Reverse (S) Curve Between Parallel Tangents
Two opposite arcs that shift an alignment onto a parallel line, like a crossover or lane shift.
Example
You enter
- First radius R1 (ft) 500
- Second radius R2 (ft) 500
- Offset between parallel tangents p (ft) 60
You get
- Central angle I (each arc) 19.948 deg
- Arc1 tangent (ft) 87.93
- Tangent-point distance 341.17 ft
- Total length (ft) 348.17
No published worked example is on file for this calculation. It is checked against a hand-worked derivation of its method.
Details, formula, and sources
ScopeTwo arcs turning opposite ways that meet at a point of reverse curvature (PRC) and return the alignment to a line parallel to where it began - a railroad crossover or a lane shift. Because the tangents are parallel, both arcs sweep the same central angle I = arccos(1 - p/(R1+R2)), where p is the perpendicular offset between the tangents. Each arc T = R tan(I/2), L = R I_rad; the tangent points are (R1+R2) sin I apart. Shifting between tracks 60 ft apart on 500 ft radii turns each arc 19.9 deg, 348 ft of curve over a 341 ft reach. With equal radii the arcs mirror. A reverse curve leaves no superelevation runout, so AASHTO/AREMA want a tangent or spiral at the PRC above low speeds. The engineer of record governs.
Formula or tableparallel tangents: both arcs sweep the same central angle I = arccos(1 - p/(R1+R2)) for a perpendicular offset p; T = R tan(I/2) and L = R I_rad per arc; tangent-point distance = (R1+R2) sin I.
Source and editionReverse (S) curve geometry between parallel tangents per Ghilani & Wolf, Elementary Surveying, and the AASHTO A Policy on Geometric Design of Highways and Streets (the Green Book), by name; first-principles circular-curve trig.
Free accessThe reverse-curve offset geometry (equal central angle from the tangent offset, semi-tangents, arc lengths) is standard published route-surveying math; the design of record and the engineer of record govern the alignment.
What governsEstimate. Engineer of record governs the design and acceptance. Verify against the project structural drawings and the manufacturer's published capacity / chart.
API field namesr1_ft, r2_ft, offset_ft, central_angle_deg, arc1_tangent_ft, distance_ft, total_length_ft
- Equal central angle I = arccos(1 - p/(R1+R2)); both arcs sweep I so the tangents stay parallelAASHTO / Ghilani & Wolf
- Semi-tangents and lengths T = R tan(I/2); L = R I_rad per arc; tangent-point distance (R1+R2) sin Ifirst-principles circular-curve trig
- Scope opposite-curvature arcs between parallel tangents; insert a tangent/spiral at the PRC for superelevation runoutscope of this tile