While horizontal alignment defines the $X-Y$ coordinate path of a road, the vertical alignment defines the elevation profile ($Z$-axis). It consists of straight gradient lines (tangent grades) connected by vertical curves. The primary objective is to provide a smooth, safe, and comfortable transition between different grades while minimizing earthwork costs. A well-designed vertical alignment must also ensure adequate sight distances for safe stopping and passing maneuvers, maintain proper drainage of surface water, and satisfy aesthetic and environmental considerations.

Unlike horizontal curves which are circular, vertical curves in highway engineering are almost exclusively parabolic. Parabolas are preferred because they provide a constant rate of change of grade, which translates to uniform centrifugal force and comfortable riding characteristics.

Vertical curves are classified into two broad categories based on the intersection of the tangent grades:

The geometry of a parabolic vertical curve is defined by its length ($L$) and the algebraic difference in intersecting grades ($A$).

Stopping Sight Distance (SSD) is the minimum distance required for a vehicle traveling at design speed to stop before reaching a stationary object in its path. It is the sum of two components:

The perception-reaction time ($t$) is typically assumed to be $2.5$ seconds by AASHTO for design purposes. The braking distance depends on the vehicle's initial speed ($V$), the deceleration rate ($a$, typically $3.4 \text{ m/s}^2$), and the longitudinal gradient of the road ($G$).

Passing Sight Distance (PSD) is the minimum sight distance required on a two-lane, two-way highway to safely overtake a slower vehicle without colliding with an oncoming vehicle. It is significantly longer than SSD and is composed of four distances:

The most critical factor in determining the minimum length of a crest vertical curve is ensuring that a driver can see an object over the hill in time to stop safely. The required length ($L$) depends on the Stopping Sight Distance ($S$), the algebraic difference in grades ($A$), the height of the driver's eye ($h_1$, typically $1.08 \text{ m}$), and the height of the object ($h_2$, typically $0.60 \text{ m}$).

The formulas differ depending on whether the required sight distance ($S$) is less than or greater than the curve length ($L$).

When $S \lt L$ (Most common for highways):

For sag curves, the critical design parameter is generally headlight sight distance at night. The vehicle's headlights must illuminate the road ahead for a distance equal to or greater than the Stopping Sight Distance ($S$).

The design formulas depend on the height of the headlights ($H$, typically $0.60 \text{ m}$) and the upward divergence angle of the headlight beam ($\beta$, typically $1^\circ$).

For drainage purposes or clearance calculations, it is often necessary to locate the exact highest point on a crest curve or lowest point on a sag curve. This point occurs where the tangent grade is zero.

The horizontal distance ($x$) from the Point of Vertical Curve (PVC) to the high/low point is given by:

On two-lane, two-way highways, drivers must occasionally overtake slower vehicles by briefly entering the opposing lane. This requires significantly more sight distance than simply stopping.