Chapter 2 of 12

Curves & Traversing

In the DSSSB AE Civil syllabus under Surveying · 2 parts

Curves — Simple, Compound, Reverse, Transition & Vertical · Theodolite Surveying & Traversing

CIVILGYAN TEST SERIES · civilgyantests.com — free study notes for civil engineering exams

📑 Contents (17 sections)

Part 1 of 2

Curves — Simple, Compound, Reverse, Transition & Vertical

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).

Part 2 of 2

Theodolite Surveying & Traversing

Last reviewed 16 Sept 2026 · 10 min read

The theodolite

A theodolite measures horizontal and vertical angles accurately. It is also used for prolonging lines, setting out angles and curves, levelling (trigonometric) and, with stadia hairs, measuring distances (tacheometry).

Types

Transit theodolite — the telescope can be revolved through 180° in a vertical plane about its horizontal axis (standard today).
Non-transit theodolite — telescope cannot be transited (obsolete).
By reading system: vernier theodolites (least count commonly 20″), micrometer/optical theodolites (1″ or better), electronic digital theodolites (with digital angle display); total stations combine electronic theodolites with EDM.

Main parts

Trivet and tribrach with levelling screws, lower plate (carrying the horizontal circle) with lower clamp and tangent screw, upper plate (carrying verniers/reading system) with upper clamp and tangent screw, plate levels, standards (A-frame) supporting the horizontal (trunnion) axis, telescope, vertical circle with vertical circle vernier/index and altitude bubble, plumb bob or optical plummet, tripod.

Technical terms

Term	Meaning
Vertical axis	Axis about which the instrument rotates in a horizontal plane
Horizontal (trunnion) axis	Axis about which the telescope rotates in a vertical plane
Line of collimation	Line through the intersection of cross hairs and optical centre of objective
Centring	Setting the vertical axis exactly over the station mark
Transiting (plunging, reversing)	Rotating the telescope 180° about the horizontal axis
Swinging	Rotating the telescope about the vertical axis — right swing (clockwise) or left swing (anticlockwise)
Face left (telescope normal)	Vertical circle on the left of the observer when sighting
Face right (telescope inverted)	Vertical circle on the right of the observer
Changing face	Transiting and swinging so that the face changes

Temporary adjustments

Setting up over the station with tripod.
Centring — plumb bob or optical plummet over the station mark; shifting head for fine centring.
Levelling up — using plate levels and foot screws: bubble central parallel to two foot screws, then perpendicular using the third screw; repeat until central in all positions.
Focusing — eyepiece for cross hairs, objective for the object — elimination of parallax.

Measurement of angles

Horizontal angle (simple method)

Set the vernier to 0°, sight the first station using the lower clamp; release the upper clamp, sight the second station, read the angle. Repeat on the other face and take the mean.

Repetition method

Used to measure a single horizontal angle to a finer degree of accuracy than the least count.

Measure the angle once and do not reset the vernier; with the lower clamp, sight the first station again.
Using the upper clamp, sight the second station — the reading accumulates the angle twice.
Repeat for a set number of repetitions (e.g. 3 on face left and 3 on face right, with left and right swings).
Angle = (final reading) ÷ (number of repetitions) — adding full circles if the reading passes 360°.

Errors eliminated/reduced: errors of eccentricity of verniers, errors due to inadequate least count (reading error distributed), errors of graduations (different parts of the circle used), errors of collimation and trunnion axis (face left + face right), errors due to slip partly — accuracy improves.

Reiteration (direction) method

Used when several angles are to be measured at a station: directions of all stations are read successively from a reference station, closing back on it (the horizon is closed; sum of angles = 360°). Repeated with the circle set at different initial readings on both faces.

Vertical angles

Angle of elevation (above the horizontal) or depression (below). Measured using the vertical circle with the altitude bubble central; face left and face right readings are averaged to eliminate index error.

