What is a vertical curve in road design?

A vertical curve is a smooth parabolic transition between two different roadway grades (slopes) along the centerline profile. It is used at summits (crest curves) and valleys (sag curves) to provide comfort, drainage, and adequate sight distance for drivers.

What is the K value of a vertical curve?

The K value of a vertical curve is defined as the curve length L divided by the algebraic grade difference A (in percent): K = L / A. It is a measure of the curve's flatness. Design standards like AASHTO specify minimum K values to ensure adequate stopping and passing sight distance.

How do I know if my curve is crest or sag?

If the roadway profile goes from an upgrade to a downgrade (or vice versa) forming a summit, the vertical curve is a crest curve. If it forms a valley between two grades, it is a sag curve. In this calculator you can explicitly choose crest or sag; the formulas for sight distance differ between the two.

Which sight distance should I use for vertical curve design?

For most highway and street designs, the controlling sight distance is the stopping sight distance (SSD), which ensures a driver can see and stop for an object in the roadway. Passing sight distance (PSD) is used on two-lane rural highways where passing maneuvers are allowed. Many agencies provide tabulated SSD and PSD values by design speed; you can enter those directly into this calculator.

Can this calculator compute elevations along the vertical curve?

Yes. Once you enter the curve type, grades, length, and PVC elevation, the tool computes the elevation at the PVI, PVT, and any station offset you specify. It uses the standard parabolic equation y(x) = g1·x + (A·x²)/(2L) relative to the PVC.

Vertical Curve Calculator (Crest & Sag)

Design and analyze highway and roadway vertical curves. Compute curve length, K value, elevations, and check stopping/passing sight distance for crest and sag curves.

Design Vertical Curve for Required Sight Distance

Enter grades and desired sight distance to compute the minimum curve length and K value. Supports both crest and sag curves using standard parabolic formulas.

Curve type

Design for

Initial grade g₁ (%)

Upgrade positive, downgrade negative (e.g., +2.0, -1.5).

Final grade g₂ (%)

Algebraic difference A = |g₂ − g₁| is used in design.

Design speed (km/h)

Used to suggest stopping sight distance (SSD). You can override SSD manually.

Stopping sight distance SSD (m)

Typical SSD at 80 km/h ≈ 130–150 m depending on standard.

Passing sight distance PSD (m, optional)

For two-lane highways. Used only for information.

Given curve length L (m)

Used when “Design for available sight distance” is selected.

Compute Elevations Along a Vertical Curve

Use this panel when the curve length is already fixed. Enter PVC station and elevation, grades, and length to get elevations at PVI, PVT, and any intermediate station.

PVC station (m)

Reference station of the Point of Vertical Curvature.

PVC elevation (m)

Initial grade g₁ (%)

Final grade g₂ (%)

Curve length L (m)

Offset from PVC x (m)

0 ≤ x ≤ L. Elevation at this point will be computed.

Vertical Curve Basics

In highway and roadway design, a vertical curve provides a smooth transition between two different longitudinal grades along the centerline profile. Vertical curves are almost always designed as simple parabolas because they provide a constant rate of change of grade, which is comfortable for drivers and easy to compute.

Crest vs. Sag Vertical Curves

Crest curve: forms a summit between an upgrade and a downgrade. The critical sight distance is usually the line of sight over the crest.
Sag curve: forms a valley between two grades. At night, the critical sight distance is governed by the headlight beam and curve geometry.

Key Points and Notation

PVC – Point of Vertical Curvature (beginning of curve)
PVI – Point of Vertical Intersection (intersection of tangents)
PVT – Point of Vertical Tangency (end of curve)
g₁ – initial grade (%), positive for upgrade, negative for downgrade
g₂ – final grade (%)
A – algebraic grade difference, A = |g₂ − g₁| (%)
L – length of vertical curve (m)
K – flatness parameter, K = L / A (m/%). Larger K means a flatter curve.

Parabolic vertical curve equation
Let x be the horizontal distance from the PVC (0 ≤ x ≤ L), and let grades be in decimal (e.g., 2% = 0.02).

Elevation at distance x from PVC:

y(x) = y_PVC + g₁·x + (A·x²) / (2L)

where A = g₂ − g₁ (in decimal).

Design for Stopping Sight Distance (SSD)

Most design manuals (AASHTO, state DOTs, etc.) specify minimum K values or minimum curve lengths to provide adequate stopping sight distance (SSD) for a given design speed. This calculator implements the standard parabolic formulas for crest and sag curves.

Crest Vertical Curve – Minimum Length for SSD

For a crest curve, the driver’s line of sight is limited by the roadway surface itself.

Case 1: L ≥ S (long curve)

L = (A · S²) / (200 · (h₁ + h₂) )

Case 2: L < S (short curve)

L = 2S − (200 · (h₁ + h₂)) / A

where:

A = |g₂ − g₁| in percent
S = required sight distance (m)
h₁ = driver eye height (typically 1.08–1.2 m)
h₂ = object height (typically 0.6 m for SSD)

Sag Vertical Curve – Minimum Length for SSD

For sag curves at night, sight distance is controlled by the headlight beam. A common simplified formula is:

Case 1: L ≥ S

L = (A · S²) / (200 · (h₃ + S·tan θ))

Case 2: L < S

L = 2S − (200 · (h₃ + S·tan θ)) / A

where:

A = |g₂ − g₁| in percent
S = required sight distance (m)
h₃ = headlight height (≈ 0.6 m)
θ = upward headlight beam angle (≈ 1°)

Stopping Sight Distance from Speed

If your design manual does not provide SSD directly, a common approximation (SI units) is:

SSD ≈ 0.278 · V · t + V² / (254 · (f + G))

V = speed (km/h)
t = perception–reaction time (s), often 2.0–2.5 s
f = coefficient of friction (≈ 0.35–0.4)
G = grade (decimal). For conservative SSD, G is often taken as 0.

Worked Example

Given:

Crest curve, design speed V = 80 km/h
Grades: g₁ = +2.0%, g₂ = −1.0% ⇒ A = 3.0%
SSD from table: S = 140 m

Assume: h₁ = 1.1 m, h₂ = 0.6 m, so h₁ + h₂ = 1.7 m.

Check long-curve case (L ≥ S):

L = (A · S²) / (200 · (h₁ + h₂))
  = (3.0 · 140²) / (200 · 1.7)
  ≈ (3.0 · 19600) / 340
  ≈ 58800 / 340
  ≈ 173 m

So a crest vertical curve of length L ≈ 175 m (K ≈ 58 m/%) would satisfy the SSD requirement. The calculator performs this computation instantly and also checks the available sight distance for any alternative length you enter.

Practical Tips

Always compare your computed K values with the minimum K values in your governing design manual.
For urban streets, drainage and comfort may control over sight distance, especially for very flat grades.
For sag curves in areas with good lighting, some agencies allow relaxed criteria compared to headlight control.
Round curve lengths to practical stationing (e.g., to the nearest 5 or 10 m) and re-check sight distance.