Horizontal curves provide a smooth, gradual transition between two intersecting straight lines (tangents) in the horizontal plane of a route alignment. They are primarily designed as circular arcs of specific radii, ensuring safe and comfortable vehicle operation at designed speeds.

• Simple Curve: A single circular arc connecting two tangents.
• Compound Curve: Two or more circular arcs of different radii turning in the same direction, with a common tangent and point of compound curvature (PCC).
• Reversed Curve: Two circular arcs turning in opposite directions, with a common tangent at the point of reversed curvature (PRC).
• Spiral (Transition) Curve: A curve with a continuously changing radius, providing a gradual transition from a straight tangent (infinite radius) to a circular curve of constant radius.

The Degree of Curve ($D$) defines the sharpness of the curve. There are two standard definitions:

• Arc Basis (Metric Standard): The central angle subtended by a $20 \text{ m}$ (or $100 \text{ ft}$) arc.
• Chord Basis (Railway Standard): The central angle subtended by a $20 \text{ m}$ (or $100 \text{ ft}$) chord.

Route alignment is tracked using stationing. The station of the PC is found by subtracting the Tangent distance from the PI station. The station of the PT is found by adding the Curve Length to the PC station. The PI station is never calculated by going backwards from the PT.

To lay out a simple curve in the field using a total station or theodolite set at the PC, the curve is divided into smaller segments. A sub-chord ($c$) is used to connect full stations. The deflection angle ($\delta$) from the tangent to any point on the curve is half the central angle ($\theta$) subtended by the arc to that point.

Because full stations rarely fall exactly on the PC and PT, surveyors must calculate an initial sub-chord from the PC to the first full station, full chords between intermediate stations, and a final sub-chord from the last station to the PT.

To ensure safe stopping sight distance ($S$) on a horizontal curve, a clear line of sight must be maintained across the inside of the curve. If the sight distance is less than the curve length ($S < L$), the minimum clearance distance from the centerline of the inside lane to an obstruction (like a wall, building, or cut slope) is defined by the middle ordinate ($M$).

The most common field method for setting out (staking) a simple curve is the deflection angle method using a total station or theodolite set up at the PC.

The deflection angle to any point on the curve is equal to half the central angle subtended by the arc from the PC to that point. The surveyor turns the calculated cumulative deflection angle from the tangent line and measures the corresponding chord distance to set the stake.

When instruments are unavailable or precision requirements are low, curves can be set out using linear measurements alone:

• Offsets from the Tangent: Perpendicular distances ($y = R - \sqrt{R^2 - x^2}$) are measured from established points ($x$) along the tangent line to locate points on the curve.
• Offsets from the Long Chord: Perpendicular offsets are measured from the long chord connecting the PC and PT to locate curve points.

Compound curves involve two circular curves ($R_1$ and $R_2$) meeting at a Point of Compound Curvature (PCC). The centers of the two curves lie on the same side of the common tangent passing through the PCC.

Reversed curves involve two circular arcs turning in opposite directions meeting at a Point of Reversed Curvature (PRC). The centers of the curves lie on opposite sides of the common tangent. They are generally avoided on high-speed highways due to the sudden change in centrifugal force and superelevation requirements, but are common in railway switchbacks and slow-speed urban roads.

• Parallel Tangents: If the initial back tangent and final forward tangent are parallel, the central angles of both curves are exactly equal ($I_1 = I_2$). The perpendicular distance ($d$) between the tangents is $d = R_1(1 - \cos I_1) + R_2(1 - \cos I_2)$.
• Converging/Diverging Tangents: The total angle between the main tangents is determined by the difference or sum of the individual intersection angles depending on geometry.

Spiral curves are introduced between straight tangents and circular curves to provide a gradual introduction of centrifugal force and superelevation. The most common type is the clothoid (Euler spiral), where the radius decreases linearly as the length along the curve increases ($R \times L = \text{constant}$).