Vertical Curve Calculator
Enter two grades and a curve length to instantly compute every key station (PVC, PVI, PVT), the algebraic grade difference, the K-value, the high or low point location, and the elevation at any station along the parabolic curve. An elevation profile chart updates in real time. Works in both feet and metres.
What is a vertical curve?
A vertical curve is the parabolic transition used in road and railway design wherever the profile grade changes from one slope to another. Unlike a sharp corner, the parabola distributes the grade change smoothly over a horizontal distance called the curve length (L), making the ride comfortable and giving drivers adequate sight distance. Two types exist: a crest curve occurs when the grade decreases (g2 < g1, e.g., going over a hill), and a sag curve occurs when the grade increases (g2 > g1, e.g., going through a valley). Both are designed as equal-tangent symmetric parabolas in standard highway practice, meaning the point of vertical intersection (PVI) sits exactly at the midpoint of the curve horizontally.
How to use this calculator
Enter the entering grade (g1) and exiting grade (g2) as signed percentages (positive = uphill, negative = downhill), the horizontal curve length (L), and the PVI station and elevation. The calculator instantly returns the PVC and PVT stations and elevations, the algebraic grade difference A = |g2 - g1|, the K-value (L / A), and the location and elevation of the high or low point. Set the "station for spot elevation" field to any station between PVC and PVT to read the parabolic elevation at that point. Optionally enter a design speed to check whether the K-value meets the AASHTO Green Book minimum for your curve type.
Formulas used
The core equation is the symmetric parabolic elevation formula: E(x) = E_PVC + (g1/100) * x + r/2 * x^2, where x is the horizontal distance from the PVC, r = (g2 - g1) / (100 * L) is the rate of change of grade per unit length, and E_PVC is the elevation at the beginning of the curve. Setting dE/dx = 0 gives the high or low point at x_apex = -(g1/100) / r = -g1 * L / (g2 - g1), which exists inside the curve only when the apex falls between 0 and L. The K-value equals L / |A| and represents the horizontal distance required for each 1% change in grade; it is the fundamental measure of how gradual the curve is. Minimum K values from the AASHTO Green Book ensure adequate stopping sight distance (crest) or headlight sight distance at night (sag).
Practical design guidelines
For crest curves, stopping sight distance is the controlling criterion: the driver must see an object 2.0 ft high from a 3.5 ft eye height. For sag curves, headlight sight distance governs at night: headlights at 2.0 ft height with a 1-degree upward beam must illuminate the road for the stopping distance. A K-value above 167 ft/% (metric: about 51 m/%) can cause poor pavement drainage on sag curves, so special drainage provisions are needed at high K values. Many agencies also set an absolute minimum curve length of about three times the design speed in mph (e.g., 150 ft at 50 mph) to prevent abrupt visual breaks, regardless of what the K formula requires.
AASHTO minimum K-values by design speed
| Design speed (mph) | Stop. sight dist. (ft) | Min K - Crest | Min K - Sag |
|---|---|---|---|
| 15 | 80 | 3 | 10 |
| 20 | 115 | 7 | 17 |
| 25 | 155 | 12 | 26 |
| 30 | 200 | 19 | 37 |
| 35 | 250 | 29 | 49 |
| 40 | 305 | 44 | 64 |
| 45 | 360 | 61 | 79 |
| 50 | 425 | 84 | 96 |
| 55 | 495 | 114 | 115 |
| 60 | 570 | 151 | 136 |
| 65 | 645 | 193 | 157 |
| 70 | 730 | 247 | 181 |
| 75 | 820 | 312 | 206 |
| 80 | 910 | 384 | 231 |
Minimum rate-of-curvature K (ft/% or m/%) from the AASHTO Green Book. Use the larger of the minimum crest and sag K, multiplied by |A|, to get minimum curve length.
Frequently asked questions
What does the K-value mean in vertical curve design?
K is the horizontal distance, in feet or metres, needed for the grade to change by 1%. A higher K means a more gradual curve. It is calculated as K = L / |A|, where L is the curve length and A is the absolute algebraic grade difference. AASHTO provides minimum K values for each design speed and curve type to ensure adequate sight distance.
What is the difference between a crest and a sag vertical curve?
A crest curve happens when the grade decreases (g2 < g1), forming a hill shape. The driver must be able to see a stopped vehicle over the crest in daylight, so stopping sight distance drives the minimum K. A sag curve happens when the grade increases (g2 > g1), forming a valley. There is no line-of-sight problem in daylight, but at night the headlight beam must reach far enough down the road, so headlight sight distance sets the minimum K instead.
How do I find the high point or low point of a vertical curve?
The high or low point occurs where the instantaneous grade equals zero. For an equal-tangent parabolic curve, the distance from the PVC is x = -g1 * L / (g2 - g1), where grades are entered as signed percentages. If x falls between 0 and L, the point is on the curve; otherwise the extreme elevation is at one of the endpoints. Substitute x back into the parabolic elevation formula to get the elevation at that station.
What is PVC, PVI, and PVT?
PVC stands for Point of Vertical Curvature - the station where the parabolic curve begins. PVI is the Point of Vertical Intersection, where the two grade tangent lines meet; it sits at the horizontal midpoint of an equal-tangent curve. PVT is the Point of Vertical Tangency - the station where the curve ends and the exiting grade continues. Stations and elevations for all three are computed from the PVI location and the two grades.
Why do some road agencies use a minimum curve length regardless of K?
A very short curve, even if it satisfies sight distance, can appear as an abrupt "kink" to drivers at high speeds and creates maintenance problems at expansion joints. Most agencies therefore require a minimum length equal to roughly three times the design speed in mph (e.g., at 60 mph, at least 180 ft), regardless of the grade change. This ensures a visually smooth alignment and adequate riding comfort.
What happens when the high/low point falls outside the curve?
When the calculated apex distance x is negative or greater than L, the high or low point lies outside the PVC-PVT limits. This means both grades point the same way throughout the curve (e.g., g1 = 2% and g2 = 4% both go uphill, so the elevation keeps rising through the sag curve with no low point inside it). In that case the extreme elevation within the curve is simply at one of the endpoints.