In 1569, a flemish cartographer named Gerardus Mercator found a way of mapping the curved surface of the earth onto a flat surface. This projection, now known as the Mercator projection, is a map, which means that it preserves angles. To see how the Mercator projection works, we consider the follwing parametrization of a sphere: where -π ≤ u ≤ π and -π ≤ v ≤ π. To create a flat map of the sphere, we wrap a circular cylinder around the sphere and project onto it. After we map the sphere onto the cylinder, we can then cut the cylinder lengthwise and unroll it, which will give us a flat map. The parametrization of this cylinder can be written as: We can use the fact that the map is conformal (i.e. angle-preserving), to solve for the function f(v). The partial derivative vectors for the sphere are: Source.