Geodesics on an ellipsoid

As geodesic main tasks are understood in geodesy two major types of coordinate transformation, namely those of rectangular to polar coordinates and vice versa.

First and second main task

The first main task (polar ⇒ right angles ) corresponds to the transmission of measurements ( direction and distance) to the planar coordinate system of a plan or a map.

As rectangular coordinates in addition to the plane coordinates of a map (x, y) to understand the geographic or geodetic coordinates on the Erdellipsoid because these coordinate lines intersect (latitude and longitude ) on the Earth's surface at right angles.

The second main task ( ⇒ rectangular polar coordinates) corresponds, eg, the computation of direction and distance between two measurement points. The position of topography or other features are usually specified as Gauss-Krueger or UTM coordinates in meters as (x, y or Northern, Eastern).

Point distances and accuracy

Are the mutual distances D between two points - or their coordinate differences dx, dy - no greater than about 5 km, then they can be converted directly into the other forms of coordinates:

Up to distances of a few kilometers, this is about to centimeter accuracy. For larger distances, the projection distortion must be taken into account, and from around 20 km you have to go on to more complicated formulas:

Depending on the complexity of computing surface used ( plane, sphere, reference ellipsoid, geoid ) must therefore be designed differently, the mathematical formulation of the main tasks. In the plane - as it satisfies some of the simple measurement of land and for not too accurate maps - it is limited to plane angle functions - as in the example above formula.

The analogous problem on the globe already needed some formula rows from the Spherical trigonometry, while the exact solution of the problem on the Erdellipsoid even requires a formula apparatus of about 1 page. In geodesy, geophysics or long-range navigation such calculations on doubly curved surfaces are unavoidable. The same task on the geoid or complicated shaped celestial bodies such as Mars or some asteroids is even only iteratively solved. Several geodesic active mathematicians of the last centuries ( for example, Gauss, Bessel, Legendre, Laplace, Hilbert ) or recently Grafarend and others have worked for appropriate solutions.