Degree of curve, elements of simple curve

Degree of curve

The degree of curve $D$ is the angle subtended at the centre by a standard arc (or chord) of fixed length $C$, usually 30 m in metric practice (100 ft in foot units). It expresses the sharpness of the curve: a bigger $D$ means a smaller radius.

$$D = \frac{180^\circ\,C}{\pi R}\quad\Rightarrow\quad R = \frac{1718.87}{D}\ \ (C = 30\ \text{m})$$

Elements of a simple circular curve

              V (IP)
             /\
          T /  \ T
           / Δ  \
      T1  /      \  T2
       o'--------'o
         \ M  E /
          \ __ /
          R \  / R
             O

• T₁ (PC) and T₂ (PT): the points where the curve begins and ends.
• Deflection angle $\Delta$: angle between the forward extension of the back tangent and the forward tangent; it equals the angle subtended at the centre by the curve.
• Tangent length: $T = R\tan(\Delta/2)$.
• Length of curve: $L = \pi R\Delta/180$.
• Long chord: $LC = 2R\sin(\Delta/2)$.
• External (apex) distance: $E = R(\sec(\Delta/2) - 1)$, from IP to the mid-point of the curve.
• Mid-ordinate (versed sine): $M = R(1 - \cos(\Delta/2))$, from the mid-point of the long chord to the mid-point of the curve.
• Chainages: T₁ = IP − $T$; T₂ = T₁ + $L$.

Example: $\Delta = 40^\circ$, $D = 4^\circ$ gives $R = 429.72$ m, $T = 156.40$ m and $L = 300.00$ m.