Vertical curves in highway design: What is the mathematical shape generally adopted for vertical curves used to connect different gradients on highways?

Introduction / Context:
Vertical curves provide smooth transitions between grades for comfort, sight distance, and drainage. The geometric choice influences the uniformity of rate of change of grade and headlight/sight distance calculations on crest and sag curves.

Given Data / Assumptions:

• Standard highway practice per IRC/AASHTO principles.
• Need a curve with constant rate of change of grade.
• Crest (summit) and sag vertical curves both considered.

Concept / Approach:
A simple parabola yields a constant rate of change of grade, making it ideal for calculating comfort and sight distance. Parabolic geometry also simplifies setting out using chord/slope offsets and supports linear headlight sight distance approximations for sag curves at night.

Step-by-Step Solution:
Define the desired property: constant rate of change of grade.Identify curve type that satisfies this property: the parabola.Adopt the parabolic equation y = ax^2 + bx + c in profile design and compute offsets accordingly.

Verification / Alternative check:
Most design manuals present formulae for length of vertical curves based on stopping/passing sight distance that explicitly assume a parabolic profile. Field setting-out methods also match parabolic offset relationships.

Why Other Options Are Wrong:

• Elliptical, circular, spiral: not standard for vertical curves; spirals are used horizontally as transition curves, not vertically.
• All the above: incorrect because practice standardizes on parabolic shapes for vertical alignment.

Common Pitfalls:

• Confusing the use of spirals (horizontal) with vertical applications.
• Using too short a curve, reducing stopping sight distance on crests.

Final Answer:
parabolic