Vertical curves

Where two gradients meet, a vertical curve smooths the change so that vehicles ride comfortably and drivers can see far enough ahead. A summit (crest) curve joins a rising gradient to a falling one, or a steeper rise to a flatter one; its main constraint is sight distance, because the crest hides the road beyond. A valley (sag) curve joins a falling gradient to a rising one; its constraints are comfort, from the vertical acceleration as the vehicle is pushed into its springs, and headlight sight distance at night.

Vertical curves are parabolic, because a parabola has a constant rate of change of grade, which gives uniform comfort and makes calculations simple. The algebraic difference of grades, N, together with the curve length, defines the curve. For summit curves, the length must provide stopping sight distance (and overtaking sight distance on undivided roads where overtaking is permitted), using standard driver eye height and object height. For valley curves, headlight sight distance usually governs, with a check for comfort.

Elevations along the curve follow from the tangent levels minus the offset y = N x² / (2L). The highest or lowest point lies where the gradient is zero, important for placing drains on sag curves, which must not be at a low point without an outlet. Combining horizontal and vertical curves needs care: a sharp bend hidden just beyond a crest is dangerous.