Optical distance principle, tangential horizontal distance

Principle: in optical distance measurement the distance to a point is found with the instrument alone, from a small angle subtended at the instrument by a known length (or from the length subtended by a known angle), instead of taping the distance. It rests on the geometry of a thin isosceles triangle:

$$D=\frac{s}{2}\cot\frac{\beta}{2}\approx\frac{s}{\beta}$$

where $s$ is the base (known length) and $\beta$ the parallactic angle at the instrument.

 Instrument                       Staff / bar
     O  <-------------- D -------------->  A
      \  beta                              | s
       \_________________________________  B

Depending on which quantity is fixed:

1. Stadia method: the angle $\beta$ is fixed by two stadia hairs in the telescope, and the intercept $s$ on the staff varies. $D=Ks+C$.
2. Subtense method: the length $s$ is fixed (a subtense bar, 2 m long) and the angle $\beta$ is measured with the theodolite. $D=\frac{s}{2}\cot\frac{\beta}{2}$.
3. Tangential method: the vertical angles to two targets a known distance apart on a staff are measured. $D=s/(\tan\theta_1-\tan\theta_2)$.

The horizontal distance and the difference in elevation are then found from the distance and the vertical angle, so detail surveys and contouring can be done faster than by chaining, especially in rough ground.

Horizontal distance by the 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 staff is too far away for the stadia hairs 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, and $V$ the vertical height of the lower target above (or below) the axis.

Horizontal distance and vertical distance

When both angles are angles of elevation, $\theta_1$ (upper target) and $\theta_2$ (lower target), with the vanes $s$ apart:

$$D=\frac{s}{\tan\theta_1-\tan\theta_2},\qquad V=D\tan\theta_2$$ $$\text{RL of staff station}=\text{RL of axis}+V-h$$

where $h$ is the height of the lower vane above the ground at the staff station. When both are angles of depression, $\alpha_1$ (lower target) and $\alpha_2$ (upper target), with $\alpha_1>\alpha_2$:

$$D=\frac{s}{\tan\alpha_1-\tan\alpha_2},\qquad \text{RL}=\text{RL of axis}-D\tan\alpha_2-s-h$$

The method needs only the vertical circle of the theodolite and can be used at long distances, but the angles must be measured carefully, because an angle error $d\theta$ is magnified in $D$ by the small difference $(\tan\theta_1-\tan\theta_2)$ in the denominator.