Chapter 8 of 9

Tacheometry & triangulation

In the TNPSC AE Civil syllabus under Engineering Survey · 2 parts

Tacheometric Surveying · Triangulation & Trilateration

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📑 Contents (19 sections)

Part 1 of 2

Tacheometric Surveying

Last reviewed 16 Sept 2026 · 8 min read

Principle and uses

Tacheometry (tachymetry) is a branch of angular surveying in which horizontal distances and vertical distances (elevations) are determined optically from instrumental observations, without chaining.

Uses:

Contouring and topographic surveys, especially in rough, hilly or broken terrain and across rivers, swamps and valleys where chaining is difficult or impossible.
Filling in details in large-scale surveys; preliminary route surveys (roads, railways, canals).
Checking distances measured by other methods.

Accuracy is lower than good chaining (typically about 1 in 500 to 1 in 1000 for stadia distances) but sufficient for many topographic purposes; the method is fast.

Instruments

Tacheometer — a transit theodolite whose diaphragm carries two stadia hairs (equidistant above and below the central horizontal hair) in addition to the cross hairs. Features: high magnifying power, bright image, and an anallactic lens (in external focusing telescopes) — or an internal focusing telescope — so that the additive constant is zero.
Stadia rods — long staves (3–4 m) with bold graduations for long sights; ordinary levelling staves for short sights.
Subtense bar — a horizontal bar (commonly 2 m long between targets, made of invar) set at a known length perpendicular to the line of sight.

Anallactic lens

An additional convex lens placed between the eyepiece and objective of an external focusing telescope so that the stadia distance is measured from the centre of the instrument — the additive constant becomes zero and the distance equation becomes $D = K s$ . (Introduced by Porro.) Internal focusing telescopes achieve an additive constant close to zero without it.

Methods of tacheometry

Method	Principle
Stadia method — fixed hair	Stadia hairs fixed; the staff intercept varies with distance — most common
Stadia method — movable hair (subtense)	Stadia hairs movable by micrometer screws to fixed targets on the staff; distance from the hair movement
Tangential method	No stadia hairs; two vertical angles are observed to two targets a known distance apart on the staff
Subtense bar method	Horizontal angle subtended by a bar of known length
Electronic (EDM/total station)	Modern replacement for optical tacheometry

Stadia method — distance equation

FormulaHorizontal line of sight, staff vertical

D = \frac{f}{i} s + (f + d) = K s + C

$s$ = staff intercept (difference between upper and lower stadia hair readings)
$K = \frac{f}{i}$ = multiplying constant (usually made 100)
$C = f + d$ = additive constant ( $f$ = focal length of objective, $d$ = distance from objective to the vertical axis; $i$ = stadia hair interval)
With an anallactic lens (or internal focusing): $C = 0$ → $D = K s = 100 s$

Determination of constants

Measure staff intercepts $s_{1}$ and $s_{2}$ at two accurately measured distances $D_{1}$ and $D_{2}$ :

D_{1} = K s_{1} + C, D_{2} = K s_{2} + C \Rightarrow K = \frac{D _{2} - D _{1}}{s _{2} - s _{1}}, C = D_{1} - K s_{1}

Inclined sights

FormulaStaff held vertical (most common)

For vertical angle $θ$ (elevation or depression):

D = K s cos^{2} θ + C cos θ

V = K s \frac{sin 2 θ}{2} + C sin θ

$D$ = horizontal distance; $V$ = vertical distance between the horizontal axis of the instrument and the point on the staff cut by the central hair.

FormulaStaff held normal to the line of sight

D = (K s + C) cos θ \pm h sin θ, V = (K s + C) sin θ

( $h$ = central hair reading; + for angle of elevation, − for depression, in $D$ .)

Reduced levels

Angle of elevation:

R L_{s t a f f s t a t i o n} = R L_{in s t r u m e n t s t a t i o n} + h_{i} + V - h

Angle of depression:

R L_{s t a f f s t a t i o n} = R L_{in s t r u m e n t s t a t i o n} + h_{i} - V - h

( $h_{i}$ = height of instrument axis above its station; $h$ = central hair reading.) When the instrument is referred to a benchmark, $R L = H I \pm V - h$ , where HI = RL of the instrument axis.

Part 2 of 2

Triangulation & Trilateration

Last reviewed 16 Sept 2026 · 9 min read

Principle

Triangulation is a method of establishing horizontal control in which the area is covered by a network of triangles. Only one side (the base line) is measured directly with high precision, and all angles of the triangles are measured; the other sides are computed by the sine rule, and the coordinates of stations follow.

Purposes:

Establishing accurate control points for topographic and engineering surveys over large areas.
Geodetic surveys — determining the size and shape of the earth.
Locating inaccessible points; setting out long tunnels, bridges, dams.

Triangulation vs traversing: triangulation suits hilly and undulating areas where long lines can be sighted and chaining is difficult; traversing suits flat, built-up or wooded areas. Trilateration measures all sides (with EDM) instead of angles; modern control uses trilateration, combined networks and GNSS.

