Forward cut

This article explains the plan forward cut ; for the spatial forward section see photogrammetry .

Forward incision for point determination

Triangle construction ABN over the angles φ and ψ

The plane forward cut is a trigonometric method for determining points in geodesy . This is done by measuring the direction from two locations A and B to a new point N. The coordinates of the two points A and B must be known.

Due to the clear representation as the intersection of two straight lines, the term "forward cut" or "forward incision" is also explained.

The calculation is carried out by dissolving the triangle ABN or by calculating the intersection point N of the two rays that run from the respective standpoints A and B to the new point.

First the triangle angles and are calculated from the measured directions: ${\displaystyle \psi }$ ${\displaystyle \phi }$

${\displaystyle \psi =r_{AB}-r_{AN},\quad \phi =r_{BN}-r_{BA}}$

From the given coordinates and the points A and B, the direction angle and the base distance can be calculated (attention: the coordinate system is given geodetically with the y-axis to the right and the x-axis up): ${\displaystyle y_{A},x_{A}}$ ${\displaystyle y_{B},x_{B}}$ ${\displaystyle \phi _{AB}}$ ${\displaystyle s_{AB}}$

${\displaystyle \phi _{AB}=\arctan {\frac {y_{B}-y_{A}}{x_{B}-x_{A}}},\quad s_{AB}={\sqrt {(y_{B}-y_{A})^{2}+(x_{B}-x_{A})^{2}}}}$

The sides of the triangle can be calculated with the sine law :

${\displaystyle a={\frac {s_{AB}\sin \psi }{\sin(\psi +\phi )}},\quad b={\frac {s_{AB}\sin \phi }{\sin(\psi +\phi )}}}$

The following applies to the direction angles at points A and B:

${\displaystyle \phi _{AN}=\phi _{AB}-\psi ,\quad \phi _{BN}=\phi _{BA}+\phi }$

The calculation of the coordinates of the new point is now done by polar attachment: ${\displaystyle y_{N},x_{N}}$

${\displaystyle y_{N}=y_{A}+b\sin \phi _{AN},\quad x_{N}=x_{A}+b\cos \phi _{AN}}$

As a test, you can now calculate the coordinates from point B:

${\displaystyle y_{N}=y_{B}+a\sin \phi _{BN},\quad x_{N}=x_{B}+a\cos \phi _{BN}}$

A variant of the forward cut is the sideways cut , where one of the measurements in point A or B is replaced by an angle measurement in the new point itself.

Forward incision to determine length

Cutting ahead to determine length

The forward incision to determine length is a trigonometric method of determining length. This is done by measuring the direction from two locations A and B to the two end points C and D of a route. The distance between A and B must be known.

As can be seen in the figure, the angles , , and , and the length of the path a, known. ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle \delta }$

The calculation is carried out by resolving the triangles ABC, ABD and finally ACD.

First, the distance e is calculated in the triangle ABC using the sine law:

${\displaystyle {\frac {e}{\sin \delta }}={\frac {a}{\sin(180^{\circ }-\alpha -\delta )}}}$

so

${\displaystyle e={\frac {a\cdot \sin \delta }{\sin(180^{\circ }-\alpha -\delta )}}}$

Now one can calculate the distance d in the triangle ABD using the sine law:

${\displaystyle {\frac {d}{\sin \gamma }}={\frac {a}{\sin(180^{\circ }-\beta -\gamma )}}}$

so

${\displaystyle d={\frac {a\cdot \sin \gamma }{\sin(180^{\circ }-\beta -\gamma )}}}$

Now you have two sides and an angle in the triangle ACD and you can calculate the desired side c with the cosine law:

${\displaystyle c^{2}=d^{2}+e^{2}-2de\cdot \cos(\beta -\alpha )}$

See also

• Reverse cut , cutting method