Simple curve by offsets from long chord

Offsets from the long chord is a chain-and-tape method (no theodolite needed) in which the curve is located by perpendicular offsets measured from the long chord joining the two tangent points T₁ and T₂. It is used for short curves.

        T1 ______________ T2   <- long chord
          \  |  |  |  |  /
           \ |  |  |  | /     offsets O_x
            '-.__.__.-'

Principle

Let $O$ be the centre, $R$ the radius, $\Delta$ the deflection angle.

• Long chord: $LC = 2R\sin\dfrac{\Delta}{2}$
• Mid-ordinate (at the middle of the long chord): $O_0 = R\left(1-\cos\dfrac{\Delta}{2}\right)$
• Offset at a distance $x$ from the middle of the long chord:

$$O_x = \sqrt{R^2 - x^2} - (R - O_0)$$

The term $R - O_0 = R\cos(\Delta/2)$ is the distance from the centre to the long chord, so $O_x$ is the height of the curve above the chord at that point.

Procedure

1. Locate T₁ and T₂ by measuring the tangent length $T = R\tan(\Delta/2)$ from the IP; join them to get the long chord.
2. Measure the long chord, mark its mid point M.
3. Divide the long chord on both sides of M at equal intervals (say 5 or 10 m): $x = 0, 5, 10, \dots$.
4. Compute $O_x$ for each $x$; the offsets are symmetrical on both sides of M.
5. Set out each offset perpendicular to the long chord (using a cross-staff or optical square) and fix the curve pegs.
6. The offsets at T₁ and T₂ are zero; the offset at M is the mid-ordinate $O_0$ (check).

Use: short curves where the chord is accessible and level; accuracy is limited by the perpendicular setting.