Vertical Curve Calculator (Parabolic Crest & Sag Curves)

How it works

BVC = PVI − L/2, EVC = PVI + L/2 · elevation at x from the BVC: y = y_BVC + g₁·x + (g₂ − g₁)·x²/(2L) · high/low point at x = −g₁·L/(g₂ − g₁) · A = g₂ − g₁ (in %) · K = L/|A| · middle ordinate e = A·L/800 (A in %).

Highway and street profiles use equal-tangent parabolic curves between two grades because the rate of change of grade is constant, which gives a smooth ride and simple calculations. The curve starts at the BVC half the curve length before the point of vertical intersection (PVI) and ends at the EVC half a length after it; the offset from the tangent grows with the square of the distance. Crest curves (grade decreasing) are governed by stopping sight distance, sag curves by headlight sight distance and comfort; AASHTO publishes minimum K values for each design speed — for example K = 26 m/% crest and 30 m/% sag at 80 km/h.

Worked example

A +3% grade meets a −2% grade at a PVI at station 1+500.00, elevation 100.00 m, with a 200 m curve. BVC = 1+400.00 at 100 − 0.03 × 100 = 97.000 m and EVC = 1+600.00 at 98.000 m. At station 1+450 (x = 50 m): y = 97 + 0.03 × 50 − 0.05 × 50²/(2 × 200) = 98.188 m. It is a crest curve with A = −5%, K = 40.00, and its high point is at x = 0.03 × 200/0.05 = 120 m, station 1+520.00, elevation 98.800 m.

Sources: Vertical curve — equal-tangent parabolic curves · FHWA — Flexibility in Highway Design, Chapter 7: Vertical Alignment

Frequently asked questions