Distance Between Two
    Points
 Pacing
 Chaining
 Temperature
 Standardisation
 Catenary
 Slope
 Tension
 Tacheometry
 Stadia System
 Measurement of
    Tacheometric Constants
 Refraction and Curvature
 Theodolite
    Tachaeometry
 Field Tachaeometry
 Electronic Tachaeometry

Theodolite Tacheometry

The theory discussed so far, in The Stadia System, Measurement of Tacheometric Constants and Refraction and Curvature, all applies to the situation where the staff is held vertically and the line of sight of the telescope is horizontal. It is very seldom, however, that this situation occurs in practice. Generally a theodolite is sighted to a level staff held vertically (by use of a staff bubble), which gives rise to the situation below.

Since the staff is not at right angles to the line of sight of the instrument, the intercept cut on the staff by the stadia hairs will be too large. Let the actual distance between upper and lower stadia be s and the required projection of it at right angles to IQ be s₁

\ D = Cs₁ + K, but s₁ = s cos q

In practice, the slope distance D is not often required. What we really want is S, the horizontal distance and V the vertical distance between the trunnion axis of the telescope and the point of the staff cut by the centre hair.

So now the horizontal distance S = Cs cos² q + K cos q, and the vertical component is given by V = Cs cos q sin q + K sin q. In practice these can be reduced to:

S = 100 s cos² q and

V = 100 s cos q sin q

The difference in height between the two points is given by:

DH =HI + V - CL,

and the Relative Level (R.L.) of the point is given by

RL = RL_A + HI + 100 s cos q sin q - CL

The use of these formulae gives the three dimensional location of the point. It is quite easy to determine the coordinates of the point if the bearing is measured as well as the staff intercepts and vertical angle, which of course is the standard field procedure.