Back to Curves and Surfaces homepage
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.
