The Mercator Projection
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 conformal map, which means that it preserves angles. To see how the Mercator projection works, we consider the follwing parametrization of a sphere:
X(u,v) = (cos(u)sin(v), sin(u)sin(v), cos(v)),
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:
Y(u,v) = (cos(u),sin(u),f(v))
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:
Xu(u,v) = (-sin(u)sin(v), cos(u)sin(v), 0)
Xv(u,v) = (cos(u)cos(v), sin(u)cos(v), -sin(v))
So, the metric coefficients for the sphere are
g11(u,v) = sin2(v) g22(u,v) = 1 g12(u,v) = 0
For the cylinder, the partial derivative vectors are:
Yu(u,v) = (-sin(u), cos(u), 0)
Yv(u,v) = (0, 0, f'(v))
So, the metric coefficients for the cylinder are
g11*(u,v) = 1 g22*(u,v) = f'(v)2 g12*(u,v) = 0
Because the map is conformal, the following conditions must be satisfied:
g11(u,v) = λg11*(u,v)
g22(u,v) = λg22*(u,v)
g12(u,v) = λg12*(u,v)
The third condition is immediately satisfied, since g12(u,v) = g12*(u,v) = 0, but the first equation implies that λ = sin2(v). Plugging this value into the second condition, we get f'(v)2 = 1/sin2(v). We then solve for f(v) by integrating:
f(v) = ∫f'(v)dv = ∫(1/sin(v))dv = ln(tan(v/2))
The parametric equation for the cylinder is, therefore,
Y(u,v) = (cos(u), sin(u), ln(tan(v/2))