About Horizontal Curve Calculator (Highway Circular Curve)
The horizontal curve calculator designs a simple (circular) highway curve. It first finds the minimum radius for a chosen design speed using the AASHTO point-mass equation R_min = V^2 / (127*(e + f)) in SI units (V in km/h) or R_min = V^2 / (15*(e + f)) in US units (V in mph), where e is the superelevation rate and f the side-friction factor.
It then computes the complete geometry of a circular arc connecting two tangents that meet at the point of intersection (PI) with a deflection angle Delta: tangent length T, arc length L, external distance E, middle ordinate M, long chord LC, the degree of curve D, and the stationing of the point of curvature (PC) and point of tangency (PT). It also flags whether the chosen radius satisfies the minimum.
How It Works
- Choose the unit system (SI metres / km/h, or US feet / mph) and enter the design speed, superelevation rate e, and side-friction factor f.
- The calculator evaluates the minimum radius R_min = V^2 / (g*(e + f)), where g = 127 for SI and g = 15 for US units.
- Enter the chosen radius R and the deflection angle Delta; the simple-curve geometry follows from T = R*tan(Delta/2), L = R*Delta (in radians), E = R*(sec(Delta/2) - 1), M = R*(1 - cos(Delta/2)), and LC = 2*R*sin(Delta/2).
- The degree of curve is D = 1746.4 / R (SI, 100 m arc) or 5729.58 / R (US, 100 ft arc), and the stationing is PC = PI - T and PT = PC + L.
Worked Example
A 100 km/h highway curve uses superelevation e = 0.06 and side friction f = 0.12, giving R_min = 100^2 / (127*0.18) = 10000 / 22.86 = 437.4 m. A radius R = 400 m would be below this minimum, so it is flagged inadequate. For the geometry of an R = 400 m curve with a deflection angle Delta = 30 deg: T = 400*tan(15 deg) = 107.18 m, L = 400*(30*pi/180) = 209.44 m, E = 400*(sec(15 deg) - 1) = 14.11 m, M = 400*(1 - cos(15 deg)) = 13.63 m, and LC = 2*400*sin(15 deg) = 207.06 m. With the PI at station 1000, PC = 1000 - 107.18 = 892.82 and PT = 892.82 + 209.44 = 1102.26. The degree of curve is D = 1746.4/400 = 4.366 deg.
Formulas
- Minimum radius from design speed
R_min = V^2 / (127*(e + f)) [SI, km/h] or V^2 / (15*(e + f)) [US, mph]- Simple circular-curve geometry
T = R*tan(Delta/2); L = R*Delta(rad); E = R*(sec(Delta/2)-1); M = R*(1-cos(Delta/2)); LC = 2*R*sin(Delta/2)- Degree of curve (arc definition)
D = 1746.4 / R [SI, 100 m] or 5729.58 / R [US, 100 ft]- Stationing
PC = PI - T; PT = PC + L
Standards & References
- AASHTO A Policy on Geometric Design of Highways and Streets (Green Book)
- Point-mass equation R_min = V^2/(g*(e+f)), g = 127 (SI) or 15 (US)
- Arc definition of degree of curve: 1746.4/R (SI 100 m), 5729.58/R (US 100 ft)
Frequently Asked Questions
What is the minimum radius of a horizontal curve?
The minimum radius is the smallest curve radius that keeps a vehicle at the design speed from skidding, given the available superelevation and side friction. It is R_min = V^2 / (127*(e + f)) in SI units (V in km/h) or V^2 / (15*(e + f)) in US units (V in mph). Higher speed needs a larger radius; more superelevation or friction allows a smaller one.
What is the difference between the degree of curve and the radius?
The radius R is the geometric radius of the circular arc, while the degree of curve D is the central angle subtended by a standard 100 m (SI) or 100 ft (US) arc. They are inversely related: D = 1746.4/R in metric or 5729.58/R in US units. A sharp curve has a small radius and a large degree of curve.
How are the PC and PT stations found?
The point of curvature PC is located one tangent length back from the point of intersection: PC = PI - T. The point of tangency PT is one curve length ahead of the PC along the arc: PT = PC + L. The tangent T and length L come from the radius and the deflection angle.
What is the external distance and the middle ordinate?
The external distance E is the distance from the PI to the midpoint of the curve, E = R*(sec(Delta/2) - 1). The middle ordinate M is the distance from the midpoint of the long chord to the midpoint of the curve, M = R*(1 - cos(Delta/2)). Both describe how far the arc departs from the tangents and are used for clearance and stakeout.