Highway geometric design – length–radius relation for an ideal transition curve For an ideal transition curve (clothoid/spiral), the curvature varies linearly with the curve length measured from the tangent point. What is the correct proportional relationship between the transition length l and the instantaneous radius r?

Introduction:
Transition curves provide a gradual change from straight alignment (infinite radius) to a circular curve (constant finite radius), improving comfort and safety by controlling lateral acceleration and its rate of change (jerk). The classic ideal transition is the clothoid (spiral), widely used in highway and railway design.

Given Data / Assumptions:

• Ideal transition curve with linear curvature law.
• Curvature k defined as 1/r.
• Length l measured from the tangent point along the transition.

Concept / Approach:

For a spiral, curvature increases uniformly with length: k ∝ l. Since k = 1/r, it follows that 1/r ∝ l, or equivalently r ∝ 1/l, giving l ∝ 1/r. This ensures that the lateral acceleration (V^2/r) grows linearly with distance/time at constant speed and the jerk remains constant, satisfying comfort criteria.

Step-by-Step Solution:

Verification / Alternative check:

Design formulas for spiral length based on allowable rate of change of centrifugal acceleration also imply curvature increasing linearly with l, consistent with l ∝ 1/r.

Why Other Options Are Wrong:

Direct proportionality to r or r^2 contradicts the linear curvature law; inverse-square relations are not used for transition design; independence from r is incorrect.

Common Pitfalls:

Confusing the spiral with a cubic parabola in Cartesian form; mixing up the relations among curvature, radius, and length.

Final Answer:

l ∝ 1 / r