Closed loop/link traverse, misclosure terms

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.

Angular misclosure

The difference between the sum of the observed interior angles and the theoretical value $(2n-4)\times90^\circ$ in a closed loop (or between the computed and known bearing of the last line in a link traverse).

Relative closing error

The linear error of closure divided by the total length of the traverse:

$$\frac{e}{\Sigma L}=\frac{\sqrt{(\Sigma\text{Lat})^2+(\Sigma\text{Dep})^2}}{\Sigma L}=\frac{1}{N}$$

It is expressed as 1 in $N$ and compared with the specified precision.

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.