Article — Vertical Curve Calculator (AASHTO)
Vertical Curve Calculator — Highway Geometry per AASHTO
A vertical curve is the parabolic transition between two roadway grades. Length is calculated as L = K × A, where A is the absolute difference between entry grade and exit grade (in percent) and K is the AASHTO design parameter for stopping sight distance or comfort.
Vertical curves prevent abrupt grade changes that would launch vehicles over hilltops or bottom them out in valleys. The classic example: a road climbing at 3% suddenly becoming a 2% descent would create a sharp point that drivers cannot see over and that compresses the suspension on impact. A parabolic curve smooths the transition with constant rate of grade change.
Vertical curve basics
Every vertical curve has three control points: PVI (point of vertical intersection, where the two grade tangents would meet if extended), PVC (point of vertical curvature, where the curve begins), and PVT (point of vertical tangency, where the curve ends). The curve itself is a parabola, not a circular arc — the parabola gives constant rate of grade change, which keeps vertical acceleration uniform and ride comfortable.
For a symmetric curve, PVC and PVT sit equidistant from the PVI: each is L/2 away. The grade transitions linearly from G1 at PVC to G2 at PVT, passing through 0% at the highest point of a crest curve or the lowest point of a sag curve.
The vertical curve length formula
L = K × A, where L is curve length in feet, K is the design parameter in feet per percent, and A is |G1 − G2| in percent. AASHTO publishes K-tables by design speed and curve type. For a 60 mph crest curve, K minimum is 151. For a 60 mph sag curve, K minimum is 136. The crest value is larger because sight distance over a hilltop is more restrictive than headlight throw at night.
Example: a road transitions from G1 = +3% to G2 = −2% at 60 mph. A = 5%. L_min = 151 × 5 = 755 ft. The PVC is 755/2 = 377.5 ft before the PVI; the PVT is 377.5 ft after. The road actually starts curving 377.5 ft before what looks like the hilltop on a profile drawing.
AASHTO uses a parabola, not a circular arc, for vertical curves. The parabola produces constant rate of grade change (and therefore constant vertical acceleration), which feels smoother to drivers. A circular arc would have changing curvature — and a noticeable kink at the entry and exit.
Crest vs sag vertical curves
A crest curve goes over a high point — G1 > G2, like the top of a hill. The curve length is governed by stopping sight distance: drivers must be able to see far enough past the crest to brake for stopped traffic. The taller the driver's line of sight clears the road, the longer the curve has to be.
A sag curve goes through a low point — G1 < G2, like a valley bottom. Length is governed by either headlight sight distance (at night, the upward-aimed headlight beam controls how far ahead the driver can see) or by rider comfort (vertical g-force shouldn't exceed about 0.03 g). At slow speeds (under 40 mph) headlight distance usually governs; at high speeds (60+ mph) comfort sometimes governs but rarely.
K-value and design speed
K rises rapidly with design speed because both sight distance and comfort scale with V². At 30 mph, a crest curve needs K ≥ 19; at 70 mph, K ≥ 247 — more than 13 times longer for the same grade change. A 3-degree grade transition at 30 mph fits in 60 ft; at 70 mph it needs 740 ft, more than two football fields.
The math: SSD = 1.47 × V × 2.5 + 1.075 × V² / 11.2 (where V is in mph). At 30 mph SSD is 200 ft; at 60 mph it is 570 ft; at 70 mph it is 730 ft. K must be large enough to produce a curve at least as long as the SSD on a crest, so K scales as V² approximately.
Use the rate of grade change as a sanity check. AASHTO recommends r ≤ 1% per 100 ft for comfort on rural highways and r ≤ 0.5% per 100 ft for high-speed Interstates. A curve with A = 5% and L = 250 ft has r = 2%/100 ft — too steep for comfort.
Sight distance on vertical curves
Stopping sight distance is the foundation of crest curve design. AASHTO assumes a driver's eye height of 3.5 ft (passenger car) and an object height of 2.0 ft (a deer, an obstacle, or a stopped vehicle's taillight). The crest must be long enough that the line of sight from eye to object doesn't get blocked by the road profile itself.
The formula relates K to these heights: K_crest = SSD² / (200 × (√3.5 + √2.0)²) for SSD < L. The result is the K value that produces a curve barely long enough for the driver to brake. Real-world design uses the published K-tables, which add a small safety margin and round to convenient numbers.
Parabolic vertical curve elevation
To stake a curve, you need elevations at every station along it. The parabolic formula: Elev(x) = E_PVC + G1 × x/100 + (A × x²) / (200 × L), where x is the distance in feet from the PVC.
Example: PVC at elevation 1500.00 ft, G1 = +3%, A = 5%, L = 305 ft. Elevation at x = 100 ft past PVC: E = 1500 + 3 × 100/100 + (5 × 100²) / (200 × 305) = 1500 + 3 + 0.82 = 1503.82 ft. The parabolic term (the third part) is what makes the road curve smoothly rather than break sharply at the PVC.
Common vertical curve mistakes
The most common design error is using the wrong K-value — confusing crest with sag, or selecting a value for the wrong design speed. Always double-check against the latest AASHTO tables; values have shifted slightly between editions as research updated reaction-time and object-height assumptions.
The second most common: assuming a symmetric curve when the topography demands asymmetry. If the road grade approaches the PVI from a cut and leaves into a fill, the L1 (PVC to PVI) may need to be shorter than L2 (PVI to PVT) to avoid excessive earthwork. Asymmetric curves use two parabolas joined at the PVI; both must meet the same K constraints.
- L = K × A = the master formula for all vertical curves
- PVC and PVT = L/2 from PVI on a symmetric curve
- Crest at 60 mph = K ≥ 151 per AASHTO (sight distance)
- Sag at 60 mph = K ≥ 136 per AASHTO (headlight + comfort)
- SSD at 60 mph = 570 ft, basis for crest K-table
- Rate of change = r ≤ 1%/100 ft for comfort, ≤ 0.5%/100 ft for high-speed
Sag curves create natural low points where water collects. Without engineered drainage, a sag curve becomes a seasonal pond and an aquaplaning hazard. Modern AASHTO practice limits the minimum sag K-value based on drainage as well as comfort and sight distance — too-flat sag curves cannot shed water fast enough during heavy rain.
The drainage rule of thumb: minimum slope of 0.5% (longitudinal) on any roadway. A sag curve with G1 = −0.1% and G2 = +0.1% (A = 0.2%) creates a near-zero grade for hundreds of feet — water has nowhere to go. Designers either steepen the approaches or shorten the curve to keep the longitudinal slope above 0.5% along its full length.