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The following picture shows the Transverse Mercator coordinates on the sphere, defined by the parametrisation \(p:\mathbb{R} \times (0, 2\pi) \rightarrow \mathbb{R}^3\) \[ p(u, v) = \left( \text{sech}(u) \cos(v), \tanh(u) \text{sech}(u) \sin(v) \right) . \] In a simlar way to the Mercator projection, this is more naturally a parametrisation of the sphere minus two points by a cylinder. This time the cylinder is oriented along the \(y\) axis, which means that the two missing points from the parametrisation are on the equator, rather than the north and south poles as for the Mercator projection. In this way the transverse Mercator projection gives more detail around the north and south poles, although now the equator is distorted.
