Leveling

Under Verebnung a simplified calculation method is spherical triangles understood they have on plane triangles returns. The method is used in geodesy and partially in spherical trigonometry when two to three triangular angles are measured in nature or otherwise known. It gives an accuracy in the millimeter range for triangle sides of a length of up to about 100 km , i.e. H. approx. ${\displaystyle \alpha ,\beta ,\gamma }$ ${\displaystyle 10^{-8}.}$

The calculation process is as follows:

1. approximate calculation of the triangular area (provisional values ​​are sufficient for this) ${\displaystyle {\tilde {A}}}$
2. Calculation of the spherical excess by which the sum of the angles of a spherical triangle exceeds the value of 180 °: ${\displaystyle \epsilon }$

${\displaystyle \epsilon =\alpha +\beta +\gamma -180^{\circ }}$

from the approximate triangular area : ${\displaystyle {\tilde {A}}}$

${\displaystyle {\tilde {A}}={\frac {\epsilon }{180^{\circ }}}\cdot \pi R^{2}\quad \Leftrightarrow \quad \epsilon ={\frac {\tilde {A}}{\pi R^{2}}}\cdot 180^{\circ },}$

where is the radius of the sphere . ${\displaystyle R}$

3. Reduction of all measured (spherical) triangular angles by ${\displaystyle {\frac {\epsilon }{3}}}$

4. Calculation of the sides of the triangle using plane trigonometry .

In the case of a very small spherical triangle (small compared to the entire surface of the earth ) the sum of the angles only slightly exceeds the value of 180 °. So has z. For example, an equilateral triangle with sides 21 km long has a spherical excess of only 1  " (about ten times the modern measurement accuracy ). If the triangle covers almost half the spherical surface (three angles of almost 180 °), the sum of the angles is only slightly smaller than 540 ° and the excess therefore almost 360 °.

The direct connection between excess and triangular area becomes clear in the eighth of a sphere (equilateral triangle with three spherical angles of 90 ° each), where is. Such a triangle connects z. B. a quarter of the equator with the North Pole . According to the calculation rule above, the three angles are to be reduced by in each case , so that 60 ° results in the angle that a flat equilateral triangle has. ${\displaystyle \epsilon =90^{\circ }}$ ${\displaystyle {\tfrac {\epsilon }{3}}={\tfrac {90^{\circ }}{3}}=30^{\circ }}$

See also

• Curvature of the earth
• Spherical triangle , unit sphere
• spherical astronomy
• Network propagation , survey network