Theodolite theory – effect of centering error on observed angles Centering error (imperfect placement of the instrument over the station) produces what kind of error in the measured horizontal angles?

Introduction / Context:
Accurate centering of a theodolite over a ground station is essential to minimize angular errors. If the instrument is off the true station point, rays sighted to targets subtend slightly different angles than intended. This question explores how such centering error propagates into the measured horizontal angles.

Given Data / Assumptions:

• The instrument is displaced laterally by a small distance e from the true station.
• Sights are taken to targets at various bearings and distances.
• Small-angle approximations are valid for surveying-level precision analysis.

Concept / Approach:

The angular error due to centering depends on the geometry: the greater the sight length, the smaller the angular deviation caused by a fixed lateral shift at the instrument. Furthermore, the sign and magnitude depend on the pointing direction relative to the displacement vector. Hence, errors vary with azimuth and are inversely proportional to target distance, so distant targets reduce centering effects.

Step-by-Step Solution:

Verification / Alternative check:

Adjustment theory derives the same relationship, and field practice reduces centering sensitivity by taking long sights and balancing directions.

Why Other Options Are Wrong:

Equal in all angles / direction-independent: Contradicts geometry; effect clearly depends on pointing and distance.

None of these: Incorrect because option (c) states the known relationship.

Common Pitfalls:

Ignoring that longer sights mitigate centering error; assuming centering only affects orientation but not angle magnitudes between any two rays.

Final Answer:

Error that varies with the direction of pointing and inversely with the length of sight