.0 suggesting that the local scale has been exaggerated 200% compared to the scale along the equator.
C. Measurement Units
Just as the scale factor on a projection can vary, the units by which you measure on a projection can also vary widely as well. Be aware that there are two fundamental units by which ArcGIS defines measurement activities. The Map Units are those units in which the coordinate system is created. For example, the country92.shp file that is available through the ESRI dataset has Decimal Degrees as the map units. This means that the units by which the coordinates were computed are decimal degrees. The Display Units correspond to those units that are displayed in the lower righthand corner of the screen as the mouse moves over the mapped area or when using the measurement tool. By default, if the map units are known, then the display units are set to same as the map units. The user can change the display units to whatever they are most comfortable with. Figure 5 below shows the Data Frame Properties window where the Display units can be changed. Users need to be sure to check the Data Frame Properties window so as to make sure that the correct units are set for their measurement activities.
Figure 5. The Data Frame Properties window where the map and display units are available.
D. Assigning a Coordinate System
Coordinate systems in ArcGIS take on one of two definitions: Geographic or Projected.
Geographic coordinate systems are those systems that store their coordinate values in degrees of longitude and latitude. Geographic coordinate systems have the following definitions:
Name: (e.g., ‘GCS_WGS_1984’)
Datum: (e.g., ‘D_WGS_1984’)
Spheroid: (e.g., ‘WGS_1984’)
Prime Meridian (e.g., ‘Greenwich’)
Units (e.g., ‘degrees)
 The coordinate system name specifies a unique identifier that describes the coordinate system as being geographic (i.e., GCS). The WGS_1984 identifies that the ‘model’ of the earth on which the coordinate system is based is from the World Geodetic Reference System of 1984 (‘WGS_1984’).
 The datum, specified by the ‘D’, also identifies the model of the earth’s shape as being WGS_1984.
 The spheroid identifies the reference ellipsoid that is used to define earth’s shape – WGS_1984. For geographic coordinate systems, the name, datum, and spheroid should typically all have the same earth model definition.
 The Prime Meridian identifies the location of the prime meridian from which the longitude values were derived. In most cases, the Primer Meridian is Greenwich. However, various countries use other Prime Meridian location (e.g., Ferro, the western most edge of the Canary Islands).
 The Units specify that the latitude and longitude coordinate values are in degree format.
Projected coordinate systems are fundamentally different from geographic coordinate systems. Projected coordinate systems have the following definitions:
Name: (e.g., ‘NAD_1983_UTM_Zone_17N’)
Geographic Coordinate System Definition
Projection: (e.g., ‘Transverse_Mercator’)
Various projection parameters
False_Easting: (e.g., ‘500000.000000’)
False_Northing: (e.g., 0.000000’)
Central_Meridian: (e.g., 81.000000’)
Scale_Factor: (e.g., 0.999600’)
Latitude_Of_Origin: (e.g., 0.000000’)
Units: (e.g., ‘Meters’)
 In a projected coordinate system, the name refers to the coordinate system definition. In this case, the coordinate system definition is the Universal Transverse Mercator (UTM) based on the North American Datum of 1983. This projected coordinate system specifies that this geographic area of coverage corresponds to Zone 17. The map projection is the Transverse Mercator.
 There is a false easting established of 500,000 upon which all other easting values are based.
 The false northing value is 0.0. This value is used to measure locations that are north of the equator.
 The Central Meridian is set to 81° W (approximately 30 miles west of the Ohio and Pennsylvanian border). Each zone will have its own Central Meridian.
 The scale factor of the Central Meridian is set to 0.9996, which is a reduction of scale along this line so as to give a more accurate scale factor across the projection rather than if the scale factor was left to 1.0 along the Central Meridian. The latitude of origin is set to 0.0°.
 The Units are in meters meaning that the coordinate values are recorded in meters.