How this instrument works
A vertical curve is the smooth parabolic transition civil engineers design into a road, driveway or site profile wherever two straight grades of different slope meet — a crest where the road tips over the top of a hill, or a sag where it dips through a low point. Connecting two grades with a sharp angle instead of a curve would create an abrupt, uncomfortable and potentially unsafe change in direction for a moving vehicle, so highway design standards specify a smooth parabolic curve instead, long enough to keep the transition gradual and, on crests, to preserve enough sight distance to see hazards ahead.
The curve begins at a point called the BVC (beginning of vertical curve) and ends at the EVC (end of vertical curve), with the two grades — g1 entering the curve and g2 leaving it — meeting at a theoretical point called the PVI if the curve weren't there. Elevation anywhere along the curve is a parabola: elevation starts at the BVC's known elevation, rises or falls initially at the g1 grade rate, and then a curvature term — proportional to the square of the distance from the BVC and scaled by how much the grade changes from g1 to g2 — bends that straight-line path into the smooth parabolic transition.
This formula, published in AASHTO's 'A Policy on Geometric Design of Highways and Streets' (the Green Book), is standard practice for road, driveway and site-grading design: an engineer given the two grades, the chosen curve length and a starting elevation can compute the design elevation at any station along the curve, which is exactly what a survey crew needs staked out on the ground and what a grading plan needs shown on paper.
- Enter the grade the road is on before the curve into Initial grade g1 (%) — positive for uphill, negative for downhill.
- Enter the grade the road transitions to after the curve into Final grade g2 (%) — again positive for uphill, negative for downhill.
- Enter the curve's total length into Curve length L (ft) — the distance from the BVC to the EVC.
- Enter how far along the curve you need a reading into Distance from BVC (ft) — must be between 0 and the curve length.
- Enter the known starting elevation into Elevation at BVC (ft), then read Elevation at station for the design elevation at that point on the curve.
Worked example — a 400-ft crest curve, +2% to −3%
Enter 2 into Initial grade g1 (%), -3 into Final grade g2 (%), 400 into Curve length L (ft), 200 into Distance from BVC (ft), and 100 into Elevation at BVC (ft) — a 400-foot crest vertical curve read at its exact midpoint. Elevation at station reads 101.500 ft.
By hand: (g2−g1)/100 = (−3−2)/100 = −0.05, divided by 2×400 = 800 gives −0.0000625, and multiplying by x² = 200² = 40,000 gives −2.5. The straight-line term is g1×x = 0.02×200 = 4. Adding it all up: 100 + 4 − 2.5 = 101.5 ft — the curve's designed elevation at its midpoint, lower than the straight-grade projection because the road is already curving over toward the downhill −3% grade.
Questions
What do BVC, EVC and PVI stand for?
BVC is the beginning of vertical curve, where the parabolic transition starts; EVC is the end of vertical curve, where it finishes and the new grade takes over in a straight line; PVI is the point of vertical intersection, the theoretical point where the two straight grades would meet if the curve weren't inserted between them. This instrument measures the station offset (x) from the BVC, which is where the elevation formula's x=0 starting point sits.
Why is the curvature term proportional to x squared rather than x?
Because a vertical curve is specifically a parabola, and a parabola's defining feature is that its curvature term scales with the square of the distance from a reference point — that's what makes the elevation profile bend smoothly rather than following a straight line. The (g2−g1)/(2L) coefficient scales how sharply that x² term bends the curve, based on how much the grade changes and how long the curve is.
What happens if g1 and g2 are equal?
The (g2−g1) term becomes zero, which eliminates the entire curvature term and leaves a plain straight-line grade calculation: elevation = elevBVC + g1×x. This is a useful sanity check — equal grades mean there's no actual direction change to curve through, so the 'vertical curve' formula degenerates back into simple straight-line slope math.
Is a crest curve calculated differently from a sag curve?
No — the identical formula handles both; the difference is simply in the sign and relationship of g1 and g2. A crest curve typically has a positive g1 transitioning to a less positive or negative g2 (the road curving downward over a high point), while a sag curve has a negative g1 transitioning to a less negative or positive g2 (the road curving upward through a low point). The formula's algebra handles either case automatically once the correct signed grades are entered.
Where does the curve length L come from — is it something I calculate or choose?
Curve length is typically a design decision, chosen by an engineer to satisfy minimum sight-distance and rider-comfort requirements published in AASHTO's Green Book, based on the road's design speed and the algebraic difference between g1 and g2. This instrument takes the curve length as a given input rather than deriving it, since choosing L involves standards and judgment beyond a single formula.