Classification of triangulation

Specification (typical)	First order (primary)	Second order (secondary)	Third order (tertiary)
Length of base line	8–12 km	1.5–5 km	0.5–3 km
Length of sides of triangles	16–150 km	8–65 km	1.5–10 km
Average triangular closure	Less than 1″	3″	6″
Maximum triangular closure	Not more than 3″	8″	12″
Accuracy of base measurement	1 in 300 000	1 in 150 000	1 in 75 000

First-order — geodetic, national control, very high accuracy.
Second-order — fills in the primary network.
Third-order — provides control for topographic and engineering surveys.

Triangulation figures

Figure	Features
Chain of single triangles	Simple, rapid, economical; few checks — accuracy depends on figure shape; used for narrow strips (e.g. valleys, routes)
Braced (geodetic) quadrilaterals	Quadrilateral with both diagonals observed — best and most accurate figure, many checks, strongest
Centred polygons (centred figures)	Polygons (quadrilaterals, pentagons, hexagons) with a central station — cover large areas, good checks but slower
Combinations	Networks of the above as required

Well-conditioned triangle

The accuracy of a computed side depends on the angles opposite to known and unknown sides (errors in sines of small angles are large). A triangle is well-conditioned if no angle is smaller than about 30° or greater than about 120°. The best shape is an equilateral triangle; mathematically, the error in the computed side of an isosceles triangle is minimum when the base angles are about 56°14′.

Strength of figure

FormulaStrength of figure (U.S. Coast and Geodetic Survey)

R = \frac{D - C}{D} \sum (δ_{A}^{2} + δ_{A} δ_{B} + δ_{B}^{2})

$D$ = number of directions observed (excluding the known side)
$C$ = number of geometric conditions $= (n^{'} - s^{'} + 1) + (n - 2 s + 3)$ , where $n$ = total lines, $n^{'}$ = lines observed in both directions, $s$ = total stations, $s^{'}$ = occupied stations
$δ_{A}$ , $δ_{B}$ = tabular differences of log sines (per second) of the distance angles (angles opposite the known and computed sides)

A smaller R means a stronger figure; the best route (chain of triangles) through a network is the one with the smallest total R.

Field work in triangulation

Reconnaissance — examination of terrain, selection of stations and base line, checking intervisibility, heights of signals, access, materials.
Station marking and erection of signals and towers.
Measurement of the base line (and base net extension).
Measurement of horizontal angles (and vertical angles for heights).
Astronomical observations at selected stations — azimuth, latitude and longitude.
Computations and adjustment.

Selection of stations

Stations must be intervisible with adjacent stations; form well-conditioned triangles.
Easily accessible, on firm ground, requiring low signal heights, providing a line of sight well clear of the ground (to reduce refraction), with minimum clearing of vegetation.

Intervisibility and height of stations

Because of the earth's curvature (and refraction), stations far apart may not be visible even over flat ground.

FormulaIntervisibility

Distance to the visible horizon from a height $h$ (m):

D = 3.855 h km, h = 0.06728 D^{2} m (D in km)

(including the effect of refraction.)

For stations A (height $h_{A}$ ) and B separated by distance $D$ : $D_{A} = 3.855 h_{A}$ ; the required height at B is $h_{B} = 0.06728 (D - D_{A})^{2}$ .

The line of sight should clear the intervening ground by at least about 3 m to reduce grazing refraction.

Signals and towers

Signals mark stations for sighting: luminous signals — heliotropes and heliographs (reflect sunlight), lamps and lights for night observations; opaque signals — pole signals, target signals, pole and brush, stone cairns, beacons.
Towers — elevate the instrument and signal above obstacles: scaffolds, masonry pillars, and steel Bilby towers (two independent towers — the inner one for the instrument and the outer for the observer — so that the observer's movements do not disturb the instrument).
Phase correction — for cylindrical opaque signals illuminated by the sun on one side, the observer tends to bisect the bright portion, introducing an error; a correction is applied to the observed direction depending on the observation method (bright portion or bright line).

Base line measurement

Site requirements: fairly level or uniformly sloping ground, free from obstructions, well-conditioned connection to the network, ends intervisible, suitable length.
Equipment: historically invar tapes/wires under standard tension with corrections for temperature, pull, sag, slope and reduction to MSL; now precise EDM.
Base net (base extension) — the measured base is expanded through a series of well-conditioned triangles to the length of main triangulation sides.
Check bases are measured at intervals to control error accumulation.

Satellite stations and reduction to centre

When a triangulation station cannot be occupied (e.g. a church spire, chimney or lighthouse) or visibility is blocked, observations are made from a nearby satellite (eccentric) station S, a short distance $d$ from the true station C. The angles observed at S are reduced to the centre C.

FormulaReduction to centre (correction angle)

α^{''} = \frac{d sin θ}{L} \times 206265

$d$ = distance of satellite station from the true station; $L$ = distance from the true station to the observed station; $θ$ = angle at the satellite station between the direction to the true station and the direction to the observed station. Corrections to the other directions are computed similarly and applied with appropriate signs.

Spherical excess

The sum of the angles of a spherical triangle exceeds 180° by the spherical excess:

ε^{''} = \frac{A}{R ^{2} sin 1 ^{''}} \approx \frac{A}{R ^{2}} \times 206265

( $A$ = area of the triangle, $R$ = radius of the earth.) It is about 1″ for every 196 km² of area — significant only in geodetic triangulation. Each angle is reduced by one-third of the spherical excess before plane computations (Legendre's theorem).

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