Editing Mercator projection (section)

=== Generalization to the ellipsoid ===
When Earth is modelled by a [[spheroid]] ([[ellipsoid]] of revolution) the Mercator projection must be modified if it is to remain [[conformal map|conformal]]. The transformation equations and scale factor for the non-secant version are<ref>{{harvnb|Osborne|2013|loc=Chapters 5, 6}}</ref>
<math display=block>
\begin{align}
x &= R \left( \lambda - \lambda_0 \right) ,\\
y &= R \ln \left[\tan \left(\frac{\pi}{4} + \frac{\varphi}{2} \right) \left( \frac{1-e\sin\varphi}{1+e\sin\varphi}\right)^\frac{e}{2} \right] = 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{align}
</math>

The scale factor ''k'' 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''<sup>2</sup>, a correction of less than 1%. (The value of ''e''<sup>2</sup> 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.