## 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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#### This document is still in preparation.

John C. Polking
<polking@rice.edu>
Last modified: Fri Jan 28 14:33:57 CST 2000