Why Two Cases for Summit Curve Length?
The design of a summit curve revolves around a single geometric question: does the driver’s line of sight — from eye level to the object ahead — stay within the parabolic curve, or does it extend beyond the end of the curve into the descending tangent section? This creates two geometrically distinct situations, each requiring a different formula.
The two cases are: Case 1: L > S (the curve is longer than the required sight distance, so the entire line of sight falls within the curve) and Case 2: L < S (the sight distance is longer than the curve, so the line of sight extends beyond the VPT into the tangent section).
General Formula — Both Cases
The sight distance S across a summit curve is governed by two heights: h₁ (driver’s eye height above road surface) and h₂ (height of the object above road surface).
Case 1: L > S (Curve Longer than Sight Distance)
When the entire sight line falls within the parabolic curve:
L = NS² / (√h₁ + √h₂)²
Case 2: L < S (Sight Distance Longer than Curve)
When the sight line extends beyond the VPT:
L = 2S − (√h₁ + √h₂)²/N
IRC Simplified Formulas
IRC specifies standard heights for h₁ and h₂ depending on the type of sight distance being used:
| Sight Distance Type | h₁ (Driver Eye Height) | h₂ (Object Height) | (√h₁+√h₂)² |
|---|---|---|---|
| SSD (Stopping) | 1.2 m | 0.15 m | 4.4 (approx.) |
| OSD / ISD (Overtaking/Intermediate) | 1.2 m | 1.2 m | 9.6 (approx.) |
Substituting these values:
| Case | SSD Formula | OSD/ISD Formula |
|---|---|---|
| L > S | L = NS²/4.4 | L = NS²/9.6 |
| L < S | L = 2S − 4.4/N | L = 2S − 9.6/N |
Location of Highest Point on Summit Curve
On a symmetric summit curve (equal gradients on both sides), the highest point falls at the mid-point of the curve. But when the two gradients are unequal — which is the common case — the highest point shifts toward the side with the flatter gradient. Its exact position from the VPC (start of curve) is:
x₀ = n₁ × L / N
Where n₁ = first (approaching) gradient, L = length of curve, N = deviation angle = |n₁ − n₂|.
Solved Example 1 — SSD Summit Curve
Problem: SSD = 150 m. National highway at junction of upward gradient +1% and downward gradient −2%. Find summit curve length.
Solution:
N = |n₁ − n₂| = |0.01 − (−0.02)| = 0.03
Assume L > S: L = NS²/4.4 = (0.03 × 150²)/4.4 = 675/4.4 = 153.41 m
Check: 153.41 m > 150 m ✔ (assumption correct)
Highest point: x₀ = n₁L/N = (0.01 × 153.41)/0.03 = 51.14 m from VPC
Observation from IRC
The length of summit curve decreases as N (deviation angle) and/or S (sight distance) decreases. This means flatter roads require shorter vertical curves, while sharp grade changes require longer curves for the same sight distance. On very flat grades with small N, no vertical curve may be needed for sight distance — but one is still provided for comfort and smooth driving.