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Vertical curves provide smooth transitions between varying grades on highways and railroads, improving safety and comfort. 

These curves are parabolic and can be symmetrical or nonsymmetrical, depending on the lengths of the tangents.

They connect the back tangent, representing the initial grade, to the forward tangent, representing the following grade.

The transition begins at the Point of Vertical Curvature or P.V.C., where the back tangent meets the curve. The curve rises or falls gently, guided by its geometry, and eventually connects to the forward tangent at the Point of Vertical Tangency or P.V.T.

The point where the extensions of the back and forward tangents intersect is called the Point of Vertical Intersection or P.V.I.

Tangent offsets, the vertical distances from tangents to the curve, help calculate elevations and define the curve's profile.

At the curve’s midpoint, the elevation is halfway between the P.V.I. and the midpoint of a straight line joining the P.V.C. and P.V.T.

Vertical curves are carefully designed to ensure smooth alignment, proper sight distance, and balance between cutting and filling, enhancing the functionality of roadways.