Setting out simple curve by Rankine's method

Rankine's method sets out a circular curve by deflection angles measured at the first tangent point T₁ with a theodolite, and chord lengths measured with a tape.

              V (IP)
             /\
            /  \
           / .. \
     T1 o'  3 4  'o T2
          1  2

Principle

Each deflection angle is the angle between the tangent at T₁ and the chord from T₁ to the point, which equals half of the central angle of the arc:

$$\delta = \frac{1718.87\,c}{R}\ \text{minutes},\qquad \Delta_n = \delta_1+\delta_2+\dots+\delta_n$$

Procedure

1. Compute $R$, $T$, $L$ and the chainages of T₁ and T₂.
2. Find the sub-chord $c_1$ (T₁ to the first full peg), the full chords $c$ and the last sub-chord $c_n$.
3. Compute the deflection angle for each chord and the cumulative angles.
4. Set the theodolite at T₁ with the plates at zero, sight V.
5. Set the first angle $\delta_1$ and measure $c_1$ from T₁ to fix point 1.
6. Set the next cumulative angle $\Delta_2$; with the tape zero at point 1 and length $c$, swing until it meets the line of sight to fix point 2. Continue to T₂.
7. The final angle must equal $\Delta/2$ and the final point must coincide with T₂.

Merit: accurate; only one instrument station is needed. Limitations: errors accumulate and there is no check except at the end; the whole curve must be visible from T₁.