Geometric Design – Ideal Summit (Crest) Curve Form For vertical alignment at a summit (crest), which ideal mathematical curve is adopted to provide a constant rate of change of grade and comfortable sight distance transitions?

Introduction / Context:
Vertical curves connect two grades in highway design. At a summit (crest), the curve must ensure comfort, safety, adequate sight distance, and smooth appearance. The chosen curve should deliver a uniform rate of change of slope to minimize jerk on vehicles and facilitate straightforward setting-out in the field.

Given Data / Assumptions:

• Location: summit (crest) vertical curve between two grades.
• Design needs: constant rate of change of grade and adequate stopping/overtaking sight distance.
• Constructability: simple field layout and easy computation of offsets.

Concept / Approach:

A simple parabola y = ax^2 + bx + c provides a constant rate of change of grade because the second derivative is constant. This property directly meets comfort (jerk control) and visibility requirements. Additionally, parabolic curves are straightforward to stake out using equal chord offsets and allow direct sight-distance checks at the crest.

Step-by-Step Solution:

Verification / Alternative check:

Design manuals universally adopt parabolic vertical curves for both summits and sags due to the constant rate-of-grade property and computational convenience.

Why Other Options Are Wrong:

• Spiral and lemniscate are used in horizontal alignment transitions, not vertical crest curves.
• Circle gives constant radius but not constant rate of change of grade in vertical alignment, leading to non-uniform comfort.

Common Pitfalls:

Mixing horizontal transition concepts with vertical design; using inadequate length causing restricted sight distance at the crest.

Final Answer:

Parabola