Abstract
Throughout human history there has been a close connection between mathematics and cartography to the benefit of both disciplines. Yet most undergraduate students in both areas are unaware of it. In this paper, itself a cooperative effort between a cartographer and a mathematician, we hope to show that mathematical ideas available to many first-year college students can be used to analyze properties of some commonly used map projections. Specifically, our principal mathematical tool is the Calculus of functions of a single variable. After giving a proof, using only high-school algebra and trigonometry, that there can be no fixed-scale flat map of the earth, we demonstrate how the Calculus can be used to compute the scale factors along meridians and parallels for selected cylindrical and azimuthal projections. Conditions that these scale factors should satisfy for a projection to be either conformal or equal-area are then discussed, and canonical examples of each type are exhibited. Finally, we use scale factors to analyze how each projection distorts areas and angles, and discuss how the classic tool known as Tissot's indicatrix applies to this setting.
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