When to use: A simple circular curve connects two tangents through a single arc of radius R and deflection angle Ξ. From these two inputs all geometric elements are fixed: the tangent length T (PI to PC/PT), curve length L along the arc, long chord LC, middle ordinate M, external distance E, and the degree of curve D. Used to lay out roadway, rail, and pipeline alignments. All lengths in US customary feet.
This calculator computes all geometric elements of a simple circular horizontal curve from just two inputs β radius R and deflection angle Ξ β using the AASHTO equations that govern road, rail, and pipeline alignment design.
A simple circular curve connects two tangent lines through a circular arc of radius R. The deflection (intersection) angle Ξ is the angle between the two tangent directions measured at the Point of Intersection (PI). From R and Ξ all curve elements are uniquely determined:
Tangent length T = R Β· tan(Ξ/2) is the distance from the PC (Point of Curvature) or PT (Point of Tangency) to the PI along each tangent. Curve length L = R Β· Ξ (with Ξ in radians) is the arc length. Long chord LC = 2R Β· sin(Ξ/2) is the straight-line distance from PC to PT. Middle ordinate M = R Β· (1 β cos(Ξ/2)) is the distance from the midpoint of the long chord to the midpoint of the arc. External distance E = R Β· (sec(Ξ/2) β 1) is the distance from the PI to the midpoint of the arc. Degree of curve Dc = 5729.58 / R defines curvature in the arc definition used by US highway engineers.
Horizontal curve design is governed by AASHTO A Policy on Geometric Design of Highways and Streets (the "Green Book"), which tabulates minimum radii by design speed for US highways. For rail design, AREMA Manual for Railway Engineering specifies curve standards by degree of curve and speed class. State DOTs publish supplemental design manuals that adopt AASHTO minima or set more restrictive standards for specific terrain or functional classes. The Federal Railroad Administration (FRA) regulates maximum degree of curve for passenger rail by speed under 49 CFR Part 213.
Minimum radius is controlled by design speed and superelevation. AASHTO specifies absolute minimum radii ranging from 150 ft at 20 mph to 5860 ft at 80 mph (with maximum superelevation of 6%). Sight distance on horizontal curves must equal or exceed the stopping sight distance, which requires the middle ordinate M to be clear of obstructions within the right-of-way.
Spiral transition curves are added between tangents and circular curves on high-speed roadways to gradually introduce centrifugal force and superelevation runoff. AASHTO recommends spirals when speeds exceed 50 mph and radii are less than about 1500 ft. The spiral length is determined by the superelevation runoff length requirement.
Enter the deflection angle Ξ in decimal degrees and the radius R in feet. All six curve elements (T, L, LC, M, E, Dc) are computed instantly. Verify that the radius meets the AASHTO minimum for your design speed. Use T to locate the PC and PT stations from the PI station (PC = PI β T; PT = PC + L). The long chord LC and middle ordinate M are used for staking the curve in the field using the chord-offset or deflection angle methods.
AASHTO recommends a minimum radius of 700β860 ft for 45 mph design speed with 4β6% maximum superelevation on rural highways. Urban arterials at 45 mph may use flatter radii with lower superelevation. Always check your state DOT design manual, which may be more restrictive.
The arc definition (used by US highway engineers) defines Dc as the central angle subtended by a 100-ft arc: Dc = 5729.58 / R. The chord definition (used by some railroad engineers) defines Dc as the central angle subtended by a 100-ft chord: sin(Dc/2) = 50/R. For small angles the two are nearly identical, but they diverge significantly for sharp curves (Dc > 5Β°).
Given the PI station: PC = PI station β T; PT = PC + L. For example, with PI at station 25+00, T = 145.31 ft and L = 261.80 ft: PC = 25+00 β 1+45.31 = 23+54.69; PT = 23+54.69 + 2+61.80 = 26+16.49. Station arithmetic is standard surveying practice for all alignment elements.
The external distance E is the distance from the PI to the nearest point on the curve (the midpoint of the arc). It is used to check sight distance clearances near the PI and to locate obstacles within the curve's swept path. On sharp curves E can be substantial β a 500 ft radius 60Β° curve has E β 58 ft.
AASHTO recommends spiral transitions for curves with radii below about 1000β2000 ft on roads with design speeds above 50 mph, where the abrupt introduction of centripetal acceleration at the PC/PT can cause driver discomfort. Below 50 mph or on urban streets, circular curves without spirals are generally acceptable. Railroad design almost always uses spirals on main line tracks.