Other operations

Magnetic bearing of a line — using a trough or tubular compass attached to the theodolite.
Deflection angle — angle a line makes with the prolongation of the preceding line (right or left) — used in route surveys.
Prolonging a straight line — by double sighting (face left and face right, taking the mean point) to eliminate collimation error.
Setting out angles, ranging a line, locating the intersection of two lines.

Fundamental lines and permanent adjustments

Required relationship	Adjustment/test
Axis of plate levels perpendicular to the vertical axis	Plate level test — bubble central in all positions
Line of collimation perpendicular to the horizontal axis	Collimation test — prolonging a line by face left and right (spire test variants)
Horizontal axis perpendicular to the vertical axis	Spire test — sighting a high point and a low point on both faces
Axis of altitude level parallel to line of collimation (vertical circle index correct)	Vertical index test — two-peg-type test for zero reading when line of sight is horizontal
Vertical cross hair in a plane perpendicular to the horizontal axis	Cross hair test

Errors eliminated by face-left and face-right observations

Eliminated by averaging both faces	Not eliminated by changing face
Collimation error (line of sight not perpendicular to horizontal axis)	Error due to vertical axis not being truly vertical (imperfect levelling / plate level error)
Horizontal (trunnion) axis error (not perpendicular to vertical axis)	Graduation errors (reduced by using different parts of the circle)
Index error of vertical circle	Personal and natural errors
Eccentricity of verniers (by reading both verniers)

Theodolite traversing

Methods

Method	Description
Included angles method	Interior (or exterior) angles measured at each station — closed traverses; common for boundaries
Deflection angles method	Deflection angles measured — open traverses such as roads, railways, canals
Direct angle (angle to the right) method	Clockwise angles from the back station to the forward station
Fast needle method	Magnetic bearings measured with the theodolite compass, carried forward with the circle clamped
Loose needle method	Magnetic bearing observed independently at each station

Checks for a closed traverse: sum of interior angles $= (2 n - 4) \times 90°$ ; sum of exterior angles $= (2 n + 4) \times 90°$ ; sum of deflection angles (right − left) $= 360°$ . Angular misclosure is distributed equally among the angles (if all measured with equal care).

Traverse computations

FormulaLatitudes and departures

For a line of length $l$ and whole circle bearing (or reduced bearing) $θ$ :

Latitude L = l cos θ, Departure D = l sin θ

Latitude: + northing, − southing
Departure: + easting, − westing

For a closed traverse, $\sum L = 0$ and $\sum D = 0$ ideally.

FormulaClosing error

e = (\sum L)^{2} + (\sum D)^{2}, tan δ = \frac{\sum D}{\sum L}

Relative precision (accuracy) $= \frac{e}{perimeter}$ , expressed as 1 in $\frac{perimeter}{e}$ .

Balancing the traverse

FormulaBalancing rules

Bowditch's rule (compass rule) — when angular and linear measurements are of equal precision:

C_{L} = - \sum L \times \frac{l}{\sum l}, C_{D} = - \sum D \times \frac{l}{\sum l}

Transit rule — when angles are measured more precisely than lengths:

C_{L} = - \sum L \times \frac{∣ L ∣}{\sum ∣ L ∣}, C_{D} = - \sum D \times \frac{∣ D ∣}{\sum ∣ D ∣}

Other methods: graphical (Bowditch) adjustment, third rule, Crandall's method and least squares adjustment.

Coordinates

Consecutive coordinates — latitude and departure of each line relative to its starting point.
Independent (total) coordinates — coordinates of each station relative to a common origin; obtained by cumulative addition of corrected consecutive coordinates.
Gale's traverse table — a standard tabular format for computing bearings, latitudes, departures, corrections and independent coordinates.

Omitted measurements

When some measurements of a closed traverse are missing (not measured or lost), they can be computed since $\sum L = 0$ and $\sum D = 0$ provide two equations:

Length and bearing of one line omitted — the missing line closes the traverse: $L = - \sum L_{k n o w n}$ , $D = - \sum D_{k n o w n}$ .
Length of one line and bearing of another omitted, or lengths of two lines omitted, or bearings of two lines omitted — solved by trigonometric relations (sometimes after joining known points with a closing line).

