Meridian convergence

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Meridian convergence (here marked with ) ${\displaystyle \chi }$

The meridian convergence (to "meridian" ) is of particular importance for geodesy and mathematical cartography . It occurs as an angular deviation between the parallel grid lines of geodetic coordinate systems ( grid north) and the true north direction (true north) on the earth's surface at locations away from the reference meridian of the grid. ${\displaystyle \gamma }$

In other words: the meridian convergence is the angle between the north direction in any surface point C on the earth and the direction parallel to it in another surface point D. See also the tangents entered in the drawing .

The cause of the difference in angle is the curvature of the surface, which is why the grid lines ( meridians ) running in north-south direction converge at the poles , i.e. converge . On the other hand, the parallel directions in space deviate from the north direction in C as the difference in length increases .

The meridian convergence can be calculated as:

${\displaystyle {\begin{aligned}\gamma &=\Delta \lambda -\epsilon '\\&=\epsilon _{0}+\epsilon \end{aligned}}}$

With

• the difference in geographical longitudes ${\displaystyle \Delta \lambda }$
• the spherical excesses of the respective triangles or square (see illustration). ${\displaystyle \epsilon ,\,\epsilon ',\,\epsilon _{0}}$

For geodetic parallel coordinates (e.g. Gauß-Krüger coordinates ) the meridian convergence is the angle between grid north and geographic north. It can be calculated approximately as follows:

${\displaystyle \gamma \approx \Delta \lambda \cdot \sin \phi }$

${\displaystyle \gamma =y'/R\cdot \tan \phi }$    (Result in radians )

With

• latitude ${\displaystyle \phi }$
• Distance from the central meridian ${\displaystyle y'}$
• Earth radius . ${\displaystyle R}$

In the usual 3 ° meridian strips of the Gauß-Krüger projection, the meridian convergence in the overlapping area of ​​two strips can reach about 1 °. This must be taken into account when converting between direction angles and ellipsoidal or astronomical azimuths (see also solar azimuths and orientation (geodesy) ). In cartography it is noticeable as a gap when adjacent map sheets are placed next to one another.

Related terms

• Declination (geography) is the deviation from true north and magnetic north
• Needle deviation ( grivation ) is the angle between the grid north of a map and magnetic north .

Web links

• Online calculation of meridian convergence on various ellipsoids of revolution