Methods of tacheometry, tangential horizontal distance

Methods of tacheometric surveying

1. Stadia method
   • Fixed hair method: stadia hairs are at a fixed interval; the staff intercept changes with distance. Staff may be held vertical or normal to the line of sight. This is the most common method.
   • Movable hair method: the interval between the hairs is varied to read a fixed staff intercept (rarely used).
2. Non-stadia methods
   • Tangential method: two vanes on a staff and two vertical angles are measured.
   • Subtense bar method: a bar of known length and the horizontal angle subtended by it.
3. Others: Optical wedge and EDM (total station) methods are also used.

Horizontal distance by the tangential method

Two vanes on the staff are a known distance $s$ apart. The vertical angles to the vanes are measured from the instrument at A. The staff is held at B and D = horizontal distance AB.

Case 1: both angles of elevation ($\alpha_1$ lower vane, $\alpha_2$ upper vane)

                          * upper vane
                     _.-' |  s
                _.-'      * lower vane
        a2 _.-' a1  . '   |
   A _.-'-------------------- B
   |<--------- D -------->|

$$\begin{aligned} s &= D\tan\alpha_2 - D\tan\alpha_1 \\ D &= \frac{s}{\tan\alpha_2 - \tan\alpha_1} \end{aligned}$$

Case 2: both angles of depression ($\beta_1$ to the upper vane, $\beta_2$ to the lower vane, $\beta_2 > \beta_1$)

$$D = \frac{s}{\tan\beta_2 - \tan\beta_1}$$

Case 3: one elevation $\alpha$ (to the upper vane) and one depression $\beta$ (to the lower vane)

$$s = D\tan\alpha + D\tan\beta \quad\Rightarrow\quad D = \frac{s}{\tan\alpha + \tan\beta}$$

In each case the RL of the staff station is found from the vertical component $V = D\tan\alpha$ to a vane of known height: $RL_{B} = RL_{A} + h_i \pm V - h_{vane}$, with $+V$ for elevation and $-V$ for depression, and $h_{vane}$ the height of that vane above the staff foot.

The method is less accurate than the stadia method because small errors in the angles affect $D$ directly, but it needs no stadia hairs.