Compound Circular Curve
Two circular arcs of different radii turning the same way, meeting at a point of compound curvature (PCC).
Example
You enter
- First radius R1 (ft) 500
- Second radius R2 (ft) 800
- First central angle delta1 (deg) 30
- Second central angle delta2 (deg) 25
You get
- Arc1 tangent (ft) 133.97
- Arc2 tangent (ft) 177.36
- Back tangent (ft) 294.6
- Forward tangent (ft) 367.39
- Total length (ft) 610.87
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
ScopeThe interchange-ramp / curb-return geometry between the simple and spiral curves. Each arc T = R tan(delta/2), L = R delta_rad; the tangents back to the PI follow from the law of sines: back tangent t1 = T1 + (T1+T2) sin(delta2)/sin(delta), forward t2 = T2 + (T1+T2) sin(delta1)/sin(delta), delta = delta1 + delta2. A 500 ft/30-deg arc into an 800 ft/25-deg arc gives 134.0 and 177.4 ft semi-tangents, a 294.6 ft back tangent, a 367.4 ft forward tangent, 610.9 ft total. Reduces to a simple curve at equal radii/angles. Same-direction arcs (a reverse curve is separate); AASHTO limits successive radii to ~1.5:1. The engineer of record governs.
Formula or tableT1 = R1 tan(delta1/2); T2 = R2 tan(delta2/2); L = R delta_rad; back tangent t1 = T1 + (T1+T2) sin(delta2)/sin(delta); forward tangent t2 = T2 + (T1+T2) sin(delta1)/sin(delta); delta = delta1 + delta2.
Source and editionCompound circular-curve geometry per the AASHTO A Policy on Geometric Design of Highways and Streets (the Green Book) and Ghilani & Wolf, Elementary Surveying, by name; first-principles trig with the law of sines.
Free accessThe compound-curve tangent geometry (semi-tangents and the law-of-sines vertex triangle) 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, delta1_deg, delta2_deg, arc1_tangent_ft, arc2_tangent_ft, back_tangent_ft, forward_tangent_ft, total_length_ft
- Semi-tangents and lengths T = R tan(delta/2); L = R delta_rad per arcAASHTO / Ghilani & Wolf
- PI tangents t1 = T1 + (T1+T2) sin(delta2)/sin(delta); t2 = T2 + (T1+T2) sin(delta1)/sin(delta)law of sines on the vertex triangle
- Scope same-direction arcs (not a reverse curve); reduces to a simple curve at equal radii/anglesscope of this tile