How to use the horizontal curve calculator
- Choose feet or meters and enter the curve radius and the deflection angle in decimal degrees.
- Optionally enter the PI station, the distance along the road to the point of intersection, to get the PC and PT stations.
- Set how often the deflection table lists a point, and in Advanced options the length that defines the degree of curve.
- Read the tangent, curve length, long chord, external distance, middle ordinate and degree of curve, then the deflection angles from the PC.
Formula
Tangent T = R tan(delta / 2); curve length L = R x delta (in radians)
Long chord LC = 2 R sin(delta / 2)
External E = R (sec(delta / 2) - 1); middle ordinate M = R (1 - cos(delta / 2))
Deflection from the PC to a point s along the arc = s / (2 R); chord from the PC = 2 R sin(s / (2 R))
R is the radius and delta the deflection angle between the two tangents. The degree of curve by the arc definition is 180 x base / (pi x R), where base is the length that defines it, 100 for the usual feet example.
A 1,000 foot radius turning 30 degrees
With R = 1,000 ft and delta = 30 degrees, the tangent is 267.949 ft, the curve length 523.599 ft, the long chord 517.638 ft, the external distance 35.276 ft and the middle ordinate 34.074 ft. The degree of curve is 5.7296 degrees by the arc definition and 5.732 by the chord definition, using 100 ft. With the PI at station 12+50, the PC is at 9+82.05 and the PT at 15+05.65.
A deflection table every 100 ft starts at 2.8648 degrees with a 99.958 ft chord at 100 ft, and ends at 15 degrees with a 517.638 ft chord at the PT. A metric curve with a 250 m radius and a 47.25 degree deflection has a tangent of 109.352 m and a length of 206.167 m.
What the elements are
A simple horizontal curve joins two straight tangents with a circular arc. The PC is the point where the curve starts, the PT where it ends, and the PI is where the two tangents would meet. The tangent length is the distance from the PC or PT to the PI. The external distance runs from the PI to the middle of the curve, and the middle ordinate from the middle of the long chord to the middle of the curve.
Elevations along the road can be reduced with the differential leveling calculator, volumes of earth with the average end area calculator, and old deed units with the surveyor's chain converter. Grades are covered by the slope calculator.
Degree of curve
Roads were once described by the degree of curve instead of the radius. By the arc definition it is the central angle that subtends 100 feet of arc, so a larger degree means a tighter curve. By the chord definition it subtends a 100 foot chord. The two are nearly equal for gentle curves. Your agency may use a different base length, which you can enter in Advanced options.
Assumptions and limits
- The curve is a single circular arc with no spiral transitions and no superelevation.
- The deflection angle is in decimal degrees, above 0 and below 180.
- Stations are plain distances along the alignment, shown as 12+34.56 for feet.
- These are geometric elements. Design speed, sight distance and field layout rules are not modeled.
Frequently asked questions
How do I calculate the tangent length of a curve?
Multiply the radius by the tangent of half the deflection angle. A 1,000 foot radius turning 30 degrees has a tangent of 267.95 feet.
What is the PI in a horizontal curve?
It is the point of intersection, where the two straight tangents would meet if extended. The PC is a tangent length before it and the PT a tangent length after it.
What is the degree of curve?
It is a way of stating sharpness. By the arc definition it is the central angle for 100 feet of arc, so degree = 5729.58 / radius in feet.
How do I find the length of a curve?
Multiply the radius by the deflection angle in radians. A 30 degree turn on a 1,000 foot radius is 523.6 feet of arc.