Vertical Curve Calculator

A vertical curve joins two tangent grades with a parabola rather than a circular arc, because a parabola produces a constant rate of grade change and makes elevation a simple quadratic in the distance from the point of vertical curvature. With grades expressed as decimals, elevation equals the PVC elevation plus G1 times x plus one half r times x squared, where r equals G2 minus G1 divided by the curve length. Differentiating gives the instantaneous grade, so the high point on a crest or the low point in a sag falls at x equal to minus G1 divided by r. That station is where surface water collects and where inlets must be placed. The algebraic difference A equals G2 minus G1 in percent, and the ratio L divided by A, the K value, is the curve length per percent of grade change and is how design tables are published. Crest curves are controlled by stopping sight distance using a driver eye height of 3.5 feet and an object height of 2.0 feet, which together produce the constant 2158 appearing in the standard crest length formula. Sag curves are controlled at night by headlight illumination, assumed to originate 2.0 feet above the pavement with the beam diverging 1 degree upward, giving the familiar denominator of 400 plus 3.5 times the sight distance. On flat sags, rider comfort, drainage, and appearance frequently govern instead of sight distance, and any curve shorter than roughly three seconds of travel time reads as a visible kink no matter what the sight distance arithmetic permits.

Parabolic vertical curves entered practice through railway location engineering in the nineteenth century, where surveyors needed a profile transition that could be staked out from tangent offsets computed with simple arithmetic. The parabola won over the circular arc for exactly that reason: offsets from the tangent vary as the square of the distance, so a table of offsets at each station is trivial to construct, and across the flat grades used on railroads the two curve types are visually indistinguishable. Turning that geometry into safety criteria was the work of American highway agencies in the middle of the twentieth century. The American Association of State Highway Officials published sight distance policies in the late 1930s and consolidated its geometric design guidance into a single volume in 1954, tying crest curve length to stopping sight distance and sag curve length to headlight illumination. The assumed driver eye height has fallen over the decades as passenger cars became lower, from 4.5 feet in early editions to 3.75 feet and then to the 3.5 feet used today. The design object height was raised from 6 inches to 2.0 feet in the 2001 Green Book after research indicated that the small-object assumption forced uneconomically long crest curves without measurable safety benefit. The K value tabulation, which lets a designer read curve length directly from design speed and grade difference, became the standard presentation in AASHTO tables during the later twentieth century.