Three-Arc Vertical Curve for Constrained Highway Alignments

Existing symmetrical and unsymmetrical (two-arc) vertical curves may not be feasible when the highway alignment is subject to physical constraints (e.g., vertical clearance) and the beginning and end of the vertical curve are fixed. This paper presents a new three-arc vertical curve that accommodates such constraints. The unsymmetrical three-arc curve connects two tangents with unequal lengths and consists of three parabolic arcs smoothly connected at the points of common curvature. Since there are generally many three-arc curves that satisfy the constraints, the optimal three-arc curve that maximizes the smoothness of the curve was found. This was achieved by minimizing the sum of the absolute differences in the rates of change of grades at the two points of common curvature using an exhaustive search. Existing traditional and equal-arc unsymmetrical curves were found to be special cases of the three-arc curve. Another interesting special case is the three-equal-arc symmetrical curve, which is directly solved in closed-form. Sight distance characteristics of the three-arc curve were analyzed and compared with those of existing curves. It is shown that the three-arc curve improves sight distance for certain conditions. Application of the new curve is illustrated using numerical examples. The proposed vertical curve should be useful in the design of constrained vertical alignments such as complex freeway interchanges.