Vertical Curve Calculator - Highway & Road Design Tool

What is a vertical curve calculator?

A vertical curve calculator finds the elevations, K value, and key points of a vertical curve, the parabolic curve engineers use to join two road grades smoothly. Instead of a road meeting at a sharp angle where the slope changes, a vertical curve blends the two slopes into one continuous curve. This makes the ride smoother, keeps drivers able to see far enough ahead to stop safely, and helps water drain properly.

Engineers use vertical curves when designing highways, streets, railways, airport runways, and graded sites such as parking lots. Anywhere a road or track needs to change slope, a vertical curve carries out that transition.

Crest curves and sag curves

A vertical curve is one of two types, depending on how the grades (slopes) compare before and after the curve. A grade is written as a percentage: a 3% grade rises 3 meters for every 100 meters traveled.

• Crest curve: the initial grade is greater than the final grade (g₁ > g₂). The road goes up, over a high point, then down — like cresting a hill.
• Sag curve: the initial grade is less than the final grade (g₁ < g₂). The road goes down, through a low point, then up — like the bottom of a dip.
• If the two grades are equal, no curve is needed. The road is already a straight line.

Crest curves matter most for stopping sight distance: a driver must see far enough ahead to stop before hitting an obstacle. Sag curves matter most for headlight sight distance at night and for drainage, since water collects at the low point.

Key terms

• PVI (Point of Vertical Intersection): the point where the two straight grade lines would meet if there were no curve.
• PVC (Point of Vertical Curve): where the curve begins.
• PVT (Point of Vertical Tangent): where the curve ends.
• K value: the curve length divided by the change in grade. It measures how gradual the curve is.
• Station: a distance measurement along the road's centerline, used instead of plain coordinates. In this calculator, station and elevation are both entered in meters.

A curve is normally symmetric: the PVC sits half the curve length before the PVI, and the PVT sits half the curve length after it.

Vertical curve formula

The elevation at any point on the curve follows a quadratic (parabolic) equation:

$y = y_{PVC} + g_1 x + \frac{A x^2}{2L}$

• $y$ = elevation at distance $x$ from the PVC
• $y_{PVC}$ = elevation at the PVC
• $g_1$ = initial grade, as a decimal (a 3% grade is 0.03)
• $A$ = algebraic difference in grades, $g_2 - g_1$ (decimal form)
• $L$ = curve length

K value formula

$K = \frac{L}{|g_2 - g_1|}$

$g_1$ and $g_2$ here are percentages. A larger K means a flatter, more gradual curve. Design manuals such as AASHTO's "A Policy on Geometric Design of Highways and Streets" set minimum K values for highways, based on design speed and whether the curve is a crest or a sag.

If $g_1$ equals $g_2$, the grade change is zero, and K is undefined (the calculator reports it as infinite).

PVC and PVT

$Station_{PVC} = Station_{PVI} - \frac{L}{2} \qquad Elevation_{PVC} = Elevation_{PVI} - \frac{g_1 L}{200}$

$Station_{PVT} = Station_{PVI} + \frac{L}{2} \qquad Elevation_{PVT} = Elevation_{PVI} + \frac{g_2 L}{200}$

($g_1$ and $g_2$ are percentages here; dividing by 200 converts the percentage to a decimal and applies it over half the curve length.)

High point or low point

A high point (crest curve) or low point (sag curve) exists inside the curve whenever the two grades have opposite signs. Its distance from the PVC is:

$x = \frac{g_1 L}{g_1 - g_2}$

Add $x$ to the PVC station to get the high or low point's station, then use the elevation formula above to find its elevation.

How to calculate a vertical curve

1. Enter the initial grade and final grade, as percentages. Use a positive number for an uphill slope and a negative number for a downhill slope.
2. Enter the curve length, in meters.
3. Enter the PVI station and PVI elevation, both in meters.
4. Read off the curve type (crest, sag, or neither), the K value, and the PVC and PVT stations and elevations.
5. If the grades have opposite signs, the calculator also reports the high or low point.
6. To find the elevation at any other point, enter a query station. If that station falls outside the curve, the calculator reports it as outside the curve rather than a number.

Worked example: crest curve

A road climbs at a 3% grade, then descends at −2%. The PVI is at station 1000.00 m, with an elevation of 150.00 m. The curve length is 400 m.

K value: $K = 400 / |{-2} - 3| = 400 / 5 = 80.00$

PVC: station $= 1000 - 400/2 = 800.00$ m; elevation $= 150 - (3 \times 400)/200 = 144.00$ m

PVT: station $= 1000 + 400/2 = 1200.00$ m; elevation $= 150 + (-2 \times 400)/200 = 146.00$ m

High point: since 3 and −2 have opposite signs, a high point exists. $x = (3 \times 400)/(3-(-2)) = 1200/5 = 240$ m from the PVC, so its station is $800 + 240 = 1040.00$ m. Its elevation is $144 + (0.03)(240) + \dfrac{(-0.05)(240)^2}{2(400)} = 144 + 7.2 - 3.6 = 147.60$ m.

Worked example: sag curve

A road descends at −3%, then climbs at 2%. The PVI is at station 500.00 m, elevation 85.00 m. The curve length is 200 m.

K value: $K = 200 / |2-(-3)| = 200/5 = 40.00$

PVC: station $= 400.00$ m; elevation $= 85 - (-3 \times 200)/200 = 88.00$ m

PVT: station $= 600.00$ m; elevation $= 85 + (2 \times 200)/200 = 87.00$ m

Low point: $x = (-3 \times 200)/(-3-2) = -600/-5 = 120$ m from the PVC, so its station is $400 + 120 = 520.00$ m. Its elevation is $88 + (-0.03)(120) + \dfrac{(0.05)(120)^2}{2(200)} = 88 - 3.6 + 1.8 = 86.20$ m.

This low point is where surface water collects, so it is where a designer places a drainage inlet, not at the PVI.

Frequently asked questions

What is a K value in vertical curve design? The K value is the curve length divided by the absolute difference between the two grades. It shows how many meters of curve give a 1% change in grade. A higher K value means a flatter, more gradual curve.

How do I know if I need a crest curve or a sag curve? Compare the two grades. If the initial grade is greater than the final grade, it is a crest curve. If the initial grade is less than the final grade, it is a sag curve. Equal grades need no curve at all.

What happens if the initial and final grades are equal? No vertical curve is needed, since the road is already straight. The K value is undefined because it would require dividing by a grade change of zero.

How do you find the high or low point of a curve? It exists only when the two grades have opposite signs. Its distance from the PVC is $g_1 L / (g_1 - g_2)$. Add that to the PVC station for its location, then use the curve's elevation formula for its height.

Why does the low point of a sag curve matter for drainage? Water flowing down both sides of the sag collects at the low point. If a drainage inlet is placed at the PVI instead of the true low point, water can pond on the road, since the two points are rarely the same.

Can a query station fall outside the curve? Yes. The elevation formula only applies between the PVC and the PVT. A station outside that range does not have a defined elevation on the curve, so the calculator reports it as outside the curve.