Mercator projection
Generalization to the ellipsoid

When the Earth is modelled by a spheroid (ellipsoid of revolution) the Mercator projection must be modified if it is to remain conformal. The transformation equations and scale factor for the non-secant version are x = R (λ − λ 0), y = R ln = R (sinh − 1 (tan φ) − e tanh − 1 (e sin φ)), k = sec φ 1 − e 2 sin 2 φ. {\displaystyle {\begin{aligned}x&=R\left(\lambda -\lambda _{0}\right),\\y&=R\ln \left=R\left(\sinh ^{-1}\left(\tan \varphi \right)-e\tanh ^{-1}(e\sin \varphi)\right),\\k&=\sec \varphi {\sqrt {1-e^{2}\sin ^{2}\varphi }}.\end{aligned}}}

The scale factor is unity on the equator, as it must be since the cylinder is tangential to the ellipsoid at the equator. The ellipsoidal correction of the scale factor increases with latitude but it is never greater than e, a correction of less than 1%. (The value of e is about 0.006 for all reference ellipsoids.) This is much smaller than the scale inaccuracy, except very close to the equator. Only accurate Mercator projections of regions near the equator will necessitate the ellipsoidal corrections.

The inverse is solved iteratively, as the isometric latitude is involved.
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