Editing Transverse Mercator projection (section)

==Coordinates, grids, eastings and northings==
The projection coordinates resulting from the various developments of the ellipsoidal transverse Mercator are Cartesian coordinates such that the central meridian corresponds to the ''x'' axis and the equator corresponds to the ''y'' axis. Both ''x'' and ''y'' are defined for all values of ''λ'' and ''ϕ''. The projection does not define a grid: the grid is an independent construct which could be defined arbitrarily. In practice the national implementations, and UTM, do use grids aligned with the Cartesian axes of the projection, but they are of finite extent, with origins which need not coincide with the intersection of the central meridian with the equator.

The '''true grid origin''' is always taken on the central meridian so that grid coordinates will be negative west of the central meridian. To avoid such negative grid coordinates, standard practice defines a '''false origin''' to the west (and possibly north or south) of the grid origin: the coordinates relative to the false origin define '''eastings''' and '''northings''' which will always be positive. The '''false easting''', ''E''<sub>0</sub>, is the distance of the true grid origin  east of the false origin. The '''false northing''', ''N''<sub>0</sub>, is the distance of the true grid origin north of the false origin. If the true origin of the grid is at latitude ''φ''<sub>0</sub> on the central meridian and the scale factor the central meridian is ''k''<sub>0</sub> then these definitions give eastings and northings by:
: <math>
\begin{align}
    E&=E_0+x(\lambda,\varphi),\\[5px]
    N&=N_0+y(\lambda,\varphi)-k_0 m(\varphi_0).
\end{align}
</math>
The terms "eastings" and "northings" do not mean strict east and north directions. Grid lines of the transverse projection, other than the ''x'' and ''y'' axes, do not run north-south or east-west as defined by parallels and meridians. This is evident from the global projections shown above. Near the central meridian the differences are small but measurable. The difference between the north-south grid lines and the true meridians is the [[Transverse Mercator projection#Convergence|angle of convergence]].