# Exploring Error and Uncertainty Related to Datums and Projections Using ArcGIS

## Measuring Scale Distortion

In ArcMap, activate the Snapping toolbar (Figure 2.32). Figure 2.32: You can access the list of toolbars by right-clicking on the empty area above the map. Double-click or tap twice to view the image in a larger size.

On the Snapping toolbar, be sure that only the Point Snapping is active (Figure 2.33). If necessary, click the others to disable them. Point snapping will make measurements easier in a later step.

Turn on the labels for the populated places layer so that the map displays the name of each city. On the Tools toolbar, find the Measure tool (Figure 2.34).

Earlier, you created the circles on the indicatrix layer by using a buffer with a radius of five hundred kilometers. For consistency, change the units in the Measure tool to kilometers (Figure 2.35). Figure 2.35: The Measure tool can use several different unit types for both distance and area.

The first measurements need to establish an accurate baseline. To do this, change the Measurement Type to Geodesic (Figure 2.36). Recall that a geodesic measurement uses the spherical model of Earth when calculating distances. Figure 2.36: The geodesic measurement calculates the shortest distance between two points on a sphere and is accurate, regardless of the map projection.

Practice using the Measure tool on one of the circles on the indicatrix layer. Zoom into the circle closest to Alaska. It has a shape that is nearly a perfect circle. Then, with the Measure tool active, move the mouse cursor over one side of the circle and click once. Then, move the cursor to the opposite side of the circle and double-click to complete the line segment. The Measure tool records the information on the dialog box (Figure 2.37).

Don’t worry about getting it perfect. This step helps you practice using the Measure tool while also demonstrating the accuracy of a geodesic measurement. Figure 2.37: This example uses the circle near Alaska to test the accuracy of the geodesic measurement. Because the buffer distance was set to a radius of five hundred kilometers, the diameter of the circle should be one thousand kilometers. Double-click or tap twice to view the image in a larger size.

Next, measure the distance between Tokyo and Vancouver. You may need to zoom out to see both cities clearly (Figure 2.38). The point snapping setting should help with the accuracy of the measurement. Figure 2.38:  ArcMap uses the spherical shape of Earth, a three-dimensional surface when applying geodesic measurements. Double-click or tap twice to view the image in a larger size.

Open a blank Microsoft Excel workbook and record the geodesic length in kilometers between Tokyo and Vancouver. Also, record the scale factor by entering the following formula in the cell next to the distance in kilometers (Figure 2.39). Be sure to include the dollar signs in the second half of the equation.

• =B2/\$B\$2 Figure 2.39: The geodesic measurement is technically not a map projection. Its purpose on the table is to serve as a relatively accurate baseline to compare with the planar distance taken from the map projections. Double-click or tap twice to view the image in a larger size

On the Measure tool dialog box, change the Measurement Type to Planar (Figure 2.40). Figure 2.40: The planar distance uses the spatial information derived from the map projection to measure distances.

Once again, measure the distance from Tokyo and Vancouver. You should notice a slight difference in the distance value (Figure 2.41). Figure 2.41: ArcMap uses the map projection, a two-dimensional surface when applying planar measurements. Double-click or tap twice to view the image in a larger size.

Record the planar distance from Tokyo and Vancouver into your Excel table (Figure 2.42). Copy and paste the scale factor formula into the cell next to the distance in kilometers for The World from Space projection. Figure 2.42: This table records the differences in Kilometers between the three-dimensional geodesic measurement and the two-dimensional planar measurements of map projections. Double-click or tap twice to view the image in a larger size.

As you learned previously, the scale factor is the relationship between the principal scale and the actual scale (Figure 2.43). One uses the principal scale, based on the scale of the generating globe, to construct the map projection. Cartographers refer to a map scale measured locally as an actual scale. Figure 2.43: Scale Factor (SF) can serve as an indicator of accuracy and distortion throughout the map.

In this instance, you are not using scale ratios for actual and principal scale. Instead, you are dividing the planar map projection measurement by the geodesic measurement (Figure 2.44). Like the principal scale, the geodesic measurement is based on the scale of the generating globe. The results are similar. Figure 2.44: Dividing the planar map projection measurement by the geodesic measurement is another way to determine the scale factor at a specific location on the map.

A scale factor of 1 means that the planar map projection distance and the spherical geodesic distance are the same. A scale factor of less than one indicates that the planar map projection distance is less than spherical geodesic distance. Therefore, the map projection is distorting distances by making them smaller. A scale factor of greater than one means that the planar map projection distance is greater than spherical geodesic distance. Thus, the map projection is distorting distances by making them larger. Knowing the range of the scale factor throughout the map is a good indicator of error and uncertainty related to size and distance.

On your Microsoft Word document, record the answer to the following question as it applies to The World from Space projection:

• What does the scale factor indicate in terms of distortion for this map projection in the region between Tokyo and Vancouver?

Save the Excel workbook to your final folder. In later steps, you enter additional measurements and scale factors for multiple map projections.

As you can see, the difference between the geodesic measurement and the planar measurement of each map projection are significant. Understanding how map projections influence accuracy is especially important when conducting spatial analysis.