Vertical Curve Calculator

A vertical curve calculator solves the parabolic curve that joins two roadway grades. From the entry grade g1, exit grade g2, and curve length L it computes the elevations of the PVC, PVI, and PVT, the high or low point, the K-value (L/A), tangent offsets, and checks the length against AASHTO stopping sight distance requirements. Highway designers use it for vertical alignment; students use it to check homework.

With the PVC as origin, the curve elevation at distance x is y(x) = y_PVC + g1·x/100 + (g2−g1)·x²/(200·L), where g1 and g2 are in percent and x and L are in feet. The first two terms are the tangent grade line; the last term is the parabolic offset from the tangent. AASHTO vertical curves are parabolas because they give a uniform rate of change of grade.

PVC is the Point of Vertical Curvature, where the curve begins. PVI is the Point of Vertical Intersection, where the two tangent grades would meet if extended. PVT is the Point of Vertical Tangency, where the curve ends. On the usual symmetric curve the PVI sits exactly at mid-length: station_PVI = station_PVC + L/2 and elevation_PVI = elevation_PVC + g1·L/200.

A crest (summit) curve has g1 greater than g2, is concave downward, and contains a high point; its length is governed by daytime stopping sight distance over the top. A sag (valley) curve has g1 less than g2, is concave upward, and contains a low point; its length is governed by nighttime headlight sight distance, plus rider comfort and drainage at the bottom.

K = L/A is the horizontal distance in feet required for a 1% change in grade, where A = |g2 − g1| in percent. A larger K means a flatter, longer curve. Example from the AASHTO sight-distance formulas: a crest with A = 5% designed for S = 570 ft needs L ≈ 752.7 ft, so the required K is about 150.5.

Set the grade along the curve to zero: x = −g1·L/(g2−g1), measured in feet from the PVC with signed grades in percent. Example: g1 = +3%, g2 = −2%, L = 400 ft gives x = 240 ft from the PVC; with a PVC elevation of 100.00 ft the high point elevation is 103.60 ft. If x falls outside 0 to L, the curve has no high or low point.

A = |g2 − g1| is the total change in grade in percent. For an entry grade of +3% and an exit grade of −2%, A = 5. It drives everything else: K = L/A, the middle ordinate A·L/800, and the sight-distance length formulas.

AASHTO gives two cases (S and L in feet, driver eye 3.5 ft, object 2.0 ft): if S ≤ L, L = A·S²/2158; if S > L, L = 2S − 2158/A. Design procedure: assume S ≤ L, compute L, and keep it if L ≥ S, otherwise use the second formula. Example: A = 5%, S = 570 ft → L = 5×570²/2158 ≈ 752.7 ft, which is ≥ 570 ft, so the assumption holds.

AASHTO (headlight 2.0 ft, beam 1° above horizontal): if S ≤ L, L = A·S²/(400 + 3.5·S); if S > L, L = 2S − (400 + 3.5·S)/A. Example: A = 6.4%, S = 196.5 ft → L ≈ 227.6 ft. Comfort adds a second check, L ≥ A·V²/46.5 with V in mph; for A = 6.4 and V = 30 mph that is 123.9 ft.

Yes. Put g1 in A1, g2 in B1, L in C1, PVC elevation in D1, and distance x in E1: A =ABS(B1−A1); K =C1/A; curve elevation =D1+A1*E1/100+(B1−A1)*E1^2/(200*C1); high/low distance from PVC =−A1*C1/(B1−A1). This page runs the same math interactively, with a full station table and profile plot.

In AASHTO practice, almost always: half the curve length falls on each side of the PVI, which keeps the math and the staking simple. This calculator assumes symmetric (equal-tangent) parabolic curves, the standard for highway design.

US customary units throughout: feet for curve length, station, and elevation, and percent for grades (g1, g2, A). The K-value is therefore in feet per 1% grade change, and sight distances are in feet.

With tangent offsets. The vertical offset from the tangent grade line at distance x from the PVC is (g2−g1)·x²/(200·L). Example: +3% to −2% over L = 400 ft — at x = 100 ft the offset is −0.625 ft, so the curve elevation is the tangent elevation 103.00 ft minus 0.625 ft = 102.375 ft (PVC at 100.00 ft).

On curbed roadways, keep at least 0.30% grade within 50 ft of the low point so water drains off the pavement (AASHTO guidance). The grade d feet from the level point is d/K: at K = 80 the grade 50 ft out is 50/80 = 0.625%, which passes. Very flat (large-K) sag curves must be checked.

No. This is a planning and education tool. Final roadway design must follow the governing agency standards, such as the AASHTO Green Book and the applicable state DOT manual, and be prepared or reviewed by a licensed professional engineer. The formulas here follow AASHTO, but agency criteria always govern.