Surveying I– coordinate account

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Calculation of coordinates of points on a straight line

Points on the straight line are the points of measurement matrix, marked out on the line between points A and B with known coordinates.

Figure 1. Point n on the straight line AB

Search:
n (X_n, Y_n)


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

Measured:
d_An
d_AB


The task of calculating coordinates of points on the straight line should start from the coordinates designate the distance between points A and B, the following formula: D_AB = √________ Δx² + Δy², where Δx = X_B – X_A, Δy = Y_B – Y_A. n, between the length of the measured d_AB and calculated D_AB you calculated deviation f, which must meet the criteria specified in the User's G-4 such that f ≤ f_l, where f = Ιd_AB – D_ABΙ. If your deviation is within the limit, we can proceed further accounts.

We calculate northing and the easting difference:

Δx_An = d_AnΔx d_AB


Δy_An = d_An Δy d_AB

Coordinates of the point lying on a straight line are:

X_n = X_A + Δx_An
Y_n = Y_A + Δy_An

Control calculation:
1) re-calculate the coordinates of a point n on the line, this time using the coordinates of point B;
2) verify that the sum of the differences Δd abscissa is equal to the length measured d_AB:

∑Δd_i = d_AB

where Δd_i = d_i – d_i-1.

Example

Calculate the coordinates of the point n situated on the straight line AB.

Data:
A ( 623,33m; 740,87m)
B ( 729,65m; 973,56m)

Measured:
d_An = 83,89m
d_AB = 255,85m

SOLUTION

Δx_AB = X_B – X_A = 106,32m
Δy_AB = Y_B – Y_A = 232,69m


D_AB = √____________ Δx_AB² + Δy_AB²


D_AB = √________________ 106,32² + 232,69²


D_AB = 255,83m


f = Ιd_AB – D_ABΙ = Ι255,85 – 255,83Ι = 0,02m
f_l = 0,09m

The condition f ≤ f_l is satisfied.



Δx_An = d_AnΔx_AB d_AB

Δx_An = 34,86m


Δy_An = d_An Δy_AB d_AB


Δy_An = 76,30m


X_n = X_A + Δx_An = 623,33 + 34,86 = 658,19m
Y_n = Y_A + Δy_An = 740,87 + 76,20 = 817,17m

Control

1) Δx_BA = X_A – X_B = –106,32m
Δy_BA = Y_A – Y_B = –232,69m


D_AB = √____________ Δx_BA² + Δy_BA²


D_AB = √____________________ (–106,32)² + (–232,69)²


D_AB = 255,83m


f = Ιd_AB – D_ABΙ = Ι255,85 – 255,83Ι = 0,02m
f_l = 0,09m

The condition f ≤ f_l is satisfied.

d_nB =d_AB – d_An = 171,96m


Δx_nB = d_nBΔx_AB d_AB

Δx_nB = – 71,96m


Δy_nB = d_nB Δy_AB d_AB


Δy_nB = – 156,41m


X_n = X_B + Δx_nB = 729,65 – 71,96 = 658,19m
Y_n = Y_B + Δy_nB = 973,56 – 156,41 = 817,17m



2) Δd_An = 83,89m
Δd_nB = 171,96m

∑Δd_i = 83,89 + 171,96 = 255,85
So: ∑Δd_i = d_AB