Worked examples

Worked ExampleExample 1 — latitudes, departures and closing error

A closed traverse ABCDA has the following data:

Line	Length (m)	WCB
AB	250.0	60°00′
BC	180.0	150°00′
CD	260.0	240°00′
DA	180.5	333°10′

Find the closing error and relative precision.

Solution.

Line	Latitude	Departure
AB	+125.000	+216.506
BC	−155.885	+90.000
CD	−130.000	−225.167
DA	+161.065	−81.477
Sum	+0.180	−0.138

$e = 0.18 0^{2} + 0.13 8^{2} = 0.227 m$ ; perimeter = 870.5 m → relative precision ≈ 1 in 3840

Worked ExampleExample 2 — Bowditch correction

Find the Bowditch corrections to the latitude and departure of line AB in Example 1.

Solution. $C_{L} = - 0.180 \times \frac{250}{870.5} = - 0.052 m$ ; $C_{D} = + 0.138 \times \frac{250}{870.5} = + 0.040 m$ Corrected: latitude 124.948 m, departure 216.546 m.

Worked ExampleExample 3 — omitted measurement

In the traverse of Example 1, suppose the length and bearing of DA were not measured. Find them.

Solution. From AB, BC, CD: $\sum L = - 160.885$ , $\sum D = + 81.339$ For DA: $L = + 160.885$ , $D = - 81.339$ Length $= 160.88 5^{2} + 81.33 9^{2} = 180.28 m$ Reduced bearing $= tan^{- 1} (81.339/160.885) = 26°49.6'$ in the NW quadrant → WCB $= 360° - 26°49.6' = 333°10.4'$

Worked ExampleExample 4 — repetition method

An angle was measured by repetition 6 times; the initial reading was 0°00′00″ and the final reading after six repetitions was 243°28′30″. Find the angle.

Solution. Angle = 243°28′30″ ÷ 6 = 40°34′45″

Frequently tested points

Transit theodolite — telescope revolves 180° vertically; vernier least count commonly 20″.
Face left = vertical circle on the left of the observer.
Temporary adjustments: setting up, centring, levelling up, focusing (parallax).
Repetition method — single angle, higher accuracy; reiteration — several angles at a station.
Face left and face right eliminate collimation, trunnion axis and index errors; not the vertical axis error.
Spire test — horizontal axis perpendicular to vertical axis.
Deflection angles for open (route) traverses; included angles for closed traverses.
Latitude $= l cos θ$ , departure $= l sin θ$ ; closing error $(\sum L)^{2} + (\sum D)^{2}$ .
Bowditch's rule ∝ length of line (equal angular and linear accuracy); transit rule ∝ latitude/departure (angles more precise).
Consecutive vs independent coordinates; Gale's traverse table.
Omitted measurements from $\sum L = 0$ and $\sum D = 0$ .

Common MistakeCommon mistakes

Believing that changing face eliminates errors due to imperfect levelling.
Using the transit rule when lengths and angles are equally precise.
Forgetting signs of latitude and departure when bearings are in SE, SW or NW quadrants.

Revision SummaryChapter summary

The transit theodolite measures horizontal and vertical angles and performs many field operations.
Temporary adjustments and methods such as repetition and reiteration give accurate angles.
Permanent adjustments maintain the fundamental axis relationships; face-left/face-right observations eliminate several instrumental errors.
Theodolite traverses are computed through latitudes and departures, closing error and relative precision.
Traverses are balanced by Bowditch's or transit rules, converted to independent coordinates, and omitted measurements are computed from closure conditions.

Downloaded from civilgyantests.com · Not for resale. Verify every code clause and date against the official source.

Finished reading? Test yourself.

A timed chapter test from the DSSSB AE Civil series, on exactly this chapter.

Practice this chapter →