Laying Out a Designed Horizontal Angle: 18-Geom-A1 Surveying, December 2013, Q3

Question 3: Laying Out a Designed Horizontal Angle (20 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.

Given. Instrument station $A$ with an established reference (backsight) direction to $B$, and a designed angle $\beta_{\text{Design}} = 65^\circ45'$ to be turned to set the direction toward a new point $P$.

Find. A field procedure that realises $\beta_{\text{Design}}$ to high precision, i.e. fixes $P$ so that $\angle BAP = 65^\circ45'$ within tolerance.

Figure 3 — Turning the designed angle $\beta_{\text{Design}}=65^\circ45'$ from reference $AB$ to place point $P$.

Approach. This is a construction-layout task, best answered as a procedure with a small computed correction: set the angle approximately, measure precisely what was actually set, then shift the point by the linear equivalent of the angular residual.

Set the angle approximately. Set the total station over $A$, level carefully, and sight $B$ with the horizontal circle zeroed. Turn the instrument through $\beta_{\text{Design}}=65^\circ45'$ and set a temporary point $P'$ at the required distance along that line of sight. This first cut is limited by a single pointing.
Measure the set-out angle precisely by repetition. Measure $\angle BAP'$ by the method of repetition (several sets, face-left and face-right) to average out pointing and instrumental errors, obtaining a refined value $\beta'$. Let the small discrepancy be $$\Delta\beta = \beta_{\text{Design}} - \beta'$$
Compute and apply the linear correction. Shift the point perpendicular to $AP'$ by the offset that rotates the ray through $\Delta\beta$, with $D=\overline{AP'}$ and $\rho''=206{,}265''\,\text{rad}^{-1}$: $$P'P = D\cdot\frac{\Delta\beta''}{\rho''}$$ Measure $P'P$ off perpendicular (in the sense that increases or decreases the angle as required) to fix $P$, then re-measure $\angle BAP$ as a final check. For example, if $D=150$ m and $\Delta\beta=20''$, then $$P'P = 150\cdot\frac{20}{206{,}265} = \boxed{0.0145\ \text{m} = 14.5\ \text{mm}}$$

Step	Result
Angular residual	$\Delta\beta = \beta_{\text{Design}}-\beta'$
Perpendicular shift	$P'P = D\,\Delta\beta''/\rho''$
Worked example ($D=150$ m, $\Delta\beta=20''$)	$14.5$ mm

Concept & Theory

Topic. Precise layout ("staking") of a designed horizontal angle by the set-out-measure-correct method.

Key relations. Small-angle arc–offset relation $P'P = D\,\Delta\beta/\rho$ with $\rho''=206{,}265''$; angle by repetition to sharpen $\beta'$.

Why this works. A single angle turned on the circle is limited by the least count and one pointing. By measuring the angle actually set with high-order repetition and converting the residual to a perpendicular linear offset, the layout precision becomes that of the measured angle rather than the set angle. The radian relationship $P'P=D\,\Delta\beta/\rho$ is exact for the small residuals involved.

Common pitfall. Applying the offset along $AP'$ instead of perpendicular to it, or using $\Delta\beta$ in seconds without dividing by $\rho''$ (a factor-of-206,265 error).

Source. Wolf & Ghilani, Elementary Surveying §23 (construction surveys — staking angles & points) and §9 (angle measurement by repetition).

Question 3 of 7: Laying Out a Designed Horizontal Angle

Question 3: Laying Out a Designed Horizontal Angle (20 marks)