Cassini projection

Considering the earth as a sphere, the projection is composed of the operations: $${\displaystyle x=\arcsin (\cos \phi \sin \lambda )}$$ $${\displaystyle \begin{array}{cc}y& =\mathrm{atan2}(\sin \phi ,\cos \phi \cos \lambda )\\ & =\arctan \left(\frac{\tan \phi }{\cos \lambda }\right)\text{restricted to }-\frac{\pi }{2}<\lambda <\frac{\pi }{2}.\end{array}}$$ where λ is the longitude from the central meridian and φ is the latitude. When programming these equations, the inverse tangent function used is actually the atan2 function, with the first argument sin φ and the second cos φ cos λ.

The reverse operation is composed of the operations: $${\displaystyle \phi =\arcsin (\sin y\cos x)\lambda =\mathrm{atan2}(\tan x,\cos y).}$$

In practice, the projection has always been applied to models of the earth as an ellipsoid, which greatly complicates the mathematical development but is suitable for surveying. Nevertheless, the use of the Cassini projection has largely been superseded by the transverse Mercator projection, at least with central mapping agencies.

Distortions

Areas along the central meridian, and at right angles to it, are not distorted. Elsewhere, the distortion is largely in a north–south direction, and varies by the square of the distance from the central meridian. As such, the greater the longitudinal extent of the area, the worse the distortion becomes.

Due to this, the Cassini projection works best for areas with greater north–south extent than east–west. For example, Ordnance Survey maps of Great Britain used the Cassini projection from 1924 until the introduction of the National Grid.