The Mercator projection

The following picture shows the Mercator coordinates on the sphere, defined by the parametrisation

$p:(0, 2\pi) \times \mathbb{R} \rightarrow \mathbb{R}^3$

$p(u, v) = \left( \cos(u) \text{sech}(v), \sin(u) \text{sech}(v), \tanh(v) \right) .$ In Problem Class 2 we presented this as a parametrisation of an open subset of the sphere by a rectangular region

$(0, 2\pi) \times \mathbb{R}$ , but it is much better to think of it as a parametrisation of the sphere (minus the north and south poles) by a cylinder, as in the diagram below, where you can move the sliders for

$u$ and

$v$ to move the black point around the cylinder. You can then see the corresponding point (coloured in red) on the sphere.