Anti-clockwise traverse PQRSP missing data

The traverse P-Q-R-S-P is anticlockwise, so each deflection is to the left and bearing of a line = bearing of previous line − left deflection. Unknowns: length QR, bearing SP (length SP is given), deflections at P and S, and the coordinates.

Step 1: Bearings that can be fixed

• PQ = S45°W = 180° + 45° = 225°00'
• QR = 225°00' − 85°00' = 140°00'
• RS = 140°00' − 100°00' = 40°00'

Latitude and departure of PQ and RS:

$$\begin{aligned} \text{PQ}:&\ 436.80\cos225^\circ=-308.86,\quad 436.80\sin225^\circ=-308.86\\ \text{RS}:&\ 499.20\cos40^\circ=+382.41,\quad 499.20\sin40^\circ=+320.88 \end{aligned}$$

Their sums are $a=73.545$ m and $b=12.015$ m.

Step 2: Length of QR and bearing of SP

Closure requires $\text{PQ}+\text{QR}+\text{RS}+\text{SP}=0$. With QR of unknown length $x$ at $140^\circ$, the vector SP is $-(a+x\cos140^\circ,\ b+x\sin140^\circ)$ and its length is 516.084 m:

$$(a+x\cos140^\circ)^2+(b+x\sin140^\circ)^2=516.084^2$$ $$x^2+2(a\cos140^\circ+b\sin140^\circ)x+(a^2+b^2-516.084^2)=0$$ $$x^2+(-97.231)x+(-260789.4)=0\ \Rightarrow\ x=561.600\ \text{m}\ \ (\text{the negative root is rejected})$$

So QR = 561.600 m.

Then SP: latitude $=-(-356.665)=+356.665$, departure $=-(+373.005)=-373.005$.

$$L_{SP}=\sqrt{+356.665^2+(-373.005)^2}=516.084\ \text{m (agrees with the given 516.084 m)},\qquad \theta_{SP}=\tan^{-1}\frac{-373.005}{+356.665}=313^\circ 43'$$

(North-west quadrant, N 46°17' W.)

Step 3: Missing deflection angles

• At P: bearing SP − bearing PQ = 313°43' − 225°00' = 88°43' left
• At S: bearing RS − bearing SP = 40°00' − 313°43' + 360° = 86°17' left

Check: 88°43' + 85°00' + 100°00' + 86°17' = 360°00', as required for a closed anticlockwise traverse.

Step 4: Coordinates with respect to R (R = 2345 mE, 6789 mN)

Going round R to S to P to Q and back to R:

The traverse closes on R, so the work is checked.

Answer: QR = 561.600 m; SP bearing = 313°43'; deflection at P = 88°43' L, at S = 86°17' L; bearings PQ 225°00', QR 140°00', RS 40°00'. Coordinates: S (7171.409 N, 2665.880 E), P (7528.075 N, 2292.875 E), Q (7219.210 N, 1984.011 E).