Tacheometry principle, stadia distance and RL

Principle of stadia tacheometry: the telescope has two additional horizontal hairs (stadia hairs), one above and one below the central cross-hair. The rays through these hairs make a fixed small angle at the focus, so the staff intercept $s$ between them is proportional to the distance of the staff from the instrument.

 Diaphragm     Objective              Staff
   a |           |   \                  A  top
     | i         |     \  ...           |
  ---+-----------+-------F--------------+---- axis
     |           |      /               |
   b |           |   /                  B  bottom
     |<-- f --->|<--- D1 ---->|

Horizontal sight, staff vertical

From the similar triangles formed at the external focus $F$ (stadia hair interval $i$, focal length $f$, intercept $s$):

$$\frac{D_1}{f}=\frac{s}{i}\ \Rightarrow\ D_1=\frac{f}{i}\,s$$

Adding the distance of the focus from the instrument axis, $(f+d)$, where $d$ is the distance from the objective to the vertical axis:

$$D=\frac{f}{i}\,s+(f+d)=Ks+C$$

$K=f/i$ is the multiplying constant (usually 100) and $C=f+d$ the additive constant (about 0.3 m for an external focusing telescope, nearly zero for an internal focusing telescope and exactly zero for an anallatic lens).

Inclined sight, staff vertical

Let the line of sight be inclined at $\theta$ and the vertical staff intercept be $AB=s$. The intercept normal to the line of sight is $s'=s\cos\theta$ (the hair angle is small). The inclined distance from the axis is $L=Ks\cos\theta+C$. Then

$$\begin{aligned} D&=L\cos\theta=Ks\cos^2\theta+C\cos\theta\\ V&=L\sin\theta=\tfrac12Ks\sin2\theta+C\sin\theta \end{aligned}$$

Reduced level

For a staff station with axial hair reading $h$:

$$\text{RL of staff station}=\text{RL of instrument axis}\pm V-h$$

(+ for an angle of elevation, − for depression), where RL of axis $=$ RL of the instrument station $+$ height of instrument.

The principle is used for contouring, detail surveys and traversing in hilly ground.