Vertical alignment — an ideal vertical curve for smoothly joining two different gradients in highways and railways is of which mathematical form?

Introduction / Context:
Vertical curves provide a smooth transition between ascending and descending grades in road and rail profiles. The choice of curve affects riding comfort, sight distance, and drainage. Designers typically use a mathematical form that ensures a constant rate of change of gradient, which corresponds to a constant vertical acceleration for a vehicle moving at constant speed.

Given Data / Assumptions:

• The objective is uniform comfort and predictable sight distance along the curve.
• Gradients change from g₁ to g₂ over a defined length.
• Vehicle travels at approximately constant speed over the curve.

Concept / Approach:

A parabolic profile (typically a simple quadratic y = ax² + bx + c) yields a constant rate of change of slope (second derivative constant). This means the vertical acceleration experienced at constant speed is uniform along the curve, minimizing jerk and improving comfort. The parabola also simplifies computation: offsets from tangents vary with the square of chainage, and sight distance calculations become tractable.

Step-by-Step Solution:

Verification / Alternative check:

Design manuals (highways/railways) specify parabolic vertical curves for crest and sag transitions; circular arcs are reserved primarily for horizontal alignment.

Why Other Options Are Wrong:

Circular — constant curvature suits horizontal curves, not ideal for vertical grade transitions.
Elliptical/hyperbolic — unnecessary complexity and do not ensure constant rate of grade change.

Common Pitfalls:

Confusing vertical with horizontal design practice; assuming any smooth curve suffices without considering acceleration (jerk) control.

Final Answer:

parabolic