Calculation of coordinates of a point by a linear intersection
Figure 1. Linear intersection in a coordinate system
Search:
P (X_{P}, Y_{P})
Data:
A (X_{A}, Y_{A})
B (X_{B}, Y_{B})
Measured: d_{AP}, d_{BP}
To calculate the coordinates of point P on the basis of the coordinates of two points A and B, and measured the length of the sides of the AP and BP use a linear intersection method. First, we sketch by placing a point in a coordinate system (Figure 1.).
We calculate the northing and the easting difference Δx_{AB} = X_{B} – X_{A},
Δy_{AB} = Y_{B} – Y_{A}, which are necessary for determining bearing* A_{AB} oraz długości* boku AB:
D_{AB} = √____________
Δx_{AB}^{2} + Δy_{AB}^{2}
*If you do not remember
 method of calculating the length of the coordinates, click
 how to calculate the bearing from coordinates of two points, click
To calculate the angles α, β and γ in a triangle ABP use the cosine theorem:
D_{AB}^{2} = d_{AP}^{2} + d_{BP}^{2} –2d_{AP}d_{BP}cosγ ⇒ γ = arctg
D_{AB}^{2} –(d_{AP}^{2} + d_{BP}^{2})
–2d_{AP}d_{BP}
d_{AP}^{2} = D_{AB}^{2} + d_{BP}^{2} –2D_{AB}d_{BP}cosβ ⇒ β = arctg
d_{AP}^{2} – (D_{AB}^{2} + d_{BP}^{2})
–2D_{AB}d_{BP}
d_{BP}^{2} = d_{AP}^{2} + D_{AB}^{2} – 2d_{AP}D_{AB}cosα ⇒ α = arctg
d_{BP}^{2} – (d_{AP}^{2} + D_{AB}^{2})
–2d_{AP}D_{AB}
Ensure you have calculated the sum of the angles is equal to 200^{g},0000.
In the remainder of the calculations proceed as though the angular intersection.
Calculate the bearing angle of the side AP:
A_{AP} = A_{AB} –α
and northing and the easting difference based on the the length and bearing of the side AP:
Δx_{AP} = d_{AP}cosA_{AP}
Δy_{AP} = d_{AP}sinA_{AP}
The final coordinates of point P:
X_{P} = X_{A} + Δx_{AP}
Y_{P} = Y_{A} + Δy_{AP}
Control calculations are reenumeration of coordinates of a point P on the basis of point B and compare them with the coordinates of point P calculated on the basis of point A.
Calculate the bearing angle of the side BP:
A_{BP} = A_{AB} + β
and northing and the easting difference based on the the length and bearing of the side BP:
Δx_{BP} = d_{BP}cosA_{BP}
Δy_{BP} = d_{BP}sinA_{BP}
The final coordinates of point P:
X_{P} = X_{B} + Δx_{AB}
Y_{P} = Y_{B} + Δy_{AB}
