Mapping equations

The actual mapping cannot usually be visualized as a true geometric projection, directly onto the mapping plane. Rather, it is achieved through mapping equations.

To represent parts of the surface of the Earth on a flat, printed map or a computer screen, the curved horizontal reference surface must be mapped onto a 2D mapping plane. The reference surface for large-scale mapping is usually an oblate ellipsoid; for small-scale mapping it is a sphere. Mapping onto a 2D mapping plane means transforming each point on the reference surface with geographic coordinates (ϕ, λ) to a set of Cartesian coordinates (x, y) that represent positions on the map plane.

A forward mapping equation transforms the geographic coordinates (ϕ, λ) of a point on the curved reference surface to a set of planar Cartesian coordinates (x, y), representing the position of the same point on the map plane (Equation 1):

The corresponding inverse mapping equation transforms mathematically the planar Cartesian coordinates (x, y) of a point on the map plane to a set of geographic coordinates (ϕ, λ) on the curved reference surface (Equation 2):

The Mercator projection (spherical assumption) (Snyder (1987)), a commonly used mapping projection, can be used to illustrate the use of mapping equations. The forward mapping equations for the Mercator projection are (Equation 3 and 4) :^(*)

The inverse mapping equations for the Mercator projection are (Equation 5 and 6):

Footnotes: 

(*) When an ellipsoid is used as a reference surface, the equations are considerably more complicated than those introduced here. R is the radius of the spherical reference surface at the scale of the map; ϕ and λ are given in radians; λ0 is the central meridian of the projection; e = 2.7182818, the base of the natural logarithms, not the eccentricity.