“The world is round; maps are flat.” So begins an introduction to mathematical cartography (McDonnell 1979) and so also do we find an interesting problem for geometers. Coordinate charts for surfaces are named after cartographic maps. In this chapter we consider the motivating examples for coordinate charts, the classical representations of the Earth, thought of as a sphere. The subject has a history almost as old as geometry itself.

The basic problem is to represent a portion R of the globe (idealized as the unit sphere S2) on a flat surface, that is, to give a mapping Y: (R ⊂ S2) → ℝ2 that is injective and differentiable, and has a differentiable inverse. Such a mapping Y is called a map projection. A coordinate chart for the sphere, x: (U ⊂ ℝ2) → S2, with a differentiable inverse, determines a map projection by taking Y = x−1, and conversely, a map projection determines a coordinate chart.

The purposes of cartography, such as navigation or government, determine the properties of interest of a map projection. An ideal map projection is one for which all relevant geometric features of the sphere are preserved in the image.