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Conformality, the Exponential Function, and World Map Projections

From the stereographic projection the ancient Greeks used to chart the heavens to modern circle-packing techniques that can chart the inner reaches of the human brain, conformal maps have played a central role in mathematical analysis and applications for over two millennia. It seems a shame, then, that undergraduate mathematics students rarely learn about them, the more so since a basic knowledge of calculus and some facility with trigonometry are enough to make a significant start. In this paper, we use single-variable calculus to establish the conformality of the stereographic projection and the Mercator world map. Along the way, the complex exponential function arises in a natural way that establishes it as a conformal mapping—without using any complex analysis. Furthermore, the exponential function provides a link between the symmetric structures of the two conformal maps. We conclude by indicating how one can construct conformal maps of surfaces of revolution other than the sphere.