Highway/railway transition curves for automobiles: The properties expected of an ideal autogenous (self-generated) path of a vehicle—such as curvature increasing linearly with arc length for smooth steering and comfort—are best represented by which transition curve?

Introduction / Context:
In highway and railway geometric design, a transition curve is inserted between a tangent and a circular arc so that curvature changes gradually. For automobiles, the ideal (often termed autogenous) steering path requires that lateral acceleration and its rate of change remain comfortable. The clothoid (also called the Cornu or Euler spiral) closely satisfies these conditions.

Given Data / Assumptions:

• Vehicle travels from straight (zero curvature) to circular arc (constant curvature).
• Driver comfort improves if curvature varies linearly with distance.
• Steering input should increase smoothly without jerks.

Concept / Approach:
The clothoid spiral has curvature k proportional to arc length s, written conceptually as k = A * s. This yields a linear growth of centripetal acceleration and a uniform rate of change (jerk), which enhances comfort and safety. Other candidate curves do not maintain this linear curvature progression over the entire transition, leading to less favorable dynamics.

Step-by-Step Solution:

Verification / Alternative check:
Design manuals specify clothoid for transition on most modern highways and railways due to its linear k–s behavior and ease of setting out by tabulated deflection angles.

Why Other Options Are Wrong:

• True spiral (generic term) is ambiguous; in practice the specific spiral used is the clothoid.
• Cubic parabola is easier to set out but does not strictly give linear curvature with arc length.
• Bernoulli's lemniscate is suited to special purposes and is not a general transition curve for roads.

Common Pitfalls:
Confusing any “spiral” with the clothoid; assuming cubic parabola is equivalent for all speeds and comfort criteria.

Final Answer:
Clothoid spiral