Cassini projection

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.[3][4]

Cassini is known as a spherical projection, but can be generalised as an elliptical form.

Considering the earth as an ellipse, the projection is composed of these operations:

${\displaystyle N=(1-e^{2}\sin ^{2}\varphi )^{-1/2}}$

${\displaystyle T=\tan ^{2}\varphi }$

${\displaystyle A=\lambda \cos \varphi }$

${\displaystyle C={\frac {e^{2}}{1-e^{2}}}\cos ^{2}\varphi }$

${\displaystyle x=N\left(A-T{\frac {A^{3}}{6}}-(8-T+8C)T{\frac {A^{5}}{120}}\right)}$

${\displaystyle y=M(\varphi )-M(\varphi _{0})+(N\tan \varphi )\left({\frac {A^{2}}{2}}+(5-T+6C){\frac {A^{4}}{24}}\right)}$

and M is the meridional distance function.

The reverse operation is composed of the operations:

${\displaystyle \varphi '=M^{-1}(M(\varphi _{0})+y)}$

If ${\displaystyle \varphi '={\frac {\pi }{2}}}$ then ${\displaystyle \varphi =\varphi '}$ and ${\displaystyle \lambda =0.}$

Otherwise calculate T and N as above with ${\displaystyle \varphi '}$, and

${\displaystyle R=(1-e^{2})(1-e^{2}\sin ^{2}\varphi ')^{-3/2}}$

${\displaystyle D=x/N}$

${\displaystyle \varphi =\varphi '-{\frac {N\tan \varphi '}{R}}\left({\frac {D^{2}}{2}}-(1+3T){\frac {D^{4}}{24}}\right)}$

${\displaystyle \lambda ={\frac {D-T{\frac {D^{3}}{3}}+(1+3T)T{\frac {D^{5}}{15}}}{\cos \varphi '}}}$

• Cassini Grid

• Media related to Cassini projection at Wikimedia Commons
• Table of examples and properties of all common projections, from radicalcartography.net
• Ordnance Survey GeoFacts on the Cassini Projection
• Cassini dans proj4