Equidistant conic projection

Coordinates from a spherical datum can be transformed to an equidistant conic projection with rectangular coordinates by using the following formulas,^[4] where λ is the longitude, λ₀ the reference longitude, φ the latitude, φ₀ the reference latitude, and φ₁ and φ₂ the standard parallels:

${\displaystyle {\begin{aligned}x&=\rho \sin \left[n\left(\lambda -\lambda _{0}\right)\right]\\y&=\rho _{0}-\rho \cos \left[n\left(\lambda -\lambda _{0}\right)\right]\end{aligned}}}$ 

where

${\displaystyle \rho =(G-\varphi )}$ 

${\displaystyle \rho _{0}=(G-\varphi _{0})}$ 

${\displaystyle G={\frac {\cos {\varphi _{1}}}{n}}+\varphi _{1}}$ 

${\displaystyle n={\frac {\cos {\varphi _{1}}-\cos {\varphi _{2}}}{\varphi _{2}-\varphi _{1}}}}$ 

Constants n, G, and ρ₀ need only be determined once for the entire map. If one standard parallel is used (i.e. φ₁ = φ₂), the formula for n above is indeterminate, but then

${\displaystyle n=\sin {\varphi _{1}}}$ ^[5]

The reference point (λ₀, φ₀) with longitude λ₀ and latitude φ₀, transforms to the x,y origin at (0,0) in the rectangular coordinate system.^[5]

The Y axis maps the central meridian λ₀, with y increasing northwards, which is orthogonal to the X axis mapping the central parallel φ₀, with x increasing eastwards.^[5]

Other versions of these transformation formulae include parameters to offset the map coordinates so that all x,y values are positive, as well as a scaling parameter relating the radius of the sphere (Earth) to the units used on the map.^[1]

The formulae used for ellipsoidal datums are more involved.^[6]

• List of map projections

• Table of examples and properties of all common projections, from radicalcartography.net