A vertical curve is the parabolic transition road and site designers use to connect two straight grades without a sharp break in slope. This calculator applies the standard AASHTO equation used in highway and driveway design to find the elevation anywhere along a crest or sag curve, plus the high or low point that governs sight distance and drainage.
How the Vertical Curve Calculator works
The calculator uses the standard centered-form parabolic vertical curve equation: elevation(x) = PVI_elev + g1×(x − L/2)/100 + ((g2 − g1)/(200×L))×(x − L/2)², where x is measured as a distance from the PVC (start of curve). The curve is centered on the PVI, which sits at exactly the midpoint of the curve length — that's why the offset uses (x − L/2) rather than x directly. A parabola is used (rather than, say, a circular arc) because it gives a constant rate of grade change per unit length, which is what makes sight-distance and comfort calculations tractable in road design.
The high/low point — where the curve's slope equals zero — is found by differentiating the elevation equation and solving for the station where the derivative is zero: x_hl = g1×L / (g1 − g2). This point only exists between the PVC and PVT when the grades genuinely reverse sign of curvature across that range; otherwise the curve's elevation is monotonic from end to end and there's no interior high or low point to report.
Inputs and what they mean
Entry grade (g1) and exit grade (g2) are the two straight tangent slopes the curve connects, entered as signed percentages. Curve length (L) is the horizontal distance from the PVC to the PVT, in feet — longer curves change grade more gradually, which is why higher-speed roads use longer curves for the same grade change. PVI elevation anchors the whole curve vertically; it's the elevation the two tangents would share if extended to meet at the curve's midpoint. Station is the point you want to evaluate, always measured as a distance from the PVC — entering an absolute highway station number instead of a PVC-relative distance is the single most common input mistake with this formula.
Limits and edge cases
When g1 equals g2 there's no curvature at all — it's a straight tangent, and there's no distinct high or low point to solve for, which the calculator flags directly. A station entered outside the 0–L range is mathematically extrapolated beyond the PVC or PVT and is not a valid point on the physical curve, so treat that result as informational only. This tool computes geometry only: it does not check minimum curve length for stopping sight distance, K-value design standards, or drainage adequacy at a sag's low point — those require checking against the AASHTO Green Book or your governing jurisdiction's design manual, and a low point without a drain inlet is a common real-world grading defect this calculator won't catch on its own.