About Vertical Curve Calculator
The vertical curve calculator designs equal-tangent parabolic vertical curves that connect two highway tangent grades. It returns the algebraic grade difference A, the rate of vertical curvature K = L / A, the stations and elevations of the begin (BVC), intersection (PVI), and end (EVC) of the curve, and the station and elevation of the high point on a crest curve or low point on a sag curve.
Enter the incoming grade g1 and outgoing grade g2 in percent, the curve length L, and the PVI station and elevation in consistent length units. The tool evaluates the parabolic elevation equation along the curve, plots the profile, and computes the minimum curve length required to provide stopping sight distance using the AASHTO crest and sag (headlight) length formulas.
How It Works
- Enter the incoming grade g1 and outgoing grade g2 in percent (positive uphill, negative downhill).
- Enter the curve length L and the PVI station and elevation in consistent length units (feet).
- The calculator computes A = |g2 - g1|, K = L / A, and the BVC and EVC by offsetting half the length each side of the PVI along the tangents.
- It evaluates the parabola y(x) = y_BVC + (g1/100) x + (g2 - g1)/(200 L) x^2 to locate the high or low point and plot the profile, and applies the AASHTO sight-distance length formulas.
Worked Example
A crest curve connects g1 = +3% to g2 = -2% with a length L = 400 ft, PVI at station 5000 and elevation 1000 ft. The grade difference is A = |-2 - 3| = 5, so K = 400 / 5 = 80. The BVC is at station 4800, elevation 1000 - 0.03 * 200 = 994 ft; the EVC is at station 5200, elevation 1000 - 0.02 * 200 = 996 ft. The high point is at x = -g1 L / (g2 - g1) = -3 * 400 / -5 = 240 ft from the BVC (station 5040), at elevation 994 + 0.03 * 240 - 5/(200*400) * 240^2 = 997.6 ft.
Formulas
- Grade difference and rate of curvature
A = |g2 - g1|, K = L / A- Parabolic profile elevation
y(x) = y_BVC + (g1/100) x + (g2 - g1)/(200 L) x^2- High / low point offset from BVC
x* = -g1 L / (g2 - g1)- Crest length for sight distance (AASHTO, US)
S < L: L = A S^2 / 2158 | S > L: L = 2 S - 2158 / A- Sag length for sight distance (AASHTO headlight, US)
S < L: L = A S^2 / (400 + 3.5 S) | S > L: L = 2 S - (400 + 3.5 S) / A
Standards & References
- AASHTO A Policy on Geometric Design of Highways and Streets (Green Book)
- Equal-tangent parabolic vertical curve geometry
- Crest constant 2158 ft for h1 = 3.5 ft eye and h2 = 2.0 ft object height
Frequently Asked Questions
What is the rate of vertical curvature K?
K is the horizontal distance, in feet or metres, required to achieve a 1 percent change in grade: K = L / A. AASHTO publishes minimum K values by design speed, so the required curve length is simply L = K * A. Larger K gives a flatter, more comfortable curve.
How is the high point or low point of a vertical curve located?
The turning point sits where the curve grade equals zero, at x* = -g1 L / (g2 - g1) measured from the BVC. It only falls on the curve when the grades change sign; if both grades have the same sign the highest or lowest point is at an endpoint.
When do I use the S < L versus S > L sight-distance formula?
Compute the length assuming S < L first; if the result is greater than the sight distance S, that case governs. Otherwise use the S > L formula. This calculator selects the governing case automatically for both crest and sag curves.
What is the difference between a crest and a sag curve?
A crest curve has g2 less than g1 (the road bends downward, like the top of a hill) and is controlled by stopping sight distance over the crest. A sag curve has g2 greater than g1 (a valley) and is usually controlled by headlight sight distance at night.