Maps: Geometry in Geography.

MOST teachers, and some students, know that the word "mapping," used syn onymously with "function," comes from a geographical context, for the many kinds of maps in use today are actually graphs of functions from subsets of the earth's sur face to sheets of paper. It is evident that some distortion is in volved when a spherical region is repre sented on a plane sheet of paper, with the amount of distortion increasing as the re gion to be represented increases. Further investigation reveals that the function itself, and not merely its domain, influences the features of the paper map. Some functions, for example, preserve the measures of angles, while others may depict regions of equal area on the earth as regions of equal area on the paper. The general subject of maps and their properties is too extensive to elaborate on here, and the interested reader should con sult a public library. The purpose of this article is to describe in detail one particular kind of map, called polar stereographicpro jection. This map involves a good deal of tenth-grade geometry, some three-dimen sional coordinate geometry, a technique from transformation geometry, and, most important, considerable opportunity for the use of intuition. The author has found the topic to be of interest to high school juniors and seniors, who are usually sur prised that mathematics familiar to them appears in a subject that is ostensibly not very mathematical. Imagine a sphere E with center C(0, 0, 5) and radius \ resting tangent to the .xy-plane at 5(0, 0, 0). S denotes the South Pole, and the North Pole, TV, has coordinates (0, 0, l) (fig. I). The polar stereographic projection /maps E { } onto the jty-plane accord ing to the following rule: For each point P(x, y, z) of ?, except for N,j[P) = P\x\ y', 0), where P' is the point where ray NT punctures the xy-plane. This mapping, which is also of great im portance in complex variable theory, can be studied completely analytically, but it is worthwhile to begin with some intuitive analysis. A few properties fhat students eas ily discover are: (l) the parallels of latitude are sent to concentric circles about S in the .xy-plane; (2) the meridians of longitude (great circles through and S) are sent to lines through S; (3) distortion is small for the Antarctic region, but much greater as one proceeds north. Jhese properties are confirmed by viewing an actual polar stereographic map of the Southern Hemi sphere, obtainable from most libraries. We then proceed to two less obvious re sults by asking first, "What would be the image of a small (i.e., not great) circle through the North Pole?" (d in fig. I ) Such a question is not geographically important, but students readily accept the question as a geometrical challenge. Sometimes a stu dent will come up with the answer, but usually a reminder that a circle is a plane figure is necessary. (Since all rays?? lie in the plane of the circle, they will puncture the xy-plane in collinear points.) The answer to the next question is even less obvious: "What would be the image of a circle that doesn't go through the North Pole?" (C2 in fig. I ) After some stewjng and arguing, students agree that the image