Elevation on a Parabolic Vertical Curve

Vertical curves are parabolas, not circles, and for one good reason: a parabola changes grade at a constant rate, so a vehicle travelling at constant speed feels a constant vertical acceleration. The elevation at any point is the tangent grade line plus a correction term that grows with the square of the distance from the BVC. Everything is measured from the beginning of vertical curve, and the standard American profile uses equal tangents, so the BVC sits L/2 before the PVI and the EVC sits L/2 after it.

A worked example: a road entering at g₁ = −3 % meets a +2 % grade over a 400 ft sag, so A = +5 %, and the BVC elevation is 100.00 ft. At x = 100 ft past the BVC, E = 100.00 − 3.00 + 5 × 100²/(200 × 400) = 100.00 − 3.00 + 0.625 = 97.625 ft. The trap is the offset term's denominator: 200L, not 2L. The 100 that converts percent to a decimal hides inside it, and dropping it inflates the curve correction a hundredfold. Solving for x is a quadratic, since a crest crosses any given elevation on both sides of its high point.