Geodetic computing

Under geodetic calculation , the calculation of the coordinates of points in a is the Cartesian coordinate system understood. Starting points with their coordinates and determinants for unknown new points are given. These pieces of determination are usually derived from measurement data obtained in nature.

The respective coordinate system of the computing plane (for example Gauß-Krüger coordinate system or UTM coordinate system ) or a Cartesian spatial coordinate system is used as the reference system. The coordinates are either as rectangular coordinates (x, y, z) or as relative polar coordinates in the plane (distance and direction angle between 2 points) or as relative spherical coordinates in space (spatial distance, direction and height or zenith angle).

The formula systems of trigonometry and analytical geometry are mainly used for the calculations . The geometric locations can be used for geometric interpretations .

Orientation

First main task

given : The coordinate-known point A as well as the direction angle and distance to point B.

wanted : The coordinate differences to point B.

Solution : Solve a right triangle.

Special feature : reversal task for the second main geodetic task.

Second main task

given : Two known points A (Y | X) and B (Y | X).

wanted : The direction angle t from A to B and the distance between the points.

Solution : Solve a right triangle.

Special feature : In geodesy, the direction angle counts clockwise starting from the vertical axis. This differs from the counting method in mathematics.

For calculation see Orthodrome .

Polar attachment

given : The coordinate known point A.

measured : the horizontal direction to a connection point F and the horizontal direction as well as the distance to the new point N.

wanted : Coordinates of the new point N.

Solution : Calculate the direction angle from A to N using the measured horizontal directions and the direction angle from A to F.

Calculate the coordinate differences using the direction angle from A to N and the measured distance (first main task).

Calculate coordinates of N from coordinates of A and the coordinate differences.

Straight cut

given: The coordinates are known points and the straight line as well and the straight line . ${\displaystyle P_{1}=(x_{1},y_{1})}$ ${\displaystyle P_{2}=(x_{2},y_{2})}$ ${\displaystyle {\overline {12}}}$ ${\displaystyle P_{3}=(x_{3},y_{3})}$ ${\displaystyle P_{4}=(x_{4},y_{4})}$ ${\displaystyle {\overline {34}}}$

wanted: The coordinates of the new point as the intersection of the two straight lines. ${\displaystyle P_{S}=(x_{S},y_{S})}$

Solution:

a) Calculation of the auxiliary value : ${\displaystyle d}$ ${\displaystyle d={\frac {(y_{4}-y_{3})(x_{1}-x_{4})-(y_{1}-y_{4})(x_{4}-x_{3})}{(y_{2}-y_{1})(x_{4}-x_{3})-(y_{4}-y_{3})(x_{2}-x_{1})}}}$

b) Calculation of the coordinates of the point of intersection: Two cases can occur depending on : ${\displaystyle d}$

• ${\displaystyle d=0}$or indefinite (denominator can become 0). No solution. The straight lines run parallel to each other.

• ${\displaystyle d\neq 0}$The new point lies at the intersection of the two straight lines and is calculated as follows ${\displaystyle P_{S}}$

${\displaystyle x_{s}=x_{1}+d(x_{2}-x_{1})}$

${\displaystyle y_{s}=y_{1}+d(y_{2}-y_{1})}$

Special cases: consideration of parallels to a given straight line or to both given straight lines.

Plumb point

given : The coordinates known points A and B of the straight line AB and the point C.

wanted : The coordinates of the new point N, which is the plumb line of point C on straight line AB.

Solution : Calculation using the angles ANC and BNC, which are right angles .

Special case : Point C is already on the straight line AB. In this case the plumb point and point C are identical.

Bow stroke or bow cut

given : The coordinate-wise known points A and B.

measured : The distance from A to new point N and the distance from B to new point N.

wanted : The coordinates of the new point N.

Solution : The coordinates of A and B and the measured distances define two circles. The intersection of these two circles provides the position of N.

There can be three cases:

1. No solution if the circles don't intersect. This constellation exists when the sum of the measured distances is smaller than the distance between A and B or one circle lies completely within the other. In practice, this solution can only occur if there is a gross measurement error or error.
2. A solution when both circles just touch. The sum or difference of the measured distances corresponds exactly to the distance between A and B. In practice, this case is hardly feasible and if the points are positioned accordingly, a different measuring method is used, since N would be very imprecise to determine because of the grinding cut when the arc is struck.
3. Two solutions if both circles intersect at two points. This is the normal case in practice. The unambiguous solution actually sought can only be determined if the measuring arrangement is known. In practice, the numbering of points A, B and N is chosen so that the position of N is to the left of the connection from A to B.

