Vertical curves are parabolic transitions that connect different grades on highways and railroads, ensuring a smooth alignment between back and forward tangents. The back tangent represents the initial grade, while the forward tangent defines the subsequent grade. These curves can be symmetrical, with equal tangent lengths, or nonsymmetrical, with varying lengths. The key points defining a vertical curve include the Point of Vertical Intersection (P.V.I.), where the tangents meet; the Point of Vertical Curvature (P.V.C.) at the curve's start; and the Point of Vertical Tangency (P.V.T.), marking the curve's end.

Tangent offsets, vertical distances measured from the tangents to the curve, are essential for elevation and profile calculations. At the curve's midpoint, its elevation is determined by averaging the elevation at the P.V.I. with the midpoint of a straight line connecting the P.V.C. and P.V.T. This property simplifies the geometric layout of vertical curves.

The mathematical principles of vertical curves rely on parabolic properties, where tangent offsets vary with the square of the distance from the tangency points, and second differences remain equal at evenly spaced stations. These properties are essential for verifying curve calculations.

The design of vertical curves balances factors like sight distance, optimal earthwork, and suitable grades. Adjustments to the curve length or gradients ensure alignment efficiency and safety. These curves are crucial for achieving functional and economical roadway designs and meeting the demands of modern transportation infrastructure.