How to use the vertical curve calculator
- Enter the entry grade and exit grade in percent, positive for uphill in the direction of travel.
- Enter the curve length, the PVI station and the PVI elevation. Use one length unit for all of them.
- Set the interval for the table of elevations.
- Read the K value, the start and end of the curve, the high or low point and the table of elevations along the curve.
Formula
PVC station = PVI station - L / 2; PVC elevation = PVI elevation - g1 x L / 2
Elevation at distance x from the PVC: y = y(PVC) + g1 x + (g2 - g1) x^2 / (2 L)
High or low point at x = -g1 L / (g2 - g1), if that is between 0 and L
K = L / |A|, where A = g2 - g1 in percent; the curve sits (g2 - g1) L / 8 from the PVI
Grades are decimals in the elevation formula (0.03 for 3 percent). The curve is a symmetric parabola whose ends touch the two grade lines at the PVC and the PVT. A negative offset means the curve lies below the PVI, which is a crest.
A crest curve and a sag curve
A crest from +3 percent to -2 percent over 400 ft with the PVI at station 1,000 and elevation 100 has A = -5 and K = 80. The curve starts at station 800, elevation 94, and ends at station 1,200, elevation 96. The high point is at x = 0.03 x 400 / 0.05 = 240 ft, which is station 1,040, elevation 97.6. The curve is 2.5 ft below the PVI, and at 50 ft intervals the elevations run 94, 95.34375, 96.375, 97.09375, 97.5, 97.59375, 97.375, 96.84375 and 96.
A sag from -2 to +3 percent over 300 ft with the PVI at station 1,500 and elevation 250 has K = 60, a low point at station 1,470 and elevation 251.8, and sits 1.875 ft above the PVI. A metric crest from +4 to -3 percent over 120 m has K = 17.1429 and a high point at station 808.57143, elevation 44.471429.
What K tells you
K is the length of curve needed for each percent of grade change. A larger K means a flatter, gentler curve. Design standards give minimum K values for sight distance and comfort, and they differ by agency and speed, so the page reports K and leaves the limit to your standard.
For the slope of a single run, see the ramp slope calculator and the elevation profile grade calculator. The triangle calculator helps with the straight grade lines.
Stations and units
Stations are entered as plain distances, so 12+50 is entered as 1250. The unit can be feet or meters as long as the length, stations, interval and elevations all use the same one. The table always includes the start and the end of the curve, even when the interval does not divide the length evenly.
Assumptions and limits
- The curve is a symmetric parabola, with half its length on each side of the PVI.
- Grades are in percent and run in the direction of increasing station.
- The two grades must differ. If they are equal there is no curve.
- Sight distance, drainage and the design speed are not checked.
Frequently asked questions
How do I find the high point of a crest vertical curve?
Use x = -g1 L / (g2 - g1) measured from the start of the curve. With +3 percent, -2 percent and 400 ft that is 240 ft from the start.
What is the K value of a vertical curve?
K = L / |A|, the curve length per percent of grade change. A 400 ft curve with a 5 percent change has K = 80.
What is the PVI?
It is the point of vertical intersection, where the two grade lines would meet if there were no curve. The curve starts half its length before it and ends half its length after.
Can I use meters?
Yes. Use meters for the length, the stations, the interval and the elevations. The grades stay in percent.