For several reasons, it’s often not convenient to use latitude and longitude to describe a set of points (perhaps connected by straight lines to make up a coastline or country’s boundaries) on the Earth’s surface. One is that doing calculations using latitude and longitude—for example, determining the distance between two points—can involve complex operations such as products involving sines and cosines. For a similar distance calculation, if the points are represented on the Cartesian x-y plane, the worst arithmetic hurdle is a square root.

Latitude and longitude measures for many geographic applications do not work well for several aspects of mapmaking. Suppose you plot many points on the Earth’s surface—say, along the coastline of a small island that is a considerable distance from the equator—on a piece of ordinary graph paper, using the longitude numbers for x-coordinates and latitude numbers for y-coordinates. The shape of the island would look strange on the map (it would appear horizontally stretched) compared to how it would appear from an airplane. You would not get useful numbers if you measured distances or angles or areas on the plot. This is due to a characteristic of the spherical coordinate system: The length of an arc of a degree of longitude does not equal the length of an arc of a degree of latitude. Those lengths are almost equal near the equator, but the difference grows as you go further north or south from the equator. At the ...

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