Convergence of Meridians

The convergence of the meridians at any point of the earth ellipsoid is the angle γ^s between the tangent to the meridian at the point and that tangent to the ellipsoid at the point that is parallel to the plane of some initial meridian. The convergence is a function of the difference in longitude l of the indicated meridians, the latitude B of the point, and the parameters of the ellipsoid. An approximate expression for the convergence is given by the equation γ^s = l sin B.

The convergence of meridians in the plane of a geodetic projection or a cartographic projection is sometimes referred to as the Gauss convergence. It is the angle γ between the tangent to the representation of some meridian and the first coordinate axis—the axis of abscissae—of the given projection; this axis is usually a representation of the central meridian of the territory being mapped. In the case of conformal projections of the ellipsoid referred to isometric coordinates, γ is equal, except, perhaps, for the sign, to the argument of the derivative of the function of a complex variable that describes the projection under consideration. By disregarding infinitesimals of third and higher orders relative to l, the equation γ = γ^s is obtained.

The convergence of meridians must be known in order to carry out the numerical processing of the results of geodetic measurements and to solve various geodetic problems. On topographic maps the convergence can be defined as the angle of rotation of the kilometer grid of the map relative to the map’s margin.