Closed loop/link traverse, misclosure balancing

Closed loop and closed link traverse

• Closed loop traverse: the traverse starts and ends at the same station and forms a polygon. Checks: sum of interior angles $=(2n-4)\times 90^\circ$, and $\Sigma\text{Lat}=0$, $\Sigma\text{Dep}=0$.
• Closed link (connecting) traverse: the traverse starts at one known station and ends at another known station, both with known coordinates, and with known reference bearings at the two ends. Checks: the computed bearing of the last line must equal its known value, and the computed coordinates of the last station must equal the known ones.

Balancing angular misclosure in a link traverse

A link traverse starts and ends on stations of known coordinates, and the bearings of the first and last reference lines are known. The angular misclosure is therefore found by carrying the bearing through the traverse.

1. Starting from the known bearing of the first reference line, compute the bearing of every line from the observed angles. For clockwise angles, bearing of a line $=$ bearing of previous line $+180^\circ+$ angle; for anticlockwise angles, bearing of previous line $+180^\circ-$ angle (adding or subtracting $360^\circ$ as needed).
2. The computed bearing of the last reference line is compared with its known bearing:

$$\text{Angular misclosure}=\text{computed bearing}-\text{known bearing}$$

3. If it is within the allowable limit (for example $\pm k\sqrt{n}$, $n$ = number of angles), distribute it equally, the correction per angle being $-\text{misclosure}/n$. The first bearing receives one correction, the second two, and so on, so the last bearing receives $n$ times the correction and then equals the known value.

After the angles are balanced, the lengths are balanced through the linear misclosure of coordinates.