Forward cut

given : Two points A and B known in terms of coordinates.

measured : In stations A and B, the directions to the other standpoint and the directions to the new point N.

wanted : The coordinates of N.

Solution :

Calculate the direction angle from A to B (second main task).

Orientation of the directions based on the direction angle from A to B.

Every oriented direction from A or B to N describes a straight line.

Straight intersection of the two straight lines (A, N) and (B, N).

Backward cut

given: three coordinate fixed points , and . ${\displaystyle P_{1}=(x_{1},y_{1})}$ ${\displaystyle P_{2}=(x_{2},y_{2})}$ ${\displaystyle P_{3}=(x_{3},y_{3})}$

measured: The directions and the new point to the fixed points and (in this order, i.e .:. ) ${\displaystyle r_{N1},r_{N2}}$ ${\displaystyle r_{N3}}$ ${\displaystyle P_{N}=(x_{N},y_{N})}$ ${\displaystyle P_{1},P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle r_{N1}<r_{N2}<r_{N3}}$

wanted: The coordinates of . ${\displaystyle P_{N}}$

Solution: The angle between the direction to and the direction to , together with the distance between and (second main task), describes a circle on which , and lie. Likewise, the angle between the direction to and the direction to together with the distance between and (second main task) describes a circle on which , and lie. The position you are looking for results from the intersection of the two circles. ${\displaystyle P_{2}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{N}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{N}}$ ${\displaystyle P_{N}}$

A solution only exists if the new point does not lie on the circle ( dangerous circle ) that is defined by the three fixed points.

a) Calculating the angles and : ${\displaystyle \alpha }$ ${\displaystyle \beta }$

${\displaystyle \alpha =r_{N2}-r_{N1}}$

${\displaystyle \beta =r_{N3}-r_{N2}}$

Note : There are a large number of calculation rules for resolving the backward section. The best known are the solutions according to Cassini and Collins .

Height determination

→ Tower height determination

Three-dimensional determination

Polar attachment

With polar attachment, on a known point (X1, Y1, H1) the distance and the elevation angle (or the zenith angle) become the new point (Xn, Yn, Hn) and the horizontal angle of refraction between another known point (X2, Y2) and the New point measured. With the help of these determination pieces, the coordinates of the new point can be calculated.

3D arc

With this measurement method, the distances to an unknown point are measured from 3 known points (X, Y, H). With the help of these distances, the coordinates of the unknown point can be calculated. Alternatively, the three-dimensional arc section is also illustrated on the model of three intersecting spheres.

Note : This method is also used for GPS measurements, where the distances are determined from the transit times of the GPS signal from the satellites (the 3 known points) to the GPS receiver (the new point).

3D forward cut

given : Two points A and B known in terms of coordinates.

measured : From the standpoints A and B the horizontal directions to B and A. At the new point Pi the horizontal direction and the elevation or zenith angle.

wanted : Three-dimensional coordinates for the new point.

Solution : Calculation of the minimum spatial distance of the two crooked spatial lines that are spanned by points A and B with the aid of the measured variables. The solution is the bisection of this route.

Note : With the measured variables specified here, this method is the only one that also enables the calculation (or the measured variables) to be checked. Theoretically, the two straight lines must intersect at one point, the new point. In practice this will not be the case due to measurement errors , but the minimum distance must not exceed a certain value (which depends on the measurement accuracy).

3D reverse cut

Note : The three-dimensional backward section, in which three solid angles are measured to three fixed points, occurs in geodesy and photogrammetry . Its solution is quite demanding and ambiguous in closed form. It leads to the problem of the incision of three tori .

Shape determination

Circle determination

given : Three points known in terms of coordinates A, B and C.

wanted : The radius R and the center M of the circle, which is clearly defined by the points A, B and C.

Solution : Calculate the intersection of the perpendiculars.

See also

• Main geodetic task

Web links

• Online calculations with coordinates and polar measured values ​​(geodetic universal computer)