Angular vs linear precision in traversing

Why angular and linear accuracy must be matched

A traverse is only as accurate as its weaker measurement. If the angles are measured to $1''$ but the lengths are taped to 1 in 1000, the extra effort on angles is wasted, and the reverse is equally true. An angular error $\alpha$ moves a station sideways by $L\alpha$ at a distance $L$, which acts like a linear error. So both are made to give the same position error, giving a balanced and economical survey (same accuracy for both lets Bowditch's rule be used).

Expressions

Linear precision is the closing error divided by the perimeter, $1/N$. An angular error $\alpha$ (in radians) gives a sideways error $L\alpha$ over a length $L$, so the equivalent linear precision is $\alpha$ itself:

$$\frac{1}{N}=\alpha\ (\text{rad})=\frac{\alpha''}{206265}$$

Hence

$$\text{angular to linear: } N=\frac{206265}{\alpha''},\qquad \text{linear to angular: } \alpha''=\frac{206265}{N}$$

Examples

• Angular to linear: angles measured to $20''$ need distances to $1/N=20/206265$, i.e. $N=206265/20\approx10313$, about 1 in 10,000.
• Linear to angular: lengths taped to 1 in 5000 need angles to $\alpha''=206265/5000=41.3''$, about $40''$ (a $1'$ least-count theodolite is enough for 1 in 3438, and a $20''$ one for 1 in 10,000).
• A quick rule: $1'\leftrightarrow1/3438$, $1''\leftrightarrow1/206265$.