Curves — Simple, Compound, Reverse, Transition & Vertical — Surveying & Geomatics Notes

Last reviewed 16 Sept 2026 · 10 min read

Classification of curves

Horizontal curves — in plan: simple circular, compound, reverse, transition (spiral) curves.
Vertical curves — in the longitudinal section: summit (crest) and valley (sag) curves, usually parabolic.

Simple circular curve

A simple curve is a single arc of a circle connecting two straights (tangents).

Elements

Element	Symbol / formula
Radius	$R$
Deflection angle	$Δ$ — angle between the back tangent produced and the forward tangent (also the central angle)
Angle of intersection	$I = 180° - Δ$
Point of intersection (PI, vertex V)	Intersection of the two tangents
Tangent length	$T = R tan \frac{Δ}{2}$
Length of curve	$L = \frac{π R Δ}{180°}$
Length of long chord	$L_{c} = 2 R sin \frac{Δ}{2}$
Mid-ordinate (versed sine)	$M = R (1 - cos \frac{Δ}{2})$
Apex distance (external distance)	$E = R (sec \frac{Δ}{2} - 1)$

Chainages: chainage of first tangent point $T_{1}$ = chainage of PI − T; chainage of second tangent point $T_{2}$ = chainage of $T_{1}$ + L.

Degree of curve

Definition	Relation
Arc definition — angle subtended by an arc of 30 m	$R = \frac{1718.9}{D}$ (for a 30 m arc)
Chord definition — angle subtended by a chord of 30 m	$R = \frac{15}{sin ( D /2 )} \approx \frac{1719}{D}$
With 20 m arc/chord (common in India for roads)	$R \approx \frac{1146}{D}$

(In railways, the degree is commonly defined on a 30.5 m chord: $R \approx 1750/ D$ .)

Peg interval and sub-chords

The curve is set out with pegs at regular chord lengths (e.g. 20 m or 30 m — the normal chord), such that the chord is nearly equal to the arc: chord length should not exceed about R/20.
Pegs are placed at full-station chainages; the first chord from $T_{1}$ to the next full station is the first sub-chord; the last one to $T_{2}$ is the last sub-chord.

Setting out simple curves

Linear methods (tape only — short curves)

FormulaLinear methods

1. Offsets from the long chord:

O_{x} = R^{2} - x^{2} - (R - M), O_{0} = M

( $x$ measured from the mid-point of the long chord.)

2. Offsets from tangents:

Radial offsets: $O_{x} = R^{2} + x^{2} - R \approx \frac{x ^{2}}{2 R}$
Perpendicular offsets: $O_{x} = R - R^{2} - x^{2} \approx \frac{x ^{2}}{2 R}$

3. Successive bisection of arcs (versed sines).

4. Offsets from chords produced (long curves on roads):

O_{1} = \frac{C _{1}^{2}}{2 R}, O_{2} = \frac{C _{2} ( C _{1} + C _{2} )}{2 R}, O_{n} = \frac{C _{n} ( C _{n - 1} + C _{n} )}{2 R}

$C_{1}$ , $C_{2}$ , … = successive chord lengths.

Angular methods

FormulaRankine's method of tangential (deflection) angles

Each chord $C$ subtends a tangential angle at the tangent point:

δ (minutes) = \frac{1718.9 C}{R}

The total deflection angle for a point = sum of tangential angles of all preceding chords; the final total deflection equals $Δ/2$ (check).

The theodolite at $T_{1}$ is set to each total deflection angle in turn, and the chord length is measured from the previous peg to fix the next peg.

Two-theodolite method — theodolites at both tangent points set to the deflection angles; pegs at intersections of lines of sight — used where chaining is difficult (rough ground).
Tacheometric / total station methods — pegs set out by polar coordinates (angle and distance) from a station; coordinates of curve points computed in advance.

Compound curve

A compound curve consists of two (or more) arcs of different radii bending in the same direction, with a common tangent at the point of compound curvature (PCC).

FormulaCompound curve (two arcs, radii

R_{1}

and

R_{2}

, deflections

Δ_{1}

and

Δ_{2}

)

Δ = Δ_{1} + Δ_{2}, t_{1} = R_{1} tan \frac{Δ _{1}}{2}, t_{2} = R_{2} tan \frac{Δ _{2}}{2}

Tangent lengths from the main intersection point V:

T_{s} = t_{1} + (t_{1} + t_{2}) \frac{sin Δ _{2}}{sin Δ}, T_{L} = t_{2} + (t_{1} + t_{2}) \frac{sin Δ _{1}}{sin Δ}

Used where site conditions do not permit a single radius (e.g. following a valley or avoiding an obstacle).

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Curves — Simple, Compound, Reverse, Transition & Vertical