Simple circular curve by deflection angle method

A simple circular curve is set out by the deflection angle (Rankine's) method from the tangent point T1, using a theodolite and a tape. The angle between the tangent and the chord to any point on the curve is half the angle subtended by that chord at the centre.

                    IP
                   /  \
                  /    \
                 /      \
               /          \
            T1              T2
              ' .        . '
                 1  2  3
   (deflection angles d1, d2, ... are turned at T1 from the line T1-IP)

Basic formulas

For a chord $c$ and radius $R$, the tangential (deflection) angle is $\dfrac{c}{2R}$ rad:

$$\delta = \frac{1718.9\,c}{R}\ \text{minutes}$$

Elements of the curve

• Tangent length: $T = R\tan\dfrac{\Delta}{2}$
• Length of the curve: $L_c = \dfrac{\pi R\Delta}{180^\circ}$
• Chainage of T1 = chainage of IP − $T$; chainage of T2 = T1 + $L_c$

Procedure

1. Locate the IP and find T1 and T2 by measuring $T$ along the two straights from the IP.
2. Find the chainage of T1. The first peg is at the next full chainage, so the first chord $c_1$ is a sub-chord. The last chord $c_n$ is also a sub-chord. All other chords are normal (full) chords $C$ (20 m or 30 m).
3. Compute the deflection angles: $\delta_1 = \dfrac{1718.9\,c_1}{R}$, $\delta_2 = \delta_1 + \dfrac{1718.9\,C}{R}$, and so on. Each is a cumulative angle measured from the tangent T1-IP.
4. Set up the theodolite at T1 and sight the IP with the horizontal circle reading zero.
5. Turn the telescope through $\delta_1$ and measure $c_1$ from T1 along the line of sight to fix peg 1. Then set the angle $\delta_2$ and, holding the zero end of the tape at peg 1, swing the chord length $C$ until its end lies on the line of sight; this fixes peg 2. Continue in the same way.
6. Check: the last cumulative deflection angle must equal $\Delta/2$, and the last peg must fall on T2.

The method is accurate and is widely used for long curves on open ground, where all the pegs can be seen from T1. Because each peg is fixed from the previous one, errors accumulate along the curve. If the line of sight is blocked, the theodolite is shifted to a peg already set on the curve and the work continues from there.