Vertical Curve Calculator

A roadway vertical curve is a parabola that smoothly connects two grades. You start from the incoming grade (g1), the outgoing grade (g2), the curve length (L) and the PVI — the point where the two grade lines meet, given as a station and elevation.

The grade change is A = g2 minus g1; the K value is L divided by the absolute value of A; the curve starts at the PVC, half the length before the PVI, and ends at the PVT, half the length after. Elevations along the curve follow the parabola equation.

For example, g1 of +3%, g2 of -2% and a 600 ft curve give A = -5%, a K of 120 and a crest curve. This calculator works out the type, K, the PVC/PVI/PVT, the high or low point and a station-by-station profile.

The K value is the curve length divided by the absolute value of the grade change (A): K = L divided by the absolute value of A. It is the horizontal distance, in feet, needed for a 1 percent change in grade, so a higher K means a longer, gentler, flatter curve.

K is the single number that sets how a curve performs for sight distance — design standards give a minimum K for each design speed rather than a minimum length. It also locates the high or low point.

For example, a 600 ft curve with a 5 percent grade change has a K of 120. This calculator reports the K value and compares it against the AASHTO minimum for your chosen design speed.

It comes down to the sign of the grade change, A = g2 minus g1. A crest curve has a negative A — the grade drops, going from a higher grade to a lower one, like a hilltop. A sag curve has a positive A — the grade rises, going from lower to higher, like a valley or a dip.

Crest curves are usually limited by daytime stopping sight distance over the hump, while sag curves are limited by headlight sight distance at night, so the two have different minimum K values at the same speed.

For example, g1 of +3% to g2 of -2% gives A = -5%, a crest curve. This calculator identifies the type from the grades and applies the right AASHTO K minimum.

The high point of a crest or the low point of a sag is where the curve’s grade is zero. Measured from the PVC, that distance is minus g1 times L, divided by A. You then find its station and plug the distance into the parabola equation for its elevation.

The turning point only exists on the curve when the grades actually reverse sign; if both grades go the same way, the high or low point falls outside the curve.

For example, g1 of +3%, g2 of -2% over 600 ft puts the high point 360 ft past the PVC, at an elevation of about 996.4 ft. This calculator locates the high or low point’s station and elevation, or tells you it is off the curve.

The PVC (start) and PVT (end) are each half the curve length from the PVI, because a symmetrical vertical curve is centered on the grade intersection. The PVC station is the PVI station minus L over 2; the PVT station is the PVI station plus L over 2.

Their elevations follow the grades: the PVC elevation is the PVI elevation minus g1 times L over 2 (back up the incoming grade), and the PVT elevation is the PVI elevation plus g2 times L over 2 (down the outgoing grade).

For example, a 600 ft curve with its PVI at station 50+00, elevation 1000, has its PVC at 47+00, elevation 991, and its PVT at 53+00, elevation 994. This calculator gives all three points.

Elevations along a vertical curve follow the parabola, measured from the PVC. The elevation at a distance x past the PVC is the PVC elevation, plus g1 times x, plus A divided by twice the length, times x squared — with the grades as decimals.

The first two terms are the straight incoming grade line; the last term is the parabolic offset that bends it toward the outgoing grade. Working from the PVC keeps x positive across the whole curve.

For example, on a 600 ft crest curve 300 ft past the PVC (at the PVI station), the elevation works out to about 996.25 ft. This calculator builds a full station-by-station elevation table at your chosen interval.

AASHTO publishes a minimum K value for each design speed, separately for crest and sag curves, to provide stopping sight distance. The curve’s K must be at least that minimum; a higher K gives a longer, safer curve. The minimums rise steeply with speed.

For a 45 mph design, the minimum K is about 61 for a crest curve and about 79 for a sag; at 70 mph it is about 247 for crest and 181 for sag. Crest minimums grow faster because sight distance over a hump is harder to maintain at speed.

For example, a K of 120 comfortably exceeds the 61 crest minimum at 45 mph. This calculator checks your K against the AASHTO minimum for the design speed you pick.

They are the three defining points of a vertical curve. The PVI, the point of vertical intersection, is where the two grade lines would meet if extended — the apex of the grade change, and where you usually know the station and elevation. The PVC, the point of vertical curvature, is where the curve begins and leaves the incoming grade. The PVT, the point of vertical tangency, is where the curve ends and rejoins the outgoing grade.

On a symmetrical curve the PVC and PVT sit half the curve length either side of the PVI.

For example, with a PVI at station 50+00 and a 600 ft curve, the PVC is at 47+00 and the PVT at 53+00. This calculator reports the station and elevation of all three.