Roads and transportation
Horizontal Curve Calculator
Road and pipeline alignments use circular curves between tangents. Enter the radius and the deflection (intersection) angle to get the standard curve elements used for stationing and layout.
Formula
With Δ in degrees: T = R tan(Δ/2), L = πRΔ/180, LC = 2R sin(Δ/2), E = R[1/cos(Δ/2) - 1], M = R[1 - cos(Δ/2)], arc definition degree of curve D = 5,729.58/R. PC = PI - T and PT = PC + L (stationing follows the curve, not the tangents).
Worked example
R = 1,000 ft, Δ = 30 degrees, PI at station 50+00: T = 1,000 × tan 15° = 267.95 ft, L = 523.60 ft, LC = 517.64 ft, E = 35.28 ft, M = 34.07 ft, D = 5.73 degrees. PC = 47+32.05 and PT = 52+55.65.
Assumptions and limits
- Simple circular curve; spiral transitions and compound curves are not included.
- Arc definition of degree of curve (highway practice); railroads use the chord definition.
Frequently asked questions
What is the minimum radius for a highway curve?
AASHTO Green Book: Rmin = V² ÷ [15(0.01e + f)], with V in mph, e the superelevation in percent and f the side friction factor. At 60 mph with e = 8% and f = 0.12 that is 1,200 ft, and the Green Book tabulates 1,200 ft.
Why is station PT not PI plus T?
Stationing runs along the curve. The PT is the PC plus the curve length, which is shorter than two tangent lengths.
What is the degree of curve?
The central angle subtended by a 100 ft arc. It is an older way to express sharpness; D = 5,729.58/R.