Question 6 of 12: Theodolite Axis Errors — Horizontal-Axis and Line-of-Sight Collimation

Question 6: Theodolite Axis Errors — Horizontal-Axis and Line-of-Sight Collimation (16 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(1) Horizontal-axis (trunnion) error — proof that the two-face mean cancels it.

Given. The horizontal (trunnion) axis is tilted from horizontal by a small angle $\varepsilon$; a target is observed at vertical (altitude) angle $\alpha$.

Find. The resulting horizontal-reading error, and proof that averaging Face-Left and Face-Right removes it.

1. Effect of the tilt on the horizontal direction. When the telescope is elevated to a target at altitude $\alpha$, a trunnion tilt $\varepsilon$ swings the line of sight sideways. To first order the horizontal-circle reading is displaced by $$\delta = \varepsilon\,\tan\alpha .$$ The error is zero for a horizontal sight ($\alpha=0$) and grows with the steepness of the sight — it matters only for elevated or depressed targets.
2. Sign reversal between faces. Transiting the telescope (changing face) flips the trunnion axis end-for-end, so the tilt that leaned (say) the left end up now leans it down. The displacement therefore changes sign: $$\delta_{\text{FL}}=+\varepsilon\tan\alpha,\qquad \delta_{\text{FR}}=-\varepsilon\tan\alpha .$$
3. Two-face mean. Averaging the two faces (each already reduced by $180^\circ$): $$\bar r=\frac{r_{\text{FL}}+(r_{\text{FR}}\pm180^\circ)}{2}=r_{\text{true}}+\frac{(+\varepsilon\tan\alpha)+(-\varepsilon\tan\alpha)}{2}=r_{\text{true}}.$$ $$\boxed{\text{Two-face mean is free of trunnion-axis error.}}$$
4. Estimating $\varepsilon$. Sight a well-elevated target at altitude $\alpha$ on both faces. First remove the collimation error $c$ (found on a level sight, part (2)), because on an inclined sight the face difference also contains $2c\sec\alpha$. The remaining Face-Left minus Face-Right horizontal-reading difference is $$\Delta r = r_{\text{FL}}-(r_{\text{FR}}\pm180^\circ)=2\,\varepsilon\tan\alpha\quad\Rightarrow\quad \varepsilon=\frac{\Delta r}{2\tan\alpha}.$$ Example: if $\varepsilon=20''$ and $\alpha=30^\circ$, the one-face reading error is $20\tan30^\circ=\boxed{11.5''}$, and it appears as a $2\times11.5=23''$ spread between the faces — from which $\varepsilon$ is recovered.

(2) Line-of-sight (horizontal collimation) error — check and adjustment.

Given. The line of sight may not be perpendicular to the horizontal axis, by a small collimation angle $c$.

Find. How to detect $c$ and how to remove it.

1. Two-face test on a level sight. Point a well-defined target at (or near) the same elevation as the instrument on Face-Left and read $r_{\text{FL}}$; transit and read the same target $r_{\text{FR}}$ on Face-Right. If the line of sight were perpendicular to the trunnion, the two readings would differ by exactly $180^\circ$. The departure is twice the collimation: $$2c = r_{\text{FL}}-(r_{\text{FR}}\pm180^\circ),\qquad c=\tfrac12\left[r_{\text{FL}}-(r_{\text{FR}}\pm180^\circ)\right].$$ Unlike the trunnion error, the collimation error is (to first order) independent of altitude on a level sight. Example: $r_{\text{FL}}=0^\circ00'20''$, $r_{\text{FR}}=180^\circ00'40''$ give $2c=-20''$, so $c=\boxed{-10''}$.
2. Effect and mitigation by observation. As with the trunnion error, $c$ reverses sign between faces, so the mean of two faces is free of collimation error — the practical field remedy is always to observe on both faces.
3. Physical adjustment (permanent correction). When the instrument must be corrected (e.g. for single-face work), set the horizontal circle to the collimation-free mean reading (the mean of the two faces to the target), then use the reticle’s horizontal adjusting (capstan) screws to move the vertical cross-hair back onto the target. This brings the line of sight perpendicular to the horizontal axis. Re-test and iterate until $2c$ is within tolerance. In modern electronic theodolites the residual $c$ is measured and stored as a calibration constant applied automatically.