Transition–circular junction geometry – offset from tangent in terms of shift For a transition curve joining a circular curve, the perpendicular offset from the tangent to the junction point (transition–circular junction) is equal to which multiple of the shift S of the curve?

Introduction / Context:
Providing a transition curve between a tangent and a circular arc requires moving the circular arc slightly inward. The lateral movement is called the shift S. Field setting frequently uses simple relationships to compute offsets at key points such as the transition–circular junction. This question asks for the factor that relates the perpendicular offset from the tangent at the junction to the shift S.

Given Data / Assumptions:

• Standard transition curve (e.g., cubic parabola/clothoid) of length L joins a circular curve of radius R.
• Shift S is defined as S = L^2 / (24 * R) for one transition end.
• Perpendicular offset y from the tangent to a point on the transition at a distance along the curve ℓ is y = (ℓ^3) / (6 * R * L); at the junction, ℓ = L.

Concept / Approach:

At the junction point (where the transition meets the circular arc), substitute ℓ = L into the offset expression to obtain y_junction = L^2 / (6 * R). Compare this with the definition of shift S = L^2 / (24 * R). The ratio y_junction / S then equals (L^2 / (6R)) / (L^2 / (24R)) = 4. Therefore, the perpendicular offset from the tangent to the junction point equals four times the shift.

Step-by-Step Solution:

Verification / Alternative check:

Design examples in highway engineering texts consistently derive the same relationship for standard cubic-parabola transitions; numerical checks with practical L and R values produce offsets that are exactly 4S at the junction.

Why Other Options Are Wrong:

Shift, twice the shift, thrice the shift: each underestimates the true junction offset, leading to an incorrect placement of the circular arc and subsequent misfit of tangency conditions.

Common Pitfalls:

Confusing the offset at the junction with the maximum offset within the transition; using the wrong definition of S (per end vs both ends) and mixing units for L and R.

Final Answer:

Four times the shift