Vertical Curve Calculator (crest & sag)

Understanding the vertical curve calculator — concept and principle

A vertical curve is the smooth transition between two road or railway grades. A crest curve joins an uphill grade to a flatter or downhill one, and a sag curve does the opposite. This calculator gives the elevation at any distance along a parabolic vertical curve, the location of its high or low point, and its K value, which designers compare against sight-distance and comfort criteria.

The parabola drops below the grade line by (g₂ − g₁)x² ÷ 2L — here 0.8 m at 80 m (heights exaggerated).

How it is calculated

Elevation: y = y₀ + g₁·x + (g₂ − g₁)·x²/(2L), with grades g₁ and g₂ as decimals (the calculator converts from percent), L the horizontal curve length and x the distance from the start of the curve (BVC).
High or low point: x = −g₁·L/(g₂ − g₁), where the slope of the curve becomes zero. It lies on the curve only if g₁ and g₂ have opposite signs.
K value: K = L/|g₂ − g₁|, with the grades in percent; it is the horizontal length needed for each 1% change in grade.

The principle

A parabola is used because its slope changes at a constant rate along the horizontal distance. The slope starts at g₁, changes linearly to g₂ over length L, and integrating that linearly changing slope gives the quadratic term (g₂ − g₁)x²/(2L). It is easy to set out with offsets from the tangents. K measures how gentle the curve is: the larger K, the longer the curve for the same change in grade, and the more sight distance over a crest.

Worked example

Inputs: g₁ = 3%, g₂ = −2%, L = 200 m, x = 80 m, start elevation 100 m.
Tangent rise: 0.03 × 80 = 2.4 m.
Curve correction: (−0.02 − 0.03)/(2 × 200) × 80² = −0.000125 × 6400 = −0.8 m.
Elevation at x = 100 + 2.4 − 0.8 = 101.6 m.
High point: x = −3/(−5/200) = 120 m from the start (elevation 101.8 m).
K = 200/|−2 − 3| = 200/5 = 40 m per %.

Each point is the calculator re-run with only “Start elevation (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Start elevation (m)	Elevation at x (m)
25	26.6
50	51.6
75	76.6
100	101.6
125	126.6
150	151.6
200	201.6

How to read the result

For a crest, a larger K gives longer sight distance over the hump; for a sag, it governs headlight sight distance and comfort. AASHTO's tabulated minimum K values for 80 km/h are about 26 for crest curves and 30 for sag curves, so K = 40 in the example comfortably exceeds both. Drainage also matters: on very flat curves near the high or low point, water may pond unless kerbs and inlets are planned.

Common mistakes

Mixing percent and decimals in the equation or in K.
Measuring x from the intersection point (PVI) instead of the curve start.
Reporting a high point outside 0–L as if it were on the curve.

Keep in mind: Minimum K values and curve lengths depend on the design speed, eye and object heights and the national road standard. Final vertical alignment must follow your local code and be checked by a qualified highway engineer.

Frequently asked questions

What is the PVI?

The point of vertical intersection, where the two tangent grades would meet; on a symmetric curve it lies at L/2 from the start.

How much does the curve depart from the PVI?

The middle ordinate is (g₂ − g₁)·L/8 with decimal grades; in the example it is 1.25 m.

Why use parabolas instead of circular arcs?

Over the small angles of road grades they are nearly identical, but parabolas are much simpler to compute and set out.

Sources: AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book) · N. J. Garber and L. A. Hoel, Traffic and Highway Engineering, Cengage Learning

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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