Figure 39.3 shows the details of the transition curve TC.

The angle between the initial tangent TV and the common tangent CC₁ at the junction C of the transition curve and the circular curve is called the spiral angle (f_s).

Let us assume that the initial tangent TV is horizontal and the line OA is perpendicular to it. Draw CE perpendicular to OA Figure 39.3.

In the triangle EOC,

angle EOC = 90° - angle ECO = angle ECC₁ = angle CC₁D'

or angle EOC = f_s

In other words, the angle subtended by one-half of the transition curve at the centre is equal to f_s. In Figure 39.1 and Figure 39.2, the spiral angles are shown on both sides of the circular curve. The angle subtended by the shifted circular curve CC' at the centre is equal to (D - 2f_s).

In Figure 39.3, arc BC = Rf_s

As the arc CF is approximately equal to BC, thus

Therefore, the shift AB bisects the transition curve TC.

Now shift, s = AB = EA - EB

or s = Y -(R -R cos f_s)

where Y is equal to the distance CC₂

Therefore, s = Y -R (1 – cos f_s)

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