Subtense bar principle, fixed hair derivation

Working principle of the subtense bar

A horizontal bar of fixed length $l$ (usually 2 m) with targets at its ends is set perpendicular to the line of sight. The small horizontal angle $\beta$ it subtends is measured with a theodolite. From the isosceles triangle:

$$D=\frac{l}{2}\cot\frac{\beta}{2}$$

Since $\beta$ is measured in the horizontal plane, $D$ is the horizontal distance even when the sight is inclined. The bar is used to find distances over rough ground and to extend control.

Fixed hair (stadia) method, line of sight inclined, staff vertical

In the fixed hair method, the stadia hairs are fixed a constant distance $i$ apart in the diaphragm, and the intercept $s$ on the staff is read. The fixed hairs fix the parallactic angle, so $K=f/i$ is constant.

        A (top hair)
        |\
        | \  s
 O ----------\-------- L = K s' + C
        | /
        B (bottom hair)

Let the line of sight make an angle $\theta$ with the horizontal, and let the vertical staff intercept be $AB=s$. The intercept normal to the line of sight is $s'=s\cos\theta$, so the inclined distance from the instrument axis to the staff is

$$L=Ks'+C=Ks\cos\theta+C$$

Horizontal distance and height difference:

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

The reduced level of the staff station is

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

where $h$ is the axial hair reading (use $-V$ for an angle of depression).