Geometric design: for a main circular curve of radius R with a transition (spiral/cubic-parabola) of length L, what is the lateral shift (set-back) of the circular curve due to the introduction of the transition?

Introduction / Context:
Transition curves are inserted between tangents and circular curves to provide a gradual change of curvature, improving comfort, safety, and aesthetics. Introducing a transition requires a small inward “shift” of the circular curve to preserve tangency and maintain the intended geometry. This shift must be known for accurate setting-out in the field.

Given Data / Assumptions:

• Main circular curve radius: R.
• Transition curve length: L (per side).
• Standard transition geometry (spiral or cubic parabola) and small-angle approximations apply.

Concept / Approach:
For standard highway transitions, the shift of the circular arc is given by a compact expression that depends on the square of the transition length and is inversely proportional to the radius of the main curve. The widely used relation is: shift s = L^2 / (24 R). This ensures that the combined length of tangent–transition–circular–transition–tangent fits without altering tangent points beyond the intended offsets.

Step-by-Step Solution:
Start from transition geometry and condition of equal tangency at the junction of transition and circle. Use the small-angle/spiral properties: curvature increases linearly with length along the transition. Obtain the shift formula: s = L^2 / (24 R). Select the corresponding option.

Verification / Alternative check:
Check limiting behavior: if L → 0, s → 0 (no transition → no shift). If R is very large (flatter curve), s becomes smaller, consistent with intuition.

Why Other Options Are Wrong:

• L / (24 R) and L / (12 R): linear in L; do not reflect geometric dependence.
• L^3 / (24 R) or L^4 / (24 R): overstates influence of L; not used in highway transitions.

Common Pitfalls:

• Confusing shift with spiral angle or with offset from tangent at a given chainage.

Final Answer:
L^2 / (24 R).