Road Curve Setting Out: Tangent Length, Chainages and Offsets

Circular-curve elements

Let $\Delta$ be the change in tangent direction, $R$ the radius, and $B$ the chosen reference chain length. Angles in trigonometric expressions below are in degrees; $\Delta_r$ denotes the same angle in radians.

The stable half-angle form avoids subtracting nearly equal numbers for small angles. $S_{PC}=S_{PI}-T$ and $S_{PT}=S_{PC}+L$. For degree input, $R=B/D_r$ (arc definition) or $R=B/[2\sin(D/2)]$ (chord definition). Geometry follows Caltrans BDD 2E, June 2025, pp. 1–2, and FHWA, Chapters 2 and 5 (printed pp. 20–23, 47–48 and 58–60; attached pp. 35–38, 62–63 and 73–75). This calculator derives its own coordinate transformations.

Exact Rankine deflections

For a peg at arc distance $s$, $\theta_r=s/R$ and the cumulative deflection from the PC forward tangent is $\delta=\theta/2$. For an arc interval $\Delta s$, the measured chord is $c=2R\sin(\Delta s/(2R))$, with that sine argument in radians. The direct PC-to-peg chord is $2R\sin\delta$. This uses exact geometry rather than the small-chord $1718.9c/R$ minutes approximation. The final deflection is $\Delta/2$. Seconds are rounded only for display.

Perpendicular offsets

With PC as origin, the forward tangent as x axis, and positive y toward the curve, $x=R\sin\theta$ and $y=2R\sin^2(\theta/2)$. Caltrans BDD 2F, June 2025, p. 3, gives the equivalent tangent-offset construction. The table uses $\theta_n=\min(\theta,\Delta-\theta)$ from the nearer endpoint so the projection stays on the PC–PI or PT–PI tangent segment. At PT, face PI and offset onto the arc side; do not reuse a PC left/right offset sign blindly.

For the long chord, $u=2R\sin(\theta/2)\cos((\Delta-\theta)/2)$ from PC, and $v=2R\sin(\theta/2)\sin((\Delta-\theta)/2)$ perpendicular toward the arc. These follow by rotating the PC tangent coordinates through $\Delta/2$. All values in the setting-out tables are positive magnitudes; the selected direction mirrors the plan.

Peg grid and closure

Pegs fall at multiples of $B/n$ in total alignment chainage, plus PC and PT. Coincident endpoint/grid pegs are merged within $10^{-8}$ of an interval. Circle-equation, arc-sum, deflection and endpoint-chord residuals check geometric closure.

Reading the schedules

Each plus-notation chainage carries its fixed reference convention in brackets: [20 m], [30 m] or [100 ft]. These brackets are independent of the metric/US display switch. Distance and decimal-angle headers name their displayed units. Rankine distances and angles are separate tables with repeated chainage; the peg chord is measured from the previous peg, while the PC chord is measured directly from PC. DMS totals use nonbreaking spacing so one angle stays together.

One simple circular horizontal arc with constant radius; flat local geometry and continuous arc-length chainage. No station equations or transition spirals. PC is at nonnegative chainage. The PI chainage belongs to the incoming alignment. Chain conventions are explicit reference lengths, not a change of survey datum. The drawing is rotated to keep the long chord horizontal, with the selected left/right curve represented consistently; it is not a northing/easting plot. Coordinates and formulas are derived here from circle geometry and checked against published highway examples. For shallow curves, the small mid-ordinate is dimensioned in a separate detail with an explicit vertical enlargement factor; the main plan retains its true geometric proportions.

Input domain: radius 0.1–100000 m; deflection 0.1–170°; degree of curve 0.01–120°; 1–10 peg intervals per selected chain or station. A valid schedule has no more than 200 nominal arc intervals (up to 202 pegs). Increase PI chainage if the curve would begin before the origin. All lengths are horizontal; reduce slope distances and apply project grid/ground corrections separately. No speed, sight-distance, superelevation, pavement, spiral or setting-out tolerance design is included. Use approved project survey procedures and check against the standard in force where project specifications require one.

Reference-data correction: FHWA Table 5.2 lists the curve 3 tangent as 528.2879 ft. Table 5.5 gives 585.2879 ft, consistent with its radius, central angle and stationing; the reference check uses Table 5.5. Caltrans reference checks use individual published values, not a reproduction of its offset tables.