Bow cut

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The arc cut ( arc stroke ) is a method for determining points in geodesy .

With this method, a new point P is determined from two coordinated points A and B by measuring the distances from A and B to the new point.

Point determination (2D) with arc cut

The calculation is carried out by resolving the triangle ABP or by calculating the intersection of the two circles, the centers of which are given by A and B and the radii of which are identical to the distance measurements a and b.

The task of the arc cut has no solution if the sum of the distance measurements (a + b) is smaller than the distance between A and B or if the difference between the measured distances is greater than the distance AB. It has a solution if the sum of the distance measurements is equal to the distance between A and B, and otherwise has two solutions (in the picture P, P ').
With two measurements there are two solutions from which to choose the one that corresponds to the local position of A, B and P. It has to be clarified whether P is to the right or left of the line A, B. Without a decision-making authority, three measurements at three known points are required in order to achieve a clear determination. Overdetermination improves numerical accuracy.

Notes:
The arch cut can in principle be made in any plane. But usually point coordinates refer to sea level ( geoid ). If therefore

  1. the three points are not at the same height, the distances a and b must be reduced to the horizontal before the arc cut (with the cosine of the elevation angle ).
  2. If they are at a great height , the distances must also be reduced to sea level.
  3. In three-dimensional space, the so-called trispheration corresponds to the arc section (section of three spheres). she will
  4. with GPS even extended to the intersection of 4 spheres, because in addition to the distances to the satellites, a time synchronization error must be taken into account (so-called pseudo- routes ).

See also

literature

  • Franz Ackerl: Geodesy and Photogrammetry , Chapter 20. Verlag Georg Fromme, Vienna 1959
  • Heribert Kahmen : Surveyors . De Gruyter textbook, Berlin 1997