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Calculation of coordinates of a point by an angular intersection

Figure 1. Angular intersection in a coordinate system

Search:
P (X_P, Y_P)


Data:
A (X_A, Y_A)
B (X_B, Y_B)

Measured:
α, β

Calculation of coordinates of a point by an angular intersection is to create a triangle with two known points A, B, and delineated the point P and the measured angles α and β.
In the first instance on the basis of a sketch of the measuring points are made a sketch by placing points in the 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 and the length* of side AB:
D_AB = √____________ Δx_AB² + Δy_AB²

*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


Using the Sine theorem we calculate the length d_AP:

d_AP sinβ = D_AB sin(α + β)⇒ d_AP = D_AB•sinβ sin(α + β)

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_APcosA_AP
Δy_AP = d_APsinA_AP

The final coordinates of point P:

X_P = X_A + Δx_AP
Y_P = Y_A + Δy_AP

Control calculations are re-enumeration 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.

Using the Sine theorem we calculate the length d_BP:

d_BP sinα = D_AB sin(α + β)⇒ d_BP = D_AB•sinα sin(α + β)


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_BPcosA_BP
Δy_BP = d_BPsinA_BP


The final coordinates of point P:
X_P = X_B + Δx_AB
Y_P = Y_B + Δy_AB

An additional control is to determine the coordinates of the angle* γ is located between the points of APB and compare it with the value calculated from the formula: γ = 200^g,0000 − (α + β).

*If you do not remember how to calculate the angle from the coordinates, click

Example

Calculate the coordinates of point P by an angular intersection.

Data:
A (150,00m; 500,00m)
B (400,00m; 200,00m)

Measured:
α = 30^g,0000
β = 50^g,0000

SOLUTION

Δx_AB = X_B – X_A = 250,00m
Δy_AB = Y_B – Y_A = -300,00m

D_AB = √____________ Δx_AB² + Δy_AB²

D_AB = √________________ 250,00² + (-300,00)²

D_AB = 390,512m


φ_AB = arctg Δy_AB Δx_AB


φ_AB = arctg -300,00 250,00

φ_AB = -55^g,77159


Δx_AB = 250,00m >0
Δy_AB = -300,00m <0
So the quadrant bearing is in the fourth quarter A_AB = 400^g,0000 + φ_AB

A_AB = 400^g,0000 + (-55^g,77159) = 344^g,22841



d_AP = D_AB•sinβ sin(α + β)

d_AP = 390,512•sin(50^g,0000) sin(80^g,0000)

d_AP = 290,344m


A_AP = A_AB + α = 344^g,22841 + 30^g,0000 = 374^g,22841


Δx_AP = d_APcosA_AP = 290,344•cos(374^g,22841) = 266,877m
Δy_AP = d_APsinA_AP = 290,344•sin(374^g,22841) = -114,353m

The final coordinates of point P:
X_P = X_A + Δx_AP = 150,00 + 266,877 = 416,877m = 416,88m
Y_P = Y_A + Δy_AP = 500,00 + (-114,353) = 385,647m = 385,65


Control
A_BA = A_AB - 200^g,0000 = 144^g,22841

d_BP = D_AB•sinα sin(α + β)

d_BP = 390,512•sin(30^g,0000) sin(80^g,0000)

d_BP = 186,412m


A_BP = A_BA − β = 144^g,22841 - 50^g,0000 = 94^g,22841


Δx_BP = d_BPcosA_BP = 186,412•cos(94^g,22841) = 16,877m
Δy_BP = d_BPsinA_BP = 186,412•sin(94^g,22841) = 185,646m



The final coordinates of point P:
X_P = X_B + Δx_BP = 400,00 + 16,877 = 416,877m = 416,88m
Y_P = Y_B + Δy_BP = 200,00 + 185,646 = 385,646m = 385,65m




Additional control

γ = 200^g,0000 − (α + β) = 120^g,0000

Δx_PB = X_B – X_P = 400,00 – 416,877 = -16,877m
Δy_PB = Y_B – Y_P = 200,00 – 385,646 = -185,647m


φ_PB = arctg Δy_PB Δx_PB


φ_PB = arctg -185,647 -16,877


φ_PB = 94^g,22841

A_PB = 200^g,0000 + φ_PB = 294^g,22841


Δx_PA = X_A – X_P = 150,00 – 416,877 = -266,877m
Δy_PA = Y_A – Y_P = 500,00 – 385,646 = 114,353m


φ_PA = arctg Δy_PA Δx_PA


φ_PA = arctg 114,353 -266,877


φ_PA = -25^g,77161

A_PA = 200^g,0000 + φ_PA = 174^g,22839


γ = A_PB – A_PA = 294^g,22841 – 174^g,22839 = 120^g,0000