Horizontal Curve Elements Explained
Horizontal curves connect two straight (tangent) sections of roadway through a circular arc, allowing vehicles to transition direction smoothly rather than through an abrupt angle point. Every curve is fully defined by two independent quantities: the radius (or its equivalent, degree of curve) and the intersection angle formed by the two tangents.
Radius vs. Degree of Curve
Radius (R) is the geometric definition most familiar from surveying and CAD. Degree of curve (D) — the central angle subtended by a 100-ft arc — is a legacy field convention still used in stationing and construction documents: D = 5,729.578 / R. A larger D means a sharper, tighter curve; a smaller D means a flatter, gentler one. The two describe the same curve and convert directly into each other.
The Five Key Elements
Once R and the intersection angle (Δ) are known, five derived elements define the curve for layout and design: tangent length (T), the distance from the PI back to the curve's start; arc length (L), the actual length along the curve; external distance (E), how far the curve's midpoint lies from the PI; middle ordinate (M), the offset from the long chord to the curve's midpoint; and long chord (C), the straight-line distance between the curve's endpoints. Field crews use T and L for stationing the PC and PT; designers use E and M to check clearance and sight lines through the curve.
Field Layout and Sight Distance
E and M matter beyond stationing — they determine how far the curve intrudes into or away from the tangent alignment, which governs right-of-way needs, sight-line clearance around obstructions on the inside of the curve, and superelevation transition planning. AASHTO also ties minimum radius directly to design speed through the superelevation and side-friction relationship (see the Curve Superelevation calculator), so a horizontal curve's geometry and its superelevation design are always solved together, not independently.
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