Subtense bar principle and formula

A subtense bar is a horizontal bar of fixed length $l$ (usually 2 m) with targets at its ends. It is mounted on a tripod over the station, levelled, and set perpendicular to the line joining it to the instrument by sighting through a small alignment telescope on the bar. The small horizontal angle $\beta$ subtended by the two targets at the theodolite is measured (to seconds, by repetition), and the distance is computed from it. It is used to measure distances over difficult ground (hilly terrain, water, thick vegetation) without taping.

 Theodolite                       Subtense bar
     O  <------------ D ---------->  A |
       \  beta                       | l (2 m)
         \_____________________________ B

Working principle (plan view)

From the isosceles triangle $OAB$, with the bar normal to the line OM (M the middle of the bar):

$$\frac{l}{2}=D\tan\frac{\beta}{2}\ \Rightarrow\ D=\frac{l}{2}\cot\frac{\beta}{2}\ \approx\ \frac{l}{\beta}\ (\beta\ \text{in radians})$$

Line of sight inclined upward by angle $\alpha$ (derivation)

The bar is always kept horizontal, and the theodolite measures the angle $\beta$ on its horizontal circle, i.e. the angle projected onto the horizontal plane. Hence the triangle OAB above is a horizontal triangle whatever the inclination of the sight, and the distance found from it is directly the horizontal distance:

$$D=\frac{l}{2}\cot\frac{\beta}{2}\qquad(\text{independent of }\alpha)$$

The vertical angle $\alpha$ is then measured to the centre of the bar (the sighting target). From the vertical triangle formed by the instrument axis, the horizontal line and the inclined line of sight:

$$\begin{aligned} V&=D\tan\alpha=\frac{l}{2}\cot\frac{\beta}{2}\tan\alpha\\ S&=D\sec\alpha\quad(\text{inclined distance, if required}) \end{aligned}$$

If the bar is not exactly normal to the line of sight in plan but is out by an angle $\delta$, the effective length is $l\cos\delta$ and the distance becomes $D=\frac{l\cos\delta}{2}\cot\frac{\beta}{2}$.

The RL of the bar station is RL of instrument axis $+V-h$, where $h$ is the height of the bar's sighting target above the station.