Principle of optical distance measurement

Principle: in optical distance measurement the distance to a point is found from the instrument alone, by measuring a small angle subtended at the instrument by a known length (or a 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 by 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 obtained from the distance and the vertical angle, so that detail surveys and contouring can be done faster than by chaining, especially in rough ground.