10.9 THE OSCULATING CIRCLE

Figure 10.31 illustrates a transition curve T ₁PE. Through P; where the transition radius is r, a simple curve of the same radius is drawn and called the osculating circle.

Figure 10.31: Osculating circle

At T ₁ the transition has the same radius as the straight T ₁I, that is, ?, but diverges from it at a constant rate. Exactly the same condition exists at P with the osculating circle, that is, the transition has the same radius as the osculating circle, r, but diverges from it at a constant rate. Thus if chords T ₁ t = Pa = Pb = l, then:

This is the theory of the osculating circle, and its application is described in the following sections.

10.9.1 Setting Out with the Theodolite at an Intermediate Point Along the Transition Curve

Figure 10.32 illustrates the situation where the transition has been set out from T ₁ to P ₃ in the normal way. The sight T ₁ P ₄ is obstructed and the theodolite must be moved to P3 where the remainder of the transition will be set out. The direction of the tangent P ₃ E is first required from the back-angle ( ? ₃ - ? ₃).

Figure 10.32: Setting out the transition from an intermediate point

From the figure it can be seen that the angle from the tangent to the chord...