Transition-to-circle junction: if L is the length of the transition curve on each side and R is the radius of the circular curve, what is the maximum deflection angle (with the tangent) to the junction point of transition and circle?

Introduction / Context:
Transition curves provide a gradual change of curvature between tangent and a circular curve, improving comfort and safety. Survey layout often uses the deflection angle from the tangent to locate the end of transition (junction with the circular arc).

Given Data / Assumptions:

• Single transition of length L leads into a circular curve of radius R.
• We require the deflection from tangent to the transition–circle junction.
• Standard spiral/transition geometry conventions apply.

Concept / Approach:
For common transitions (e.g., spiral approximation), the angle between the tangent and the end of transition is approximately L/(2R) radians. This comes from the integral relation between curvature and length and is a well-known field formula for setting out.

Step-by-Step Solution:

Verification / Alternative check:
Check against detailed spiral equations; the commonly used approximation for fieldwork yields the same result.

Why Other Options Are Wrong:
L/R overestimates; L/3R and L/4R underestimate the standard transition deflection; “none of these” is incorrect because L/2R is correct.

Common Pitfalls:
Mixing total central angle of curve with end-of-transition deflection; unit confusion (radians vs degrees).

Final Answer:
L/2R