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
- = number of directions observed (excluding the known side)
- = number of geometric conditions , where = total lines, = lines observed in both directions, = total stations, = occupied stations
- , = 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.
Distance to the visible horizon from a height (m):
(including the effect of refraction.)
For stations A (height ) and B separated by distance : ; the required height at B is .
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 from the true station C. The angles observed at S are reduced to the centre C.
= distance of satellite station from the true station; = distance from the true station to the observed station; = 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:
( = area of the triangle, = 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).