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What Is a Vertical Curve in Road Design?
A vertical curve is the parabolic transition between two different road grades in the vertical plane. Unlike horizontal curves — which change direction left or right — a vertical curve changes the rate of grade change smoothly along the elevation profile, preventing abrupt grade breaks that would damage vehicles and reduce visibility.
Every time a road profile switches from one grade percentage to another, a vertical curve must be inserted if the algebraic grade difference (A = G₂ − G₁) exceeds the minimum threshold set by the governing road design standard. In practice, virtually all grade changes on high-speed roads require a vertical curve.
The parabola is the standard shape for vertical curves because its rate of grade change is constant — meaning the acceleration experienced by a vehicle travelling at constant speed is uniform throughout the curve.
Crest vs Sag Vertical Curves — What Is the Difference?
A crest curve (convex) occurs when the grade decreases: the road crests over a hill. The algebraic difference A = G₂ − G₁ is negative. The critical design constraint is stopping sight distance — the driver's line of sight is blocked by the crest, so the curve must be long enough that a driver can see and stop for an object on the road ahead.
A sag curve (concave) occurs when the grade increases: the road dips into a valley. A is positive. The design constraint shifts to headlight sight distance at night — the parabola must be gradual enough that headlight beams illuminate the road surface ahead to the required stopping distance.
The K-value controls both curve types, but the AASHTO minimum K is numerically lower for sag curves because headlight beam spread gives more effective sight distance than the direct line of sight used for crest design.
Vertical Curve Geometry — Crest & Sag Diagram
The diagrams below show the key elements of each curve type. Both use an equal-tangent parabola (PVC and PVT equidistant from PVI), which minimises earthwork and is the standard form in AASHTO and most national design codes.
Crest Curve — Convex (A < 0)
Sight line is blocked by the crest — SSD controls curve length
Sag Curve — Concave (A > 0)
Headlight beam angle controls curve length at night
How to Calculate Vertical Curve Length and K-Value
All vertical curve calculations start with the algebraic grade difference A and the K-value. The K-value is the length of curve per 1% change in grade. AASHTO tabulates minimum K values for each design speed and curve type — the table below shows the Green Book values.
| Output | Formula | Notes |
|---|---|---|
| Grade difference A | A = G₂ − G₁ | Signed; negative = crest |
| Curve length L | L = K × |A| | K from AASHTO design table |
| K-value | K = L / |A| | Rate of grade change, m/% |
| High/low point | x = −G₁ × L / A | Distance from PVC; only when 0 < x < L |
| Elevation at x | y = y_PVC + G₁x/100 + (A/200L)x² | Parabolic grade formula |
| PVC station | PVC = PVI − L/2 | Equal-tangent curve |
| PVT station | PVT = PVI + L/2 | End of curve |
Stopping Sight Distance on Crest and Sag Curves
The AASHTO minimum K values are derived directly from stopping sight distance requirements. For crest curves, the SSD formula uses driver eye height h₁ = 1.08 m and object height h₂ = 0.60 m, giving the constant C = 658. For sag curves, the headlight formula uses headlight height 0.60 m and upward beam angle 1°, giving the constants 120 and 3.5.
This calculator checks whether the curve length you entered (or the AASHTO-minimum length it computed) actually provides the required stopping sight distance, and displays the result as a percentage progress bar.
| Design Speed (km/h) | Crest K (m/%) | Sag K (m/%) | Design SSD (m) |
|---|---|---|---|
| 50 | 7 | 13 | 65 |
| 60 | 11 | 18 | 85 |
| 70 | 17 | 23 | 105 |
| 80 | 26 | 30 | 130 |
| 90 | 39 | 38 | 160 |
| 100 | 53 | 45 | 185 |
| 110 | 74 | 55 | 220 |
| 120 | 95 | 63 | 250 |
PVC, PVI, and PVT — Vertical Curve Stationing
Vertical curves are located by three key points. The PVI (Point of Vertical Intersection) is where the two tangent grade lines would meet if extended — it is the design input the surveyor places in the field. The PVC (Point of Vertical Curvature) is where the curve begins, at PVI − L/2. The PVT (Point of Vertical Tangency) is where the curve ends, at PVI + L/2. This equal-tangent arrangement is standard practice because it minimises earthwork volumes.
Stationing is expressed in the format km+m (e.g. 1+080.00 = 1,080 metres from the datum). Enter the PVI station in this calculator and it will compute PVC, PVT, and — where applicable — the high or low point station automatically.
Frequently Asked Questions
What is the K-value in a vertical curve?
The K-value (also called the rate of grade change) is the curve length in metres required for each 1% change in grade. A K of 16 m/% at 80 km/h means a 4% grade change requires L = 16 × 4 = 64 m of vertical curve. Higher K values produce flatter, more gradual transitions. AASHTO provides minimum K values based on design speed and whether the curve is a crest or sag.
How do you find the high point on a crest vertical curve?
The high point occurs where the parabola reaches its maximum — i.e. where the instantaneous grade equals zero. Using the equal-tangent parabolic formula, the distance from the PVC to the high point is x = −G₁ × L / A. This is only on the curve when 0 < x < L, which requires G₁ and G₂ to have opposite signs. The elevation at that point is y_PVC + G₁·x/100 + (A/200L)·x².
What is the difference between a crest curve and a sag curve?
A crest (convex) curve occurs when the grade decreases (A < 0) — the road goes over a hill. The sight-distance constraint is the direct driver line of sight, blocked by the crest. A sag (concave) curve occurs when the grade increases (A > 0) — the road dips into a valley. The constraint is headlight beam coverage at night. This leads to different AASHTO formulas and different minimum K values for each type.
How is stopping sight distance calculated on a vertical curve?
For crest curves, AASHTO uses: L = A·SSD² / 658 (when SSD ≤ L), where 658 = (√(2×1.08) + √(2×0.60))² × 100 from driver eye height 1.08 m and object height 0.60 m. For sag curves: L = A·SSD² / (120 + 3.5·SSD) (when SSD ≤ L), from headlight height 0.60 m and upward beam angle 1°. This calculator inverts these formulas to find the SSD the curve provides given L and A.
What grades require a vertical curve under AASHTO standards?
AASHTO recommends inserting a vertical curve whenever the algebraic grade difference |A| exceeds 0.5% on high-speed roads (posted speed 80 km/h and above) and 1.0% on lower-speed roads. Below these thresholds, the grade break may be acceptable without a curve. In practice, all design software inserts a curve at every grade change unless the designer explicitly overrides.
How do you calculate PVC and PVT from a PVI station?
For an equal-tangent vertical curve, PVC = PVI − L/2 and PVT = PVI + L/2. The curve is symmetrically placed around the PVI. If the PVI is at station 1+000.00 and L = 160 m, then PVC = 0+920.00 and PVT = 1+080.00. Enter the PVI station in the optional field above and this calculator computes PVC, PVT, and the high or low point station automatically.
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