Highway/railway transition geometry – lemniscate condition for being transitional throughout A lemniscate curve inserted between two tangents will act as a transition curve for its entire length if the polar deflection angle at the apex has what value in terms of the intersection (deflection) angle Δ between the tangents?

Introduction / Context:
The lemniscate is sometimes used as a transition curve in highway and railway alignment when the overall deflection between tangents is comparatively large. For a transition curve, the curvature should start from zero at the tangent and vary gradually to match the circular curve portion. This question asks for the specific polar-deflection requirement at the apex that ensures the lemniscate remains transitional throughout, without any portion behaving like a simple circular arc.

Given Data / Assumptions:

• A pair of straight tangents meet with intersection (deflection) angle Δ.
• A single symmetric lemniscate is used between the tangents.
• Pola r deflection angle at the apex is the maximum polar angle of the curve (measured from the initial tangent to the radius vector at the apex).
• We require curvature to increase and then decrease in the correct proportion so that throughout the used length it behaves as a transition (no constant curvature segment).

Concept / Approach:

Geometric relations of the lemniscate yield a fixed ratio between the deviation (deflection) angle of the alignment and the maximum polar deflection at the apex for which the curvature law remains transitional everywhere. Classical surveying results show that for a lemniscate used between tangents, transitional behavior throughout the curve requires the apex polar deflection to be one-sixth of the overall intersection angle Δ.

Step-by-Step Solution:

Verification / Alternative check:

Surveying references commonly tabulate the lemniscate condition as apex polar deflection = Δ/6 for fully transitional behavior; larger apex angles cause part of the curve to act effectively non-transitional over a segment, which is undesirable for comfort and safety at speed.

Why Other Options Are Wrong:

Δ/2, Δ/3, Δ/4: these overstate the apex polar angle, leading to nonuniform transition behavior and possible entry into a pseudo-constant-curvature region.

Common Pitfalls:

Confusing the apex polar deflection with the spiral angle; mixing up the lemniscate with the clothoid (cubic parabola) where different relationships apply; assuming any lemniscate is automatically transitional throughout without checking the apex condition.

Final Answer:

Δ/6