Using Scaling to Minimize Distortion in Latitude/Longitude Projects

As an approximation, the distance covered by one degree of latitude at the equator is equal to the distance covered by one degree of latitude at the poles, and is approximately 69 miles. This distance between degrees of latitude remains nearly constant over the globe, although it does vary slightly because the earth is not a perfect sphere. However, the distance between a degree of longitude decreases from the equator to the poles. For any latitudinal position, you can determine the length, in miles, between degrees of longitude based on the formula

Distance covered by 1° of longitude (in miles) = cosine (latitude) x 69.172.

This equation assumes a 1866 Clarke reference ellipsoid.

This table illustrates the change as you move from the equator to the poles.

Latitude	Distance Covered by One Degree of Longitude
0° (equator)	69.172 miles
30°	59.904 miles
60°	34.586 miles
90° (poles)	0 miles

Substitutions for units other than miles:

For kilometers, you can substitute the number 111.321 for 69.172 in the formula above.
For meters, you can substitute the number 111,321 for 69.172 in the formula above.
For feet, you can substitute the number 365,228 for 69.172 in the formula above.

So, how can you put this information to use? Remember that you are plotting degrees of unprojected latitude and longitude, but what you really want to show on the map are the correct distances. You must scale the longitude values correctly for the correct distances to be represented on the map. The scaling factor to apply for maps is based on the cosine of the latitude for the area you are working on.

To determine the scaling factors:

Find the latitude for the parallel through the center of the map, and determine the cosine corresponding to this latitude value. The easiest way to determine the latitude is to move the pointer to the center of the map, and read the latitude value from Coordinates section of the status bar.
Choose the Map | Project Limits command and set the X Axis Scaling you want to use on the map. You can set either the Length value or the Map Units value.
Uncheck the Set Proportional XY Scaling check box.
Multiply the X Axis Scaling - Map Units value by the cosine of the latitude determined in step 1, and enter this number into the Y Axis Scaling - Map Units field.
Click OK and the map is drawn with the scaling you want to use.

Consider a map of the state of Montana. When you plot the map on a one to one scale, the map appears stretched in the east-west direction. To understand this problem, consider that for Montana the latitude ranges from 44.36° to 49°. The latitude for the center of the map is determined from this to be 46.68°. The cosine of 46.68° is 0.686. The distance covered by one degree of longitude at this latitude is only 0.686 times the distance covered by one degree of latitude. To reduce the distortion on this map, you can correct for this difference.

Let’s say you are plotting the map at an X scale of 1" = 2 map units (longitude). For the map to be scaled appropriately, you would plot the Y scale at 1" = 1.372 map units (latitude, 2 x 0.686 = 1.372). This effectively stretches the map in the latitude (N-S) direction. Now the map distances are nearly the same in the longitude and latitude directions.

For more information on converting units, refer to Dent, Appendix A.