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Elements of simple circular curve

Short answer

Elements: Tangent length T = R tan(Δ/2), curve length L = πRΔ/180, long chord LC = 2R sin(Δ/2), external distance E = R(sec(Δ/2)-1), mid-ordinate M = R(1-cos(Δ/2)). Chainage of T1 (PC) = IP - T, chainage of T2 (PT) = T1 + L.

Find the elements of simple circular curve.

A simple circular curve is a single arc of constant radius that connects two straights (tangents) meeting at the point of intersection (IP).

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

With RR = radius and Δ\Delta = deflection angle (angle between the forward extension of the first tangent and the second tangent), the elements are:

ElementFormula
Tangent length (T₁V = VT₂)T=Rtan⁡Δ2T = R\tan\dfrac{\Delta}{2}
Length of curveL=πRΔ180L = \dfrac{\pi R\Delta}{180}
Long chord (T₁T₂)LC=2Rsin⁡Δ2LC = 2R\sin\dfrac{\Delta}{2}
External distance (apex distance)E=R(sec⁡Δ2−1)E = R\left(\sec\dfrac{\Delta}{2}-1\right)
Versed sine (mid-ordinate)M=R(1−cos⁡Δ2)M = R\left(1-\cos\dfrac{\Delta}{2}\right)
Chainage of T₁IP chainage − TT
Chainage of T₂Chainage of T₁ + LL
Angle at centreΔ\Delta (equal to the deflection angle)

Other terms: T₁ is the point of curve (PC) and T₂ the point of tangency (PT); the interior (intersection) angle is 180∘−Δ180^\circ - \Delta; the degree of curve is D=1718.87/RD = 1718.87/R for a 30 m arc.

✓ Answer checked by Claude Opus 5.5 (high effort)

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