## Cartography

It is not possible to map a portion of the sphere into the plane without introducing some distortion. There is a lot of evidence for this. For one thing you can do a simple experiment. Cut a grapefruit in half and eat one of the halves. Now try to flatten the remaining peel without the peel tearing. If that is not convincing enough, there are mathematical proofs. One of the nicest uses the formulas for the sum of the angles of a triangle on the sphere and in the plane. The fact that these are different shows that it is not possible to find a map from the sphere to the plane which sends great circles to lines and preserves the angles between them. The question then arises as to what is possible. That is the subject of these pages.

We will present a variety of maps and discuss the advantages and disadvantages of each. The easiest such maps are the central projections. Two are presented, the gnomonic projection and the stereographic projection. Then we will discuss the Mercator projection, still the most important map in navigation. Finally we will talk briefly about a map from the sphere to the plane which preserves area, a fact which was observed already by Archimedes and used by him to discover the area of a sphere. All of these maps are currently used in mapping the earth. The reader should consult an atlas, such as those published by Rand Macnally, or the London Times. On each of the charts in such an atlas the name of the projection used will be indicated. The variety of projections used may be surprising.

There is a quantitative way of measuring distortion, and how it changes from place to place on the sphere. The distortion ellipse provides a way of graphically displaying this information. We will compute and display the distortion ellipse for each of the maps we discuss.

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### Go to Cartography

John C. Polking <polking@rice.edu>