In this paper, the following formulae are derived for the cylindrical and conic projections:
Cylindrical: $T(\phi, \lambda)= (\lambda, h(\phi))$ for some function $h$.
Examples include:
- Equi-rectangular: $T(\phi, \lambda)= (\lambda, \phi)$
- Lambert's equal area: $T(\phi, \lambda)= (\lambda, \sin(\phi))$
- Gnomonic: $T(\phi, \lambda)= (\lambda, \tan(\phi))$
- Mercator: $T(\phi, \lambda)= (\lambda, \ln{(|\sec(\phi)+\tan(\phi)|)})$
Conic: $$ \begin{aligned} x&=\rho\sin(n(\theta-\theta_0) \\ y &= \rho_0-\rho\cos{(n(\theta-\theta_0))} \end{aligned} $$ where $\theta_0$ is the central meridian, $\rho_0$ is the distance between C and the circular arc representing the North pole, and $n$ is the ratio of the angle between meridians and their true angle (the cone constant).
$\theta_0$ is usually chosen by the cartographer whilst $\rho_0$ and $n$ are dictated by the desired properties of the map.
Examples include the Lambert conformal conic projection, Albers conic, Equidistant conic, etc.
Question How can we derive these general formulae for the cylindrical and conic map projections?