Ideal transition (spiral) curve – radius variation along length In an ideal transition (clothoid) curve used in highway design, the radius of curvature R at any point varies with the distance s measured from the beginning of the transition as:

Introduction / Context:
Transition curves provide a gradual change from a tangent (infinite radius) to a circular curve (constant radius). The clothoid (or spiral) is preferred because it gives a linear rate of change of curvature with distance, improving comfort and safety while simplifying setting-out.

Given Data / Assumptions:

• s = distance measured from the start of the transition.
• R = instantaneous radius of curvature at distance s.
• Clothoid property: curvature k = 1/R increases linearly with s.

Concept / Approach:
For a clothoid, k = a * s, where a is a constant depending on design speed and geometry. Hence 1/R = a * s → R = 1 / (a * s) → R is inversely proportional to s. This ensures the centrifugal acceleration changes at a constant rate (constant jerk), an ergonomic advantage for drivers.

Step-by-Step Solution:
Start from clothoid property: curvature k ∝ s.Therefore 1/R ∝ s → R ∝ 1/s.Select the option stating “R is inversely proportional to s”.

Verification / Alternative check:
Field staking often uses tabulated spiral offsets derived from the linear curvature relation.

Why Other Options Are Wrong:

• R constant: that is a circular curve, not a transition.
• R directly proportional to s: would imply decreasing curvature linearly with s, opposite of need.
• Statements tied only to the main curve radius miss the transition's internal law.

Common Pitfalls:
Interchanging “curvature” and “radius.” Remember, curvature ∝ s, hence radius ∝ 1/s.

Final Answer:
R is inversely proportional to s (curvature proportional to s)