We begin with the familiar spherical parametrisation of the unit sphere , where is a diffeomorphic map from an open neighbourhood which is obtained by rotating a semicircle on the xz-plane around the z-axis.
Interestingly, by making the change of coordinate, and
we can show that a new parametrisation of the coordinate neighbourhood can be derived.
First, we will start by putting . Then, where we have used the half-angle formulae for our tangent function.
Hence, gives a diffeomorphic map that parametrises the unit sphere.
In fact, we can also show that the coefficients of the first fundamental form in the parametrisation are isothermal, hence locally conformal to an open neighbourhood of .
Now that you know that is a conformal (angle-preserving) mapping, what can you say about the meridians (lines of constant longitude) of the unit sphere, mapped under the Mercator's projection?