Tacheometry, methods, tangential formula

Tacheometric surveying is a rapid method of surveying in which the horizontal distance and the difference in elevation of a point are found from optical observations on a staff with a tacheometer, without taping or levelling.

Methods of tacheometry

1. Stadia method: fixed hair (hairs fixed, staff intercept varies) and movable hair (hairs moved to fixed marks on the staff).
2. Tangential method: vertical angles to two targets on the staff.
3. Subtense bar method: the horizontal angle subtended by a bar of fixed length.
4. Special methods: use of the Beaman stadia arc, or an EDM/total station.

Tangential method

In the tangential method the stadia hairs are not used. The telescope is pointed at two targets (vanes) on a staff, a known distance $s$ apart, and the two vertical angles are measured. The distance is calculated from the tangents of these angles. It is used when the telescope has no stadia hairs, or when the distance is too large for the stadia intercept to be read accurately.

                   _ B  (upper target)
               _ -  |
        theta1 -    | s
    O  ----------   A  (lower target)  
        theta2 \_   |
 axis ----------\__ |       horizontal
                  D

Let $D$ be the horizontal distance of the staff from the instrument axis, $V$ the vertical height of the lower target A above (or below) the axis, and $h$ the height of the lower target above the foot of the staff (the staff station Q).

Case 1: both angles are angles of elevation

The angle to the upper target is $\theta_1$ and to the lower target $\theta_2$ ($\theta_1>\theta_2$):

$$\begin{aligned} V+s&=D\tan\theta_1,\qquad V=D\tan\theta_2\\ s&=D(\tan\theta_1-\tan\theta_2)\\ D&=\frac{s}{\tan\theta_1-\tan\theta_2},\qquad V=\frac{s\tan\theta_2}{\tan\theta_1-\tan\theta_2} \end{aligned}$$

Reduced level of the staff station Q:

$$\text{RL}_Q=\text{RL of axis}+V-h$$

Case 2: both angles are angles of depression

The angle to the lower target is $\alpha_1$ and to the upper target $\alpha_2$ ($\alpha_1>\alpha_2$). The lower target is at depth $D\tan\alpha_1$ below the axis and the upper one at $D\tan\alpha_2$:

$$\begin{aligned} s&=D\tan\alpha_1-D\tan\alpha_2=D(\tan\alpha_1-\tan\alpha_2)\\ D&=\frac{s}{\tan\alpha_1-\tan\alpha_2},\qquad V=D\tan\alpha_2\ (\text{depth of the upper target})\\ \text{RL}_Q&=\text{RL of axis}-V-s-h \end{aligned}$$

since the lower target is $s$ below the upper one, so the staff station is $V+s+h$ below the axis (equivalently, $\text{RL}_Q=\text{RL of axis}-D\tan\alpha_1-h$).

Case 3: one angle of elevation $\theta$ (upper target) and one of depression $\alpha$ (lower target)

The upper target is $D\tan\theta$ above the axis and the lower target $D\tan\alpha$ below it:

$$\begin{aligned} s&=D\tan\theta+D\tan\alpha=D(\tan\theta+\tan\alpha)\\ D&=\frac{s}{\tan\theta+\tan\alpha},\qquad V=D\tan\alpha\ (\text{depth of the lower target})\\ \text{RL}_Q&=\text{RL of axis}-V-h \end{aligned}$$