Short note: Rankine's method of curve setting

Rankine's method (also called the deflection angle method or tangential angle method) sets out a circular curve using a theodolite and a tape. The curve is located by a series of short chords, and each point is fixed by the angle at the tangent point between the tangent and the chord to that point, together with the length of the chord.

Principle

The angle between the tangent and a chord equals half the central angle subtended by that chord. For a chord $c$ on a curve of radius $R$:

$$\delta = \frac{90^\circ\,c}{\pi R}\ \text{(degrees)} = \frac{1718.87\,c}{R}\ \text{(minutes)}$$

The deflection angle of any point from the first tangent is the sum of the tangential angles up to that point: $\Delta_n = \delta_1+\delta_2+\dots+\delta_n$.

Data to compute

1. $R$, $T = R\tan(\Delta/2)$ and $L = \pi R\Delta/180$.
2. Chainages of the first tangent point T₁ = IP − T and last tangent point T₂ = T₁ + L.
3. First sub-chord $c_1$ (from T₁ to the next full peg), full chords $c$, and last sub-chord $c_n$ (to T₂).
4. Tangential angle of each chord and the cumulative deflection angles.

Field procedure

1. Locate T₁ and T₂ by measuring $T$ back and forward from the IP along the tangents.
2. Set up the theodolite at T₁, level it, and with the vernier at zero sight the IP.
3. Set the first deflection angle $\delta_1$ and measure the sub-chord $c_1$ from T₁ with the tape; the arrow end gives point 1.
4. Set the cumulative angle $\Delta_2$; swing the tape from point 1 with length $c$ until it cuts the line of sight, fixing point 2. Repeat for all points.
5. The last cumulative angle must equal $\Delta/2$, and the last point must coincide with T₂ (check). If the error is small, adjust the last points.

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

Merits: accurate, needs only one instrument setup and is suitable for long and moderate curves. Limitation: errors accumulate along the curve; obstacles on the curve require shifting the